Solitons and collapses: two evolution scenarios of nonlinear...

Abstract. Two alternative scenarios pertaining to the evolutionof nonlinear wave systems are considered: solitons and wavecollapses. For the former, it suffices that the Hamiltonian bebounded from below (or above), and then the soliton realizing itsminimum (or maximum) is Lyapunov stable. The extremum isapproached via the radiation of small-amplitude waves, a pro-cess absent in systems with finitely many degrees of freedom.The framework of the nonlinear Schr�odinger equation and thethree-wave system is used to show how the boundedness of theHamiltonianÐand hence the stability of the solitonminimizingitÐ can be proved rigorously using the integral estimate meth-od based on the Sobolev embedding theorems. Wave systemswith the Hamiltonians unbounded from below must evolve to acollapse, which can be considered as the fall of a particle in an

unbounded potential. The radiation of small-amplitude wavespromotes collapse in this case.

1. Introduction

The main goal of this review is to demonstrate, based on theHamiltonian approach, the difference between two mainnonlinear wave phenomena: solitons, or solitary waves, andwave collapseÐ the process in which a wave field becomessingular in a finite time. The singularity type depends on thephysical model. For example, the breaking of acoustic wavesis accompanied by the formation of sharp gradients (thegradient catastrophe) or, in mathematical parlance, a fold[1, 2]. In light self-focusing, which is just another example ofwave collapse, the light intensity becomes anomalously highas the focus is approached. For water waves, singularities insurface elevation have the form of wedges. In this case, thesecond derivative of surface elevation becomes infinite.

Collapses play a significant role in various branches ofphysics, not only in nonlinear optics and fluid dynamics butalso in plasma physics, physics of the atmosphere and ocean,and solid state physics. For many physical systems, collapse,as a process of singularity formation in finite time, is one ofthe most effective mechanisms of wave energy conversion toheat. For example, the collapse of plasma waves determinesthe efficiency of various collective methods of plasma heatingin problems of controlled thermonuclear fusion. In nonlinearoptics, this process is stopped because of multi-photonabsorption for moderate intensities and as a result of atom

V E Zakharov, E A Kuznetsov Lebedev Physical Institute,

Russian Academy of Sciences,

Leninskii prosp. 53, 119991 Moscow, Russian Federation;

Landau Institute for Theoretical Physics,

Russian Academy of Sciences,

ul. Kosygina 2, 119334 Moscow, Russian Federation;

Novosibirsk State University,

ul. Pirogova, Akademgorodok, 630090 Novosibirsk, Russian Federation

Tel. +7 (499) 132 67 51, +7 (495) 334 05 33. Fax +7 (495) 938 20 77

E-mail: [email protected], [email protected]

Received 14 July 2011

Uspekhi Fizicheskikh Nauk 182 (6) 569 ± 592 (2012)

DOI: 10.3367/UFNr.0182.201206a.0569

Translated by S D Danilov; edited by AM Semikhatov

REVIEWS OF TOPICAL PROBLEMS PACS numbers: 42.65.Jx, 42.65.Tg, 47.35.Fg, 47.35.Jk, 52.35.Sb

Solitons and collapses:

two evolution scenarios of nonlinear wave systems

V E Zakharov, E A Kuznetsov

DOI: 10.3367/UFNe.0182.201206a.0569

Contents

1. Introduction 5352. Main equations 537

2.1 Nonlinear SchroÈ dinger equation; 2.2 Hamiltonian structure; 2.3 Three-wave system and its reductions

3. Lyapunov stability for scalar models 5393.1 Stability of NLS solitons; 3.2 Lyapunov stability for the anisotropic Korteweg±de Vries equation

4. Solitons in the three-wave system 5424.1 Nonlinear stability of solitons in the three-wave system

5. Linear stability 5445.1 Linear stability of one-dimensional solitons of theNLS equation; 5.2 Stability of solitons for the coupling of the first

and second harmonics

6. Wave collapses 5466.1 The role of radiation; 6.2 Virial theorem; 6.3 Strong collapse; 6.4 Collapse in a boundary layer; 6.5 Weak collapse;

6.6 Black-hole regime

7. The role of dispersion in collapse 5518. Collapse in Kadomtsev±Petviashvili equations 552

8.1 Virial inequalities

9. Conclusions 555References 555

Physics ±Uspekhi 55 (6) 535 ± 556 (2012) # 2012 Uspekhi Fizicheskikh Nauk, Russian Academy of Sciences

ionization for large amplitudes. Collapse in nonlinear opticscan therefore be utilized to explore the interaction betweenlight andmatter over a wide range of intensities, including theprocesses of ionization by strong electromagnetic fields andthe formation of beams of relativistic particles. One morepossibility in optical systemsÐexploiting collapses to createultrashort pulsesÐhas long been discussed, but still awaitsits experimental implementation.

In speaking about solitons, we suppose, first, that theirlifetime, set by dissipation, is sufficiently large, and hencetheir dynamics, for example, as they interact with each other,can be treated as Hamiltonian; this is assumed virtuallyeverywhere in the review. Second, the existence of solitonsimplies their stability. We consider two approaches to theanalysis of soliton stability: by exploring stability at theinfinitesimal level, i.e., based on linearized equations and thesubsequent analysis of linear spectral problems, and byinvoking the Lyapunov method, which is the most versatilemethod that allows assessing the stability under not onlysufficiently small but also finite perturbations. According tothe Lyapunov theorem, a soliton is stable if it realizes aminimum (or maximum) of the Hamiltonian.

The problem of soliton stability in its most complete formwas discussed for the first time by us together with Rubenchikin review [3] published already more than twenty-five yearsago. The research on soliton stability has seen dramaticchanges since that time, especially in what concerns the useof integral majorizing inequalities that follow from theSobolev embedding theorems. Earlier reviews did not touchat sufficient length on the role of unstable solitons in theprocess of singularity formationÐ the wave collapse. Wedemonstrate in what follows that the wave collapse unfoldsin systems with unbounded Hamiltonians and can beinterpreted as a particle fall in an unbounded potential. Inthat case, the soliton represents a saddle point that the systemcrosses when passing to the wave collapse mode.

This review mainly deals with two models: the nonlinearSchroÈ dinger (NLS) equation and the three-wave system.These models have numerous applications in optics, plasmaphysics, fluid dynamics, etc. (see, e.g., review [4]). The NLSequation belongs to the class of universal models, it can beshown to describe the propagation of wave packets in aweakly nonlinear medium when the interaction with low-frequency (LF) acoustic waves is not essential. For example, itis well known [5, 6] that the propagation of optical solitons inoptical fibers can be described by a one-dimensional NLSequation to a good accuracy. A two-dimensional NLSequation describes stationary light self-focusing in mediawith the Kerr nonlinearity. The universality of the NLSequation is manifested, in particular, in the universality ofthe methods used to explore the stability of solitons (see, e.g.,review [3]).

The three-wave system describes solitons in media withquadratic nonlinearity, referred to as w �2�-media in optics.This system is written for the amplitudes of three wavepackets coupled via the quadratic nonlinearity. In a particu-lar case, this system describes the interaction between the firstand second harmonics. If the difference between the groupvelocities of three packets is sufficiently large, this systemreduces to the Blombergen equations [7], which can beintegrated with the help of the inverse scattering method [8].For close group velocities, as is commonly the case innonlinear optics, both the dispersion and diffraction termshave to be taken into account [9±11]. The model then

represents a vector system of the NLS type, but with aquadratic nonlinearity.

Solitons in both systems exist as a result of balancebetween nonlinear interaction and dispersion. The solitonsolutions for these two systems correspond to stationarypoints of their Hamiltonians for other integrals of motion,such as the momentum, the particle number, or the Manley±Rowe integrals, kept fixed. In other words, solitons areassociated with conditional extrema. This is a very essentialpoint. Had it not been the case, for example, had the solitonsbeen associated only with a stationary point of the Hamilto-nian, such a stationary point would be unstable for systemswith Hamiltonians unbounded from below (i.e., in theabsence of a vacuum). The last statement is in essence themain part of the Derrick arguments [12]. Luckily, solitonsviewed as stationary points of Hamiltonians correspond toconditional extrema, which allows exploring their stability byresorting to the Lyapunov theorem. In this case, for provingthe soliton stability, it suffices to show that theHamiltonian isbounded (other motion integrals being fixed). Obviously, asoliton realizing a minimum (or maximum) of a givenHamiltonian is Lyapunov stable. Such an approach was firstapplied to Korteweg±deVries (KdV) solitons by Benjamin(1972) [13], and later, to three-dimensional ion-acousticsolitons in strongly magnetized plasma by us [14]. It is nowone of the most powerful methods of exploring solitonstability.

We note that it is often easier to prove the Lyapunovstability than to solve the linear stability problem. In the lattercase, the completeness of all eigenfunctions of the linearizedproblem must be proved, which is an extremely difficult task.In this review, we pay special attention to embeddingtheorems and show how they lead to integral estimates, andthen use them to prove the boundedness of Hamiltonians.

When a Hamiltonian is bounded and a soliton realizes itsextremum, we can speak about the energy principle. A solitonrealizing a minimum shows up as an attractor. In particular,under a collision of two such solitons, the formation of asingle, more powerful soliton with lower energy is favorablefrom the standpoint of energy. However, in this coalescence,not only energy but also other motion integrals, for example,the number of particles, must also be preserved. This ispossible only in very special cases, for example, for integrablemodels. Nonintegrable systems are characterized by inelasticscattering and the formation of a soliton with a largeramplitude as a result. This process is accompanied by theradiation of waves in the nonsoliton sector, which evolve intolinear waves far from the scattering region owing todispersion. Here, radiation plays the role of friction, leadingto the formation of amore powerful soliton. Large-amplitudesolitons survive in such systems in the presence of multiplescattering, being in equilibrium with radiation (the linearwaves). Solitons in this case behave like a peculiar kind ofdrop, called statistical attractors by Yan'kov [15, 16], forwhich radiation plays the role of vapor.

If a Hamiltonian is unbounded, its unboundednessindicates that solitons must correspond to saddle points ofthe Hamiltonian and be unstable entities. The systembehavior cannot be stationary in this case. Two variants arepossible: either the system tends to completely spread outbecause of dispersion, in which case nonlinear interactionbecomes insignificant and waves become linear at large times,or the system collapses, and a field singularity forms as aresult. The latter process can be viewed as an analog of the fall

536 V E Zakharov, E A Kuznetsov Physics ±Uspekhi 55 (6)

of a particle on a center in an unbounded potential. Based onthe fact that the Hamiltonian is unbounded, we can ascertainthe role of radiation in the process of collapse. As was firstshown by Zakharov (1972) using the collapse of Langmuirwaves as an example [17], the radiation promotes collapse.This turns out to be a general property, inherent in manycollapsing wave systems.

However, a conclusion regarding the finiteness of thecollapse time cannot be drawn from this reasoning. Onlyrigorous theorems, like the Vlasov±Petrishchev±Talanovtheorem [18], allow formulating a sufficient criterion for thecollapse as a process throughwhich a singularity develops in afinite time.

Themethod based on the Lyapunov theorem is difficult toapply when exploring the stability of local stationary points.In this case, a linear stability analysis seems to be mostefficient. In this review, following the work by Vakhitov andKolokolov [19], we derive the stability criterion for NLSsolitons. The main point of this derivation relies on theoscillation theorem for the SchroÈ dinger operator, whichestablishes a one-to-one correspondence between the levelnumber and the number of zero crossings of the respectivewave function. Importantly, this theorem is only valid for ascalar SchroÈ dinger operator and cannot be used for a vectorSchroÈ dinger operator. This indicates that the Vakhitov±Kolokolov criteria are typically only sufficient for the solitoninstability and cannot be used as a necessary condition ofstability. As we show below, the three-wave system belongs tojust that category.

For the three-wave system, the linearized operator is theproduct of two 3� 3 matrix SchroÈ dinger operators, forwhich the oscillation theorem is inapplicable. We discussthis situation in detail for solitons describing a coupled stateof the first and second harmonics, and show with thisexample how the two approachesÐ the Lyapunov methodand the linear stability analysisÐwork. In particular, wediscuss how the stability of solitons is affected by phasedetuning.

2. Main equations

We begin with the main equations, the nonlinear SchroÈ dingerequation and equations for the three-wave system.

2.1 Nonlinear SchroÈ dinger equationIn dimensionless variables, the NLS equation is written in thecanonical form

ict �1

2Dc� jcj2c � 0 : �1�

In the context of nonlinear optics, c in Eqn (1) stands for theamplitude of the envelope of the electric field with a certain(for example, linear) polarization, and the time t is acoordinate in the direction of the wave packet propagation.The second term in Eqn (1) describes both diffraction andpositive dispersion of group velocity, which is realized inoptics in the anomalous dispersion range. In the case ofnormal dispersion, the operator D is replaced by thehyperbolic operator D? ÿ q2z . The nonlinear term jcj2c inEqn (1) corresponds to the Kerr contribution to the refractiveindex.

We note that the NLS equation, which models a broadrange of nonlinear wave phenomena, plays a central role in

the theory of wave collapse. This has become especially clearafter the prediction of light self-focusing [20] and thedevelopment of the related theory [21±25] (also see [26]). Theapplications of this model of wave collapse are not exhaustedby light self-focusing: the NLS equation finds diverseapplications in many other branches of physics (see, e.g.,reviews [4, 27±29] and the references therein). Commonly, theNLS equation and its modifications result from a reduction ofthe equations of motion of a nonlinear medium to equationsfor the envelope of a quasimonochromatic wave, whichinvolves averaging the original equations over fast oscilla-tions in time and space. Accordingly, the NLS equation isvery frequently regarded as an equation for envelopes. Wementionmonograph [30], which is dedicated exclusively to theNLS equation; it touches on its numerous mathematicalaspects, in addition to physical issues.

Nonlinear SchroÈ dinger equation (1) is often called theGross±Pitaevskii equation [31, 32], which very accuratelydescribes long-wave oscillations of the condensate of aweakly nonideal Bose gas with a negative scattering length.Presently, Eqn (1) is the main model used in the research onnonlinear dynamics of Bose condensates (see, e.g., Refs [33±38]). In that case, Eqn (1) is the SchroÈ dinger equation andc isthe wave function. Accordingly, Eqn (1) describes the motionof a quantum mechanical particle in the self-consistentattracting potential U � ÿjcj2. Just the attraction is thecause of the singularity occurrence. From the quantummechanical standpoint, collapse in the NLS framework canbe interpreted as the fall of a particle on the center (the pointof collapse) [39].

Collapse in NLS equation (1) is, however, possible, notfor all space dimensions D but only for D5 2. In the one-dimensional (1D) case, as shown by Zakharov and Shabat[40] in 1971, Eqn (1) is exactly integrable with the help ofthe inverse scattering method. This result has demonstratedthat solitons, being stationary localized objects, play anessential role in the dynamics of nonlinear waves describedby the NLS equation. The solitons of the one-dimensionalcase turned out to be given by structurally stable forma-tions, i.e., those stable with respect to not only small butalso finite perturbations, like those a soliton encounterswhen scattering on other solitons. Just this very attractiveproperty of solitons underlies the applications of opticalsolitons in optical fibers [5]. This idea is now realized inpractice (see, e.g., Ref. [41]). But solitons play a differentrole in higher dimensions.

2.2 Hamiltonian structureIt is well known (see, e.g., Ref. [11]) that NLS equation (1) is aHamiltonian equation and can be written in the form

ict �dHdc �

�2�

with the Hamiltonian

H � 1

2

��jHcj2 drÿ

�jcj4 dr

�� 1

2�I1 ÿ I2� : �3�

In addition toH, this equation has two more simple integralsof motion: the number of particles

N ��jcj2 dr �4�

June 2012 Solitons and collapses: two evolution scenarios of nonlinear wave systems 537

(which coincides, up to a constant factor, with the energy ofthe wave packet) and the momentum

P � i

2

��cHc � ÿ c �Hc� dr : �5�

The conservation of N follows from the gauge symmetryc! c exp �ia�, and the conservation of P is conditionallyrelated to the symmetry under translations. These twosymmetries ensure the Galilean invariance of Eqn (1). Inparticular, a simple solution of Eqn (1),

cs � c0�r� expil2t2

; �6�

with c0�r� ! 0 as r!1, corresponds to a soliton at rest. Asolution for a moving soliton follows by applying a Galileantransformation to Eqn (6).

It can also be easily established that soliton solution (6)represents a stationary point of the HamiltonianH for a fixedparticle number (cf. Ref. [3]):

d�H� l2N� � 0 ; �7�

where l2 is the Lagrange multiplier. Solving variationalproblem (7) is equivalent to finding a solution of thestationary NLS equation

ÿl2c� Dc� jcj2c � 0 :

This implies the dependence ofN on l for soliton solution (6),

Ns � l2ÿDN0 ; N0 �� f �x�j2 dx ; �8�

where f satisfies the equation

ÿf� 1

2Df� j f j2f � 0 :

Dependence (8) proves to be crucial for the criterion of solitonlinear stability, the subject of Section 3.

2.3 Three-wave system and its reductionsWe next consider the three-waves system, which can bewritten in the form (see, e.g., Refs [10, 11])

iqc1

qtÿ o1c1 � i�v1H�c1 �

1

2oab

1 q2abc1 � Vc2c3 ; �9�

iqc2


1

2oab

2 q2abc2 � Vc1c�3 ; �10�

iqc3


1

2oab

3 q2abc3 � Vc1c�2 ; �11�

where the amplitudes of three wave packets cl�x; t��l � 1; 2; 3� are slowly varying functions of x (klLl 4 1,where kl is the carrier wave vector of the lth packet and Ll isits characteristic length), vl � qol�kl�=qkl are the groupvelocities of packets,

oabl �

q2ol�kl�qkla qklb

is the dispersion tensor, and V is the three-wave matrixelement, which can be considered real valued without the

loss of generality. It is assumed that the carrier frequenciesol � ol�kl� are close to the resonance

o1�k1� � o2�k2� � o3�k3� ; �12�

k1 � k2 � k3 : �13�

Like the NLS equation, system of equations (9)±(11) isHamiltonian:

iqcl

qt� dH

dc �l; �14�

where

H � H0 �H1 ;

H0 �X3l�1

��oljclj2 drÿ i

�c �l �vlH�cl dr

� 1

2

�Hac

�l o

abl Hbcl dr

�; �15�

H1 � V

��c1c

�2c�3 � c �1c2c3� dr : �16�

Here, H0 corresponds to linear waves: the first term in theright-hand side of Eqn (15) makes a dominant contribution tothe energy, the second term corresponds to the motion ofpackets with the group velocity, and the third term isresponsible for dispersive broadening of packets. It isnoteworthy that the dispersive term is small in the parameterDkl=kl compared to the second term. The third term in H0

should therefore be retained only when the differencesbetween the group velocities are small. For instance, such asituation can be realized in optics.

If the carrierol and kl exactly satisfy resonance conditions(12), then the terms� ol can be eliminated by the transforma-tion cl ! cl exp �ÿiolt�.

We note that three-wave system (9)±(11) allows aphysically important reduction that corresponds to theinteraction between the first �c1� and second �c2� harmo-nics. In this case, the Hamiltonian becomes

H �X2l�1

��oljclj2 drÿ i

�c �l �vlH�cl dr

� 1

2

�qac

�l o

abl qbcl dr

�� V

��c �2c 2

1 � c:c:� dr ;

and the respective equations of motion are written as

iqc1


1

2oab

1 q2abc1 � 2Vc2c�1 ; �17�

iqc2


1

2oab

2 q2abc2 � Vc 21 ; �18�

where o2 ' 2o1. If the resonance is exact, the termsproportional to ol can be eliminated by the transformationsc1 ! c1 exp �ÿio1t�, c2 ! c2 exp �ÿi2o1t�. In the one-dimensional case, the three-wave system (9)±(11) allowssimplifications. By introducing new variables

cl � ~cl�xÿ vt; t� exp �iklx� ; k1 � k2 � k3 ; �19�


it is possible to eliminate the first derivatives if the velocity vand wave numbers kl are chosen as

v � v1d1 ÿ v2d2 ÿ v3d3d1 ÿ d2 ÿ d3

; kl � dl�vÿ vl� ; dl � 1

o 00l: �20�

(We note that the equations turn out to be Galilei invariantfor d1 � d2 � d3.) As a result, system (9)±(11) in the one-dimensional case takes the form (the tilde over cl is omittedbelow and the matrix element of the three-wave coupling V isset to ÿ1)

iqc1

qtÿ O1c1 �

1

2o 001c1xx � ÿc2c3 ; �21�

iqc2

qtÿ O2c2 �

1

2o 002c2xx � ÿc1c

�3 ; �22�

iqc3

qtÿ O3c3 �

1

2o 003c3xx � ÿc1c

�2 ; �23�

where

Ol � ol � klvl � o 00l k2l

2: �24�

Here, as previously, it is assumed that the new frequencies areclose to the resonance,

O1 � O2 � O3 ;

or, in other words, the frequency detuning is small.At the next step,O2 andO3 can be eliminated with the help

of transformations

c1�x; t� ! c1�x; t� exp�ÿi�O2 � O3�t

�;

c2�x; t� ! c2�x; t� exp �ÿiO2t� ;c3�x; t� ! c3�x; t� exp �ÿiO3t� :

As a result, Eqns (21)±(23) become

iqc1

qtÿ Oc1 �

1

2o 001 c1xx � ÿc2c3 ; �25�

iqc2

qt� 1

2o 002 c2xx � ÿc1c

�3 ; �26�

iqc3

qt� 1

2o 003 c3xx � ÿc1c

�2 ; �27�

whereO � O1 ÿ O2 ÿ O3 is the frequency detuning character-izing how far the carrier frequencies are from the resonanceEqn (12).

System of equations (25)±(27) preserves its canonicalstructure with the Hamiltonian

H ��Ojc1j2 dx�

Xl

�1

2o 00l jclxj2 dx

ÿ��c �1c2c3 � c:c:� dx : �28�

In addition to H, these equations (as well as the nontrans-formed equations) have two additional integrals of motion,the so-called Manley±Rowe integrals

N1 ��ÿjc1j2 � jc2j2

�dx ; N2 �

�ÿjc1j2 � jc3j2�dx : �29�

The invariants N1 and N2 arise as a result of the averagingprocedure that removes all nonresonant terms except the onepertaining to the three-wave coupling.

The one-dimensional system (17), (18) describing theinteraction of the first and second harmonics is transformedsimilarly to Eqn (19):

iqc1

qt� 1

2o 001 c1xx � ÿ2c2c

�1 ; �30�

iqc2

qtÿ Oc2 �

1

2o 002 c2xx � ÿc 2

1 ; �31�

where the frequencies Ol are defined by Eqns (24) with klgiven by (20). Only one Manley±Rowe integral for system(30), (31) exists:

N ��ÿjc1j2 � 2jc2j2

�dx : �32�

Similar transformations for the three-wave system inEqns (21)±(23) [as well as for the coupling of the first andsecond harmonics (17), (18)] can be carried out in the case ofmany dimensions. For this, it is necessary to make the changeo 00l clxx ! oab

l q2abcl in all integrals of motion and to replace dlwith the matrix inverse to oab

l , with the velocities vlconsidered vector quantities. It is pertinent to note that forthe three-wave system, the difference between one-dimen-sional and multi-dimensional cases for D4 3 is not soessential, in contrast, for example, to the case of the NLSequation. This can be seen most vividly when exploring thestability of solitonsÐ the coupled states of three-wavepackets.

To conclude this section, we note one more importantreduction of system (9)±(11), which corresponds to stationarywaves when the time derivative in these equations is absent. Inthis case, the system of equations, which is also Hamiltonian,describes spatial solitonsÐ the distributions localized in onedirection or in a plane (see, e.g., Refs [42±44] formore details).

3. Lyapunov stability for scalar models

3.1 Stability of NLS solitonsWe begin by exploring the stability of solitons of the one-dimensional NLS equation,

ict �dHdc �� ÿ 1

2cxx ÿ jcj2c ; �33�

where theHamiltonianH is defined in (3) with the integrationperformed only with respect to the coordinate x. Asmentioned in Section 2, from the quantum mechanicalstandpoint, Eqn (33) describes the motion of a particle in theself-consistent attracting potentialU � ÿjcj2. But in the one-dimensional case, nonlinear attraction cannot overcomedispersion �� cxx�, the cause of smearing. The exact balanceof these two opposing effects leads to the existence of asolitonÐa stationary solution of Eqn (33) of the formc�x; t� � c0�x� exp �il2t=2�, where

c0�x� � l sech �lx� �34�

satisfies the stationary NLS equation

ÿl2c0 � c0xx � 2jc0j2c0 � 0 : �35�


The simplest solution (34) describes a soliton at rest. Movingsolitons are obtained from it by performing Galileantransformations.

It can be easily verified that solution (34) corresponds to astationary point of the Hamiltonian for a fixed number ofparticlesN � � jcj2 dx, because stationaryNLS equation (35)directly follows from the variational problem

d�H� l2N

2

�� 0 : �36�

It is also easy to find that the number of particles increaseslinearly with l, Ns � 2l, in these solutions. Here andhenceforth, the index s indicates that a functional (in thiscase, N ) is computed at the soliton solution.

According to the Lyapunov theorem, to prove thestability of soliton (34), it suffices to show that the solitonrealizes a minimum of the Hamiltonian. (The unbounded-ness of H from above is obvious: for a fixed N, which isessentially the normalization of the wave function, theHamiltonian can become arbitrarily large because of thekinetic energy I1 in the class of strongly varying functionsc.)

We begin with simple, essentially, dimensional, estimatesthat indicate that H is bounded from below. Let A be acharacteristic amplitude of the soliton and l be its character-istic size. Then the particle number is approximately given byN � A2l; hence, A2 � Nlÿ1. For integrals I1 and I2, we havethe estimates

I1 � A2

l� N

l 2; I2 � A4l � N 2

l;

which give the Hamiltonian

H � 1

2

�N

l 2ÿN 2

l

�:

The last expression immediately implies thatH, as a functionof the soliton size l, is positive for small l (it grows as lÿ2) andis negative and tends to zero from below for large l. It isobviously bounded from below. Its minimum is achieved for asoliton with the size ls � 2=N.

This same conclusion can be drawn by considering scaletransformations that preserve the particle number:

c�x� ! aÿ1=2c�x

a

�: �37�

As a result of transformation (37), the Hamiltonian becomesa function of the scaling parameter a,

H�a� � 1

2

�I1a 2ÿ I2

a

�: �38�

The functionH�a� reaches a minimum at the point a � 1,which corresponds to a soliton (Fig. 1):

qHqa

��a� 1

� 0) 2I1s � I2s � 4l3

3; Hs � ÿ 1

2I1s � ÿ l3

3:

�39�

We can additionally verify that the soliton realizes aminimum of H with respect to phase transformations

c0�x� ! c0�x� exp �iw�x��, which also preserve N:

H � Hs � 1

2

��qwqx

�2

c 20 dx :

The two simple transformations defined byEqns (37) and (38)therefore give a minimum of H, which hints at the solitonstability, but, strictly speaking, does not prove it.

We offer a rigorous proof. The main element of this proofrelies on Sobolev integral estimates, which follow from thegeneral Sobolev embedding theorems. The Sobolev theoremstates that the space Lp can be embedded in the Sobolev spaceW 1

2 if the dimension of the space RD over which theintegration is performed satisfies the inequality

D <2

p�p� 4� :

This implies that the norms

kuk p �� juj p dDx

�1=p; p > 0 ;

kukW 12�� ÿ

m 2juj2 � jHuj2�dDx�1=2 ; m 2 > 0 ;

are related by the inequality (see, e.g., Ref. [45])

kuk p 4MkukW 12; �40�

whereM is a positive constant. Notably, forD � 1 and p � 4,inequality (40) can be written as�1

ÿ1jcj4 dx4M1

��1ÿ1

ÿm 2jcj2 � jcxj2

�dx

�2

: �41�

Using this, we can derive a multiplicative variant of theSobolev inequality, the so-called Gagliardo±Nirenberginequality [46] (see also Refs [29, 45, 47]). With this aim, weperform scale transformations x! ax in inequality (41), afterwhich the right-hand side of (41) becomes dependent on theparameter a,�1

ÿ1jcj4 dx4M1

�am 2

�1ÿ1jcj2 dx� 1

a

�1ÿ1jcxj2 dx

�2

:

This inequality is valid for any positive a, including the valuesfor which the right-hand side of the inequality is minimum,

Hs

H

a � 1

a

Figure 1. The profile ofH�a�.


which gives the Gagliardo±Nirenberg inequality

I2 4CN 3=2I1=21 ; �42�

where I1 and I2 are the integrals defined by Eqn (3) forD � 1,and C is a new constant.

Inequality (42) can be refined if the minimum value of theconstant C is known. To find it, we need to consider allextrema of the functional

Jfcg � I2

N 3=2I1=21

; �43�

which are found from the variational problem dJ � 0. It canbe readily seen that this amounts to finding solutions ofstationary NLS equation (35),

ÿl2c� cxx � 2jcj2c � 0 :

The solution of the last equation is unique (up to a constantphase factor) and represents the one-dimensional soliton ofNLS equation (34),

c � lcosh lx

:

We therefore conclude that the best constant Cbest equals thevalue of Jfcg at soliton solution (34):

Cbest � I2s

N3=2s I

1=21s

� 2I1=21s

N3=2s

: �44�

Inequality (42) with C � Cbest immediately implies thestability proof for the one-dimensional soliton.

The substitution of Eqn (42) with C � Cbest in Eqn (3) forD � 1 gives the desired estimate for the lower bound of theHamiltonian:

H51

2

ÿI1 ÿ CbestI

1=21 N 3=2

� � Hs � 1

2�I 1=21 ÿ I

1=21s �2 : �45�

Inequality (45) becomes exact at the soliton solution, whichproves that the NLS soliton is Lyapunov stable not onlyagainst small but also against finite perturbations.

Similarly, we can prove the stability of the `ground-state'soliton (a radially symmetric solution with no zeros) for amultidimensional NLS equation with a power-like nonlinear-ity

ict �1

2Dc� s

2jcj2sÿ2c � 0 ; �46�

where s > 1.The Hamiltonian for Eqn (46) can be written in the form

H � 1

2

�ÿjHcj2 ÿ jcj2s� dDx � I1 ÿ Is ; �47�

and the ground-state soliton solution

cs � exp

�il2t2

�l1=�sÿ2�g�lr� ;

with the function g�x� satisfying the equationÿg� H2

xg� sjgj2sÿ2g � 0 ;

corresponds to a stationary point of H for a fixed particlenumber N,

d�H� l2

2N

�� 0 :

As previously, performing the scaling transformation

cs�r� ! aÿD=2cs

�r

a

�;

which preserves the total number of particles, we find that thecorresponding Hamilton function

H�a� � 1

2

�I1a 2ÿ Is

a �sÿ1�=D

�

is bounded from below and attains a minimum only if

sÿ 1

D< 2 :

This minimum (for a � 1) corresponds to the ground-statesoliton solution, which hints at its stability. To rigorouslyprove the stability, we have to resort to the related multi-plicative inequality for Is, which follows from Eqn (40),

Is ��jcj2s dDx4Ms

��jcj2 dDx

�a��jHcj2 dDx

�b; �48�

where a � D=2� s�1ÿD=2� and b � �sÿ 1�D=2. Then, asin the case with the one-dimensional cubicNLS equation (33),we need to find the best constant Ms as a minimum of thefunctional N aI b1 =Is. It is easy to verify that this minimum isprovided by the ground-state soliton solution (46).

Substituting inequality (48) with the best constantMbest inHamiltonian (47),

H51

2

ÿI1 ÿMbestN

aI b1�;

we find that the right-hand side of the last inequality as afunction of I1 is bounded from below and reaches a minimumat the ground-state soliton solution, which just proves theLyapunov stability of this solution.

Stability criterion (45) and its generalization for�sÿ 1�=D < 2 can be interpreted as the energy principle. Ifwe ignore the integrability of the one-dimensional NLSequation (with s � 2) [40], the coalescence of two solitonsshould be favorable in terms of energy. This implies that anew soliton appearing as a result of the interaction of twosolitons must have a lower energy (the energy of the solitonbeing negative!) than the energy of the interacting solitons. Inthe general case, the interaction between solitons is accom-panied by radiation of small-amplitude waves. Why does theradiation favor the emergence of a deeper soliton?

We consider a domain G where the Hamiltonian isnegative, HG < 0. The following obvious estimate is thenvalid for the integral Is �

�G jcj2s dx:�

G

jcj2s dx4 ÿmax

xjcj2�2sÿ2 �

G

jcj2 dx :

Hence, it follows thatÿmaxGjcj2�2sÿ2 5 jHGj

NG: �49�

We note that inequality (49) is valid for an arbitrary physicaldimension D.


If small-amplitude waves are radiated from the domainG,they carry away positive energy (because the nonlinearity isnot essential for them). As a result, because of the conserva-tion of the Hamiltonian and the number of particles, jHGjincreases, and the number of wavesNG decreases. Their ratiotherefore increases owing to the radiation, and max jcj2increases accordingly. This process continues until a soli-tonÐa stationary stable state realizing the minimum of HG

for a fixed NG Ð is formed in the domain G.It is worth noting that in some, rather special cases, the

radiation can be absent. This is so if the energy and particlenumber conservation laws hold:

E�N � � E�N1� � E�N2� ;N � N1 �N2 ;

where N1 �N2� is the total number of particles for the first(second) soliton. These equations cannot be satisfied for allfunctions E�N�. For example, if E�N� � N a �a > 0�, then thesystem of equations does not have any solutions.

In the opposite case where �sÿ 1�=D > 2, the functionH�a� is unbounded from below for a! 0 and, instead of aminimum for �sÿ 1�=D < 2, a maximum is achieved, whichpoints to the instability of the soliton solution.

3.2 Lyapunov stabilityfor the anisotropic Korteweg±deVries equationThe next example to be considered here is the anisotropicKdV equation [14] derived by us in 1974:

ut � qqz

Du� 6uuz � 0 ; �50�

where D is the three-dimensional Laplace operator. In theone-dimensional case, Eqn (50) reduces to the classic KdVequation.

Equation (50) describes three-dimensional ion±acousticsolitons u � us�zÿ Vt; r?� propagating along the magneticfield vector (parallel to the z axis) in a strongly magnetizedplasma, where the plasma thermal pressure nT (with thedensity n and temperature T) is small compared to themagnetic field pressure B 2=8p, i.e., b � 8pnT=B 2 5 1. Soli-ton solutions are sought by integrating the equation

ÿVus � Dus � 3u 2s � 0 :

It can be readily seen that localized solutions (us ! 0 asr!1) are only possible if the velocity V is positive. Thesoliton solution can be written in analytic form only in theone-dimensional case, and it is then just the classic KdVsoliton

u � 2k 2

cosh2 k�xÿ 4k 2t� ; V � 4k 2 :

In the case of many dimensions, localized solutions can befound only numerically (see Ref. [14]).

For Eqn (50), as for the NLS equation, solitons corre-spond to stationary points of the Hamiltonian

H ��

1

2�Hu�2 ÿ u 3

�dr � 1

2I1 ÿ I2 �51�

for the fixed momentum P � �1=2� � u 2 dr; in other words,solitons are solutions of the variational problemd�Hÿ VP� � 0, with the velocity V entering as a Lagrangemultiplier.

The scaling transformations

u�r� ! 1

aD=2u

�r

a

�; �52�

which preserveP, lead to approximately the same dependence

H�a� � 1

2

I1a 2ÿ I2

aD=2; D4 3 ;

as in the case of the one-dimensional NLS equation (comparethis with the dependence in Fig. 1).

Accordingly, in analogy with the NLS equation, for thesoliton solution (at the point a � 1, where qH=qa � 0), wehave

I2s � 2

DI1s ; Hs � Dÿ 4

2DI1s � Dÿ 4

6ÿDVPs ;

and hence Hs < 0 for D4 3.Themultiplicative variant of the Sobolev inequality in this

case becomes

I2 4CP 1=�4ÿD�I b1 ;

where b � �1=2��10ÿ 3D�=�4ÿD�. The best value C � Cbest

is defined by the ground-state soliton (the one without zeros).Substitution of the last inequality with the best constant in(51) shows the boundedness of H for all space dimensionsD � 1; 2; 3:

H51

2I1 ÿ CbestP

1=�4ÿD�I b1 5Hs :

These inequalities become equalities when evaluated on aradially symmetric soliton solution without zeros, whichproves the stability of ion±acoustic solitons in a magnetizedplasma in the case of many dimensions [14]. As a particularresult, this also offers a stability proof for the classic KdVsolitons. For one-dimensional KdV solitons, this approachwas first used by Benjamin [13] in 1972. But at that time, theintegrability of the KdV equation via the inverse scatteringmethod had already been known owing to Gardner, Green,Cruscal, and Miura [48]. In particular, it had already beenestablished in [48] that the KdV soliton is a structurally stableobject.

4. Solitons in the three-wave system

We consider soliton solutions for system (25)±(27) in the form

c1�x; t� � c1s�x� exp�i�l1 � l2�t

�;

c2�x; t� � c2s�x� exp �il1t� ;c3�x; t� � c3s�x� exp �il2t� ;

where c1s�x�, c2s�x�, and c3s�x� are assumed to be realvalued, with no zeros (ground state) and decaying atinfinity. These functions satisfy the equations

L1c1s � ÿc2sc3s ; L1 � ÿ�l1 � l2 � O� � 1

2o 001 q

2x ; �53�

L2c2s � ÿc1sc3s ; L2 � ÿl1 � 1

2o 002 q

2x ; �54�

L3c3s � c1sc2s ; L3 � ÿl2 � 1

2o 003 q

2x : �55�


Solutions of system (53)±(55) correspond to stationary pointsof the Hamiltonian for two fixed Manley±Rowe integrals N1

and N2:

d�H� l1N1 � l2N2� � 0 : �56�

The soliton solutions exponentially decay at infinity if thefollowing three conditions are satisfied:

m 21 � d1�l1 � l2 � O� > 0 ; m 2

2 � d2l1> 0 ; m 23 � d3l2 > 0 ;

�57�

where dl � 1=o 00l .This result can be obtained differently if we determine

relations between the terms in the Hamiltonian and theManley±Rowe integrals N1 and N2 in the soliton solution(for details, see review [4]). With this aim in view, Eqn (53)should be multiplied by c1s and then integrated over x. As aresult, we obtain

ÿ�l1 � l2 � O�n1�D1 � ÿI : �58�

An analogous procedure applied to Eqns (54) and (55) gives

ÿ l1n2 �D2 � ÿI ; �59�ÿ l2n3 �D3 � ÿI ; �60�

where

nl ��jclsj2 dx ; Dl � 1

2

�o 00l

�� dcls

dx

��2 dx ; I ��c1sc2sc3s dx :

These integral relations have to be complemented by acondition that follows from the variational problem byapplying to it scaling transformations that preserve N1 andN2:

cls�x� ! aÿ1=2cls

�x

a

�:

Under such transformations, the Hamiltonian becomes afunction of the parameter a,

H�a� �Xl

��Oljclsj2 �

1

2a 2o 00l jclsxj2

�dx

ÿ 2

a 1=2

�c1sc2sc3s dx :

Using the last relation together with Eqn (56), it is straightfor-ward to obtain that

dH

da

��a�1� 0 ; or

Xl

�o 00l jclsxj2 dxÿ

�c1sc2sc3s dx � 0 :

�61�

Taking relations (58)±(60) into account and bearing in mindthe positivity of nl (by definition), we arrive at conditions (57).Hence, for the existence of solitons, all operatorsLl must havethe same sign definiteness (positive or negative). We note thatthis requirement holds for all physical dimensions D4 3.Accordingly, all matrices oab

l must have the same signdefiniteness, for example, be positive definite. In particular,if one of the matrices is not sign definite, then multi-dimensional solitons are absent. A similar situation occurs

for envelope solitons in a medium with normal dispersion(see, e.g., Ref. [49]).

It should also be mentioned that the sign definiteness ofthe matricesoab

l translates into a simple physical requirementthat Cherenkov resonances between the soliton and linearwaves be absent (see Refs [50, 51]).

4.1 Nonlinear stability of solitons in the three-wave systemWe demonstrate how the Lyapunov stability can be inferredfor soliton solutions of the three-wave system. Because wehere have three fields (the three amplitudes ci), we need toconsider two spaces, L3; 3 andW 1

2 , with the norms

kukL3; 3�� ÿjc1j3 � jc2j3 � jc3j3

�dDx

�1=3;

kukW 12��~l1

�ÿjc1j2 � jc2j2�dDx� ~l2

�ÿjc1j2 � jc3j2�dDx

� 1

2

Xl

�qac

�l o

abl qbcl d

Dx

�1=2;

where the constants ~l1; 2 > 0 and tensors oabl are supposed to

be positive definite. In this case, the Sobolev inequalitybecomes

kukL3; 3<MkukW 1

2: �62�

We note that the norm kukL3; 3and the interaction Hamilto-

nian are related by the simple inequality

kuk3L3; 35

3

2

��c �1c2c3 � c:c:� dDx : �63�

The multiplicative variant of the Sobolev inequality isobtained similarly to that of the NLS equation. In particu-lar, for D � 1, we have

J4M1�~l1N1 � ~l2N2�5=4I 1=4 ;

where

J ��c �1c2c3 � c:c:� dDx ; I � 1

2

Xl

�qac

�l o

abl qbcl d

Dx

and ~l1 and ~l2 are arbitrary positive parameters.Minimizationwith respect to ~l1 and ~l2 gives

J4C�N1N2�5=8I 1=4 : �64�

The next step consists in finding the best constant C as aminimum of the functional

Cbest � min�c�

F�c� ; F � J

�N1N2�5=8I 1=4:

It follows that this minimum is provided by the ground-statesoliton

Cbest � F �cs� : �65�

Using refined inequality (64), we arrive at the requiredestimate for the Hamiltonian with zero detuning, O � 0,

H5 Iÿ 2I 3=4s I 1=4 5Hs�O � 0� ;


which proves the stability of the (ground-state) solitonsolution in the case of one dimension for zero detuning. Aspreviously, the last inequalities transform into equalities forthe soliton solution. Following this scheme, the solitonstability can be proved for any physical dimension D4 3.

For a nonzero detuning, the Hamiltonian in the one-dimensional case becomes

H � O�jc1j2 dx� ~H ;

where ~H coincides with the Hamiltonian at O � 0:

~H �Xl

�1

2o 00l jclxj2 dxÿ

��c �1c2c3 � c:c:� dx :

We now analyze how the frequency detuning O influencesthe stability of solitons. Obviously, the soliton existencedomain is strongly asymmetric with respect to the sign of thedetuning O. For O > 0, the straightforward inequality

H5Hs�O � 0�

holds, from which it follows that the Hamiltonian is boundedfrom below by the value of Hs at the ground-state solitonsolution with zero detuning.

For O < 0, we have

H5Hs�O � 0� ÿ jOj�jc1j2 dx ; �66�

where the integral� jc1j2 dx is always bounded from below by

min �N1;N2�. Accordingly, the estimate for H is written as

H5Hs�O � 0� ÿ jOjmin �N1;N2� :

We have thus proved the stability of ground-state solitons(those without nodes) describing a coupled state of three wavepackets. Notably, in the absence of detuning, the solitonrealizes a minimum of the Hamiltonian, which is rigorouslyproved with the help of majorizing Sobolev inequalities. Thatthe Hamiltonian of the three-wave system is bounded wasfirst demonstrated by Kanashev and Rubenchik (1981) [10]for isotropic media. Later, Turitsyn (1995) [52] showed thatthe Hamiltonian attains a minimum at the ground-statesoliton (without zeros) for zero detuning for the couplingbetween the first and second harmonics, when the dispersionoperators are Laplacians. In 1997, Berge, Bang, Rasmussen,and Mezentsev [53] demonstrated the boundedness of H fornonzero detuning for the coupling between the first andsecond harmonics. The general case of a bounded H for thethree-wave system is discussed in Refs [54, 55].

5. Linear stability

We next address the problem of the linear stability of solitonsconsidered in Sections 2±4.

5.1 Linear stability of one-dimensional solitonsof the NLS equationWe begin with the one-dimensional NLS equation, assumingthat

c�x; t� � ÿc0�x� � u� iv�exp

il2t2; c0 4 u; v :

The linearized equations for a perturbation are Hamiltonianequations

ut � 1

2

d ~H

dv; vt � ÿ 1

2

d ~H

du; �67�

where u and v are canonically conjugate quantities and ~H isthe second variation of F � H� l2N=2,

~H � vjL0jv�� ujL1ju

�; �68�

with L0 � l2 ÿ q2x ÿ 2c 20 and L1 � l2 ÿ q2x ÿ 6c 2

0 . The firstterm in ~H is the mean value of L0, playing the role of kineticenergy; the second term,

ujL1ju

�, plays the role of potential

energy.The stability or instability of a soliton is therefore

determined by the properties of the operators L0 and L1.The first property of L0 follows directly from the stationaryNLS equation,

L0c0 � l2c0 ÿ q2xc0 ÿ 2c 30 � 0 ; �69�

which means that c0 is the ground state of L0 (has no zeros).Accordingly,

vjL0jv

�5 0. The stability or instability of

solitons is therefore determined by the sign of the potentialenergy.

The operator L1 has an eigenfunction with zero eigenva-lue,

L1qc0

qx� 0 :

This is a neutrally stable mode that correspond to the shift ofthe soliton and has a single node �x � 0�. Therefore, theSchroÈ dinger operatorL1 has the only lower eigenvalueE < 0,which corresponds to the ground state f0. This apparentlysuggests that

ujL1ju

�< 0 and, correspondingly, that an

instability develops. But this is not the case because u isconstrained by hujc0i �

�uc0 dx � 0, which is a consequence

of the conservation of the number of particles. For thisreason, the stability or instability of a soliton is determinedfrom the solution of the spectral problem

L1jfi � Ejfi � Cjc0i �70�

(where C is a Lagrange multiplier) under the additionalconstraint hujc0i � 0.

Expanding jfi in a series in the complete set of eigenfunc-tions ffng of L1 (L1fn � Enfn),

f �Xn

Cnfn ;

we find

Cn � Chfnjc0iEn ÿ E

; C1 � 0 :

Inserting these expressions into the solvability conditionhfjc0i � 0, we obtain the dispersion relation

f �E� �Xn

0 hc0jfnihfnjc0iEn ÿ E

� 0 : �71�

The prime at the sum indicates that the term with the energyE1 � 0 is excluded (the translation mode f1 � qc0=qxcorresponds to the energy E1).


We next consider the energy range between the ground-state energy E0 < 0 and the first positive level E2. In thisinterval, f �E� increases monotonically �qf=qE > 0� fromÿ1at E � E0 to�1 at E � E2. If f �E� is negative at E � 0, thenthe dispersion equation does not have negative eigenvaluesand the soliton is stable. If f �0� > 0, then there exists a singleeigenvalue E < 0 and hence the soliton is unstable.

To find f �0�, note that

f �0� �X0n

hc0jfnihfnjc0iEn

� c0jLÿ11 jc0

�:

Differentiating the stationary NLS equation

ÿl2c0 � q2xc0 � 2jc0j2c0 � 0

with respect to l2, we then obtain

L1

�qc0

ql2

�� ÿc0 ;

or c0jLÿ11 jc0

� � ÿ�c0

�� qc0

ql2

�� ÿ 1

2

qNs

ql2:

As a result, we arrive at the so-called Vakhitov±Kolokolovcriterion [19]: if

qNs

ql2> 0 ; �72�

then the soliton is stable; for the other sign of the derivative, it isunstable.

This criterion is valid for any dimension D and thedependence of the nonlinear term on jcj2 �i.e., if jcj2 in theNLS equation is replaced with an arbitrary function f �jcj2��.

The criterion has a simple physical interpretation. Thequantity e � ÿl2=2 for the stationary NLS equation repre-sents the energy of the bound state, the soliton. If, on adding a`single' particle, the energy of the bound state decreases, thesituation is stable. If, on adding a `single' particle, the levelÿl2 shifts to the continuum spectrum, the soliton is unstable.

The dependence of Ns on l can easily be established for acubic NLS equation,Ns / l2ÿD, which implies the stability ofone-dimensional solitons of the cubic NLS equation and theinstability of three-dimensional ones. The case of twodimensions is degenerate.

5.2 Stability of solitons for the couplingof the first and second harmonicsWe consider system of Eqns (30) and (31) that describes theinteraction of the first and second harmonics and show howthe procedure for deriving the Vakhitov±Kolokolov criterioncan be applied in this case. Soliton solutions for the couplingof the first and second harmonics c1�x; t� � c1s�x� exp �il2t�and c2�x; t� � c2s�x� exp �2il2t� are found from the equa-tions for the amplitudes c1s and c2s:

ÿ l2c1 �1

2o 001 c1 � ÿ2c2c1 ; �73�

ÿ 2l2c2 ÿ Oc2 �1

2o 002 q2xc2 � ÿc 2

1 : �74�

Here, the solutions c1s and c2s are assumed to be real valuedand have no nodes, i.e., correspond to the ground-state

soliton solution (the index s at c1 and c2 is omitted). Weconsider small perturbations on the background of thissoliton solution by writing

c1�x; t� � �c1s � u1 � iv1� exp �il2t� ;c2�x; t� � �c2s � u2 � iv2� exp �2il2t� :

The linearization of system (30), (31) leads to the linear(Hamiltonian) equations

ut � 1

2

d ~H

dv; vt � ÿ 1

2

d ~H

du; �75�

where ~H, just as for the NLS equation, is the second variationof F � H� l2N,

~H � vjL0jv�� ujL1ju

�: �76�

Here, u and v are vectors with components u1, u2 and v1, v2,and N is defined in Eqn (32). The second-order differentialoperators L0 and L1 are in this case 2� 2 matrix operators

L0; 1 �l2 ÿ 1

2o 001 q

2x � 2c2 ÿ2c1

ÿ2c1 2l2 ÿ 1

2o 002 q

2x ÿ O

0B@1CA :

Both operators, just as in theNLS case, are self-adjoint. Fromthe standpoint of quantum mechanics, these operatorscorrespond to the SchroÈ dinger operator for a nonrelativisticparticle of spin S � 1=2 in an inhomogeneous magnetic field.It is common knowledge that the oscillation theorem is notapplicable to such operators. The ground-state eigenfunctiondoes not have zeros, but the correspondence between thenumber of zeros and the level number, inherent in the scalarSchroÈ dinger operator, is absent here. This lack of correspon-dence, as is to become clear in what follows, does not allowdrawing the same conclusions about the soliton stability asfor the NLS equation. The Vakhitov±Kolokolov criterion forsuch matrix systems can only be used as a sufficient conditionof instability; the same statement is also valid for the three-wave system.

As regards the properties of the operators L0 and L1, theyare similar to the properties of analogous operators for theNLS equation. The nonnegativity of L0 follows from the factthat if, instead of v1 and v2, we introduce new functions w1 andw2 defined by the relations

v1 � c1w1 ; v2 � c2w2 ;

then hvjL0jvi can be represented asvjL0jv

� � 1

2

��o 001 c 2

1 w21 � o 002 c

22 w

22 � dx

��c 21c2�2w1 ÿ w2�2 dx :

The nonnegativity of L0 becomes obvious from this repre-sentation, and for the eigenvector of the ground state, itfollows that

w1 � c1 ; w2 � c2 ; 2c1 � c2 ;

or

v0 � c1

2c2

� �:


This eigenvector, as in the NLS case, explicitly enters theexpression dN � 2

� �c1u1 � 2c2u2� dx � 2hv0jui, which isequal to zero by virtue of the conservation of N. Just as inthe NLS case, dN � 0 is the solvability condition for linearsystem (75).

The subsequent analysis resembles that carried out for theNLS equation. To begin, we consider the eigenvalue problemfor the operator L1,

L1jfi � Ejfi � Cjv0i ; �77�

and then expand jfi in a complete set of eigenfunctionsfjfnig of L1. As a result, the solution of Eqn (77) is writtenin the form

jfi � CXn

jfnihfnjv0iEn ÿ E

:

Analogously, the solvability condition leads to the dispersionrelation

f �E � �Xn

0 hv0jfnihfnjv0iEn ÿ E

� 0 : �78�

The prime at the summation symbol indicates, as previously,the absence of the term with E � 0, because hCxjv0i � 0 andL1jCxi � 0, where

hCxj � �c1x;c2x� :

Up to this point, everything replicates the NLS case. Thedifference becomes apparent in the analysis of the functionf �E �. Now the oscillation theorem is not valid. Accordingly,there could be several levels below the level E � 0. For thisreason, the dispersion relation can have negative roots E < 0independent of the sign of the derivative qNs=ql

2. As a result,only a sufficient instability criterion can be formulated; it hasthe same form as for NLS solitons:

qNs

ql2< 0 : �79�

But we cannot decide on the stability in the general case. Thestability criterion

qNs

ql2> 0

would hold only if, for energies less than E0 � 0, the operatorL1 has only one (ground-state) level, but it is not the generalcase. Thus, the Vakhitov±Kolokolov stability criterionapplied to vector models ensures only a sufficient conditionfor soliton instability.

Nevertheless, the combination of the Vakhitov±Koloko-lov criterion and the Lyapunov method may grant the fullanswer about stability. To conclude this section, we discussone example in which, based on the combined method, it ispossible to draw a more or less definitive conclusion aboutthe soliton stability by numerically integrating system (73),(74). The dependences of H and N (for one-dimensionalsoliton solutions) on l was numerically found in [56] for anonzero frequency detuning O 6� 0. Numerical integrationhas shown that for O < 0, both dependences are monotonic:N increases with l andH decreases. As a result, it was shown

that only a single branch of soliton solutions with anunambiguous dependence H�N� exists. For O > 0, thedependence N�l� contains two branches. The first occupiesthe domain 0 < l < lmin. The function N�l� increasesmonotonically as l approaches zero. At l � lmin, thefunction N�l� attains a minimum. In agreement withcriterion (79), all this branch of soliton solutions isunstable. For l > lmin, N increases monotonically, but thelinear stability criterion cannot be applied to these solu-tions. However, the dependence H�l� allows making acertain conclusion about stability. The function H�l� has amaximum at the point l � lmin, and hence H as a functionof N has a cusp singularity at this point, which separatestwo branches of soliton solutions. The upper branchcorresponds to larger values of H than the lower one. Ifwe assume that no other soliton solutions exist in this rangeof N (which is by no means easy to establish numerically),then the lower branch belongs to the stable family of solitonsolutions.

6. Wave collapses

Wave collapse is the other variant of evolution for nonlinearwave systems, in a certain sense an alternative to the solitonscenario. As we have seen in Sections 2±4, Lyapunov stablesolitons are encountered when, for a fixed number ofparticles N or other integrals of motion, the Hamiltonianis a functional bounded from below, its lower boundcorresponding to a Lyapunov-stable soliton. A questionnaturally arises as to what happens if the Hamiltonian is notbounded from below. How does the system behave in thiscase? One possible mode of behavior is collapse, i.e.,singularity formation in finite time. The type of theemerging singularity depends on the concrete physicalproblem. In this review, we limit ourselves mainly to theanalysis of the wave collapse described by NLS equation (1)for D5 2.

It is well known (see, e.g., Refs [17, 29, 39]) that as thespace dimensionD increases, the role of nonlinear effects alsoincreases; similarly to the phase transition theory, the role ofcooperative effects becomesmore prominent with the increas-ing number of neighbors, for example, in the Ising model. Forthe NLS equation, this is seen, in particular, from how Hvaries under scaling transformations c! aÿD=2c�r=a� thatpreserve the number of particles,

H�a� � 1

2

�I1a 2ÿ I2aD

�: �80�

In one dimension �D � 1�, H�a� has a minimum thatcorresponds to a (stable) soliton. In two dimensions,I1s � I2s; therefore, H�a� � 0 for the entire soliton family.For D � 3, H�a� has a maximum instead of a minimum (inreality, a saddle point), which points to the instability of thesoliton. Additionally, H�a� turns out to be an unboundedfunction as a! 0, which is one of the collapse criteria [39] (seealso Ref. [57]). In this case, the collapse can be regarded as anonlinear stage of soliton instability.

To illustrate the foregoing, we resort to the variationalapproach. As a test function for Eqn (1), we choose c in theform

c�r; t� � aÿ3=2cs

�r

a

�exp �il2t� imr 2� ;


where a � a�t� and m � m�t� are unknown functions of time.Substituting this ansatz into the action

S � i

2

��ctc

� ÿ c:c:� dtdrÿ�Hdt

and integrating over spatial variables, we obtain a reducedaction for the two functions a � a�t� and m � m�t�. It is easy toverify that the two Lagrange equations, for a�t� and m�t�,reduce to a single second-order equation for a, which has theform of Newton's equation

C�a � ÿ qHqa

; �81�

where C � � x 2jc0�x�j2 dx plays the role of the mass of aparticle and function (80) has the meaning of potentialenergy. The behavior of a�t� depends on the total energy

E � C_a 2

2�H�a� ;

on the initial position of the `particle,' and on the dimensionD. ForD � 1, the soliton realizes a minimum of the potentialenergy H�a�, which is the cause of its stability. For D � 3, ifthe particle is initially at a maximum point, the system,depending on its motion direction (toward the center a � 0or away from it), either collapses �c!1� or expands�c! 0� because of dispersion. For the collapsing mode,which corresponds to the fall of the particle on the center, inthe vicinity of the singularity, a�t� obeys the power law

a�t� � �t0 ÿ t�2=5 ; �82�

where the collapse time t0 is finite and equal to the particle falltime in the potential H�a�. As shown in Ref. [39], such anasymptotic form of a�t� in the vicinity of the singular pointcoincides, up to a constant, with the exact collapsingsemiclassical solution, which tends to a compact distributionas t! t0,

jcj ! l��1ÿ x 2

qfor x � r

a�t� 4 1 ;

with l � �t0 ÿ t�ÿ3=5.The consideration above already invites several conclu-

sions. First, the role of nonlinearity increases with the spacedimension. Stable solitons are encountered in systems of lowdimensions, whereas in higher dimensions, we should expectexplosive events instead of solitons, which evolve intosingularities. Second, one of the collapse criteria is theunboundedness of the Hamiltonian at small scales, in whichcase the collapse can be interpreted as a fall of a particle on thecenter in a self-consistent potential which is unbounded atsmall scales [39].

6.1 The role of radiationIn reality, however, the picture of singularity formation ismore complex than that sketched above. From the verybeginning, we are dealing with a spatially distributed system,i.e., one with an infinitely many degrees of freedom. It istherefore obvious that a reduction of the NLS equation withthe help of the variational approach to the system of ordinarydifferential equations as in Eqns (81) does not account for

wave radiation. This process, as explained above, is ofprincipal importance for soliton interactions. As shown inRef. [58], it contributes crucially to the relaxation of a pulse toa soliton in the one-dimensional NLS equation integrated bythe inverse scattering method [40]. In the last case, thevariational approach [59, 60], albeit rather popular pre-sently, provides an incorrect answer by predicting solitonoscillations. A rigorous analysis shows that the initial pulserelaxes to a soliton in an oscillatory manner owing to waveradiation.Moreover, the oscillation frequency is far from thatgiven by the variational approach.

In the case where H is unbounded from below, when weanticipate collapse, the radiation of small-amplitude wavesis one of the mechanisms governing the singularity forma-tion. To demonstrate this, we can resort to the samearguments about the role of radiation in soliton formationas in Section 3.

Let the Hamiltonian be negative in a certain domain G,HG < 0. Then the inequality [cf. Eqn (49)]

maxx2Gjcj2 5 jHGj

NG

holds within the domain G. We assume that outside thedomain G, the magnitude of jcj is small. We consider therole of wave radiation from this domain. The radiation,obviously, carries away positive energy because nonlineareffects for the radiated waves are small at large distances. As aconsequence, because of radiation,HG becomes progressivelylarger in absolute value, whileNG decreases. This is the reasonwhy max jcj2 increases unlimitedly, for the Hamiltonian isunbounded from below. We can therefore state that theradiation promotes collapse. In a nonlinear wave system, theradiation therefore plays a role of friction [17, 27, 29].Moreover, the radiation accelerates the compression of thecollapsing domain, which, in particular, results in compres-sion obeying the self-similarity law

r � �t0 ÿ t�1=2 ; �83�

which differs from the semiclassical compression in (82). Thearguments above are, however, insufficient to draw aconclusion about the finiteness of the collapse time. Theanswer is provided by the virial theorem.

6.2 Virial theoremThe exact criterion of the singularity formation in the NLSframework can be inferred from the virial theorem. Inclassical mechanics, this theorem can easily be derived bycomputing the second time derivative of the inertia momentand then averaging the result to find the relation between themean values of the kinetic and potential energies of particles,whenever the interaction between the particles obeys a powerlaw.

In 1971, Vlasov, Petrishchev, and Talanov [18] found thatthe virial theorem can be applied to the two-dimensional NLSequation. The analog of the moment of inertia for the two-dimensional NLS equation is the mean square size of thedistribution hr 2i � Nÿ1

�r 2jcj2 dr. By direct calculations, it

can be verified that

d2

dt 2

�r 2jcj2 dr � 2H : �84�


BecauseH is a conserved quantity, Eqn (84) can be integratedtwice,�

r 2jcj2 dr � Ht 2 � C1t� C2 ; �85�

whereC1 andC2 are the additional integrals ofmotion, whoseexistence is related to two Noether-type symmetries: the lenstransformation (established by Talanov [22] in 1970) andscaling transformations [61, 62].

Now it can be easily seen that for any distribution with anegative Hamiltonian

H < 0 ; �86�

hr 2i vanishes in a finite time independent ofC1 andC2, which,becauseN is conserved, implies the formation of a singularityin the field c [18]. Condition (86) represents the celebratedVlasov±Petrishchev±Talanov criterion. This criterion, for-mulated in 1971, is a fundamental result in the theory ofwave collapses. It was the first rigorous result for nonlinearwave systems with dispersion, and showed that the formationof a singularity in a finite time is possible despite the linearwave dispersion, which hinders (for example, in linear optics)the appearance of point singularities, foci.

We emphasize that criterion (86) is only a sufficient, butnot a necessary condition for the onset of collapse. Forexample, if in a certain part of the system, well isolated fromthe rest, theHamiltonianH is negative, collapse would unfoldthere independent of whether the total Hamiltonian of the fullsystem is positive or negative.

6.3 Strong collapseWe note that for D � 2, a soliton is associated with theequality H � 0 with the particle number N � Ns. Moreover,for N < Ns, the inequality

H51

2I1

�1ÿ N

Ns

�> 0

holds and, as a consequence, collapse is forbidden [47]. In thiscase, the amplitude of waves tends to zero as t!1 becauseof dispersion or diffraction. It can therefore be argued that thesoliton for the two-dimensional NLS equation represents aseparatrix between the manifolds of collapsing and noncol-lapsing distributions.

It follows from relation (85) that for states withH < 0, thecharacteristic size a of the collapsing domain behaves as

a � �t0 ÿ t�1=2 ;

in full agreement with self-similarity law (83).The rigorous analysis in [63], however, shows that

a 2�t� � t0 ÿ t

log j log �t0 ÿ t�j ;

while the spatial profile c asymptotically approaches thesoliton one (see also Refs [64±68]). The collapse in suchcases is accompanied by exponentially weak radiation. Theenergy confined to the singularity coincides with N up to aconstant factor and is finite and equal to the energy of thetwo-dimensional soliton. This is why such a collapse is calledthe strong collapse [39].

We note that two-dimensional NLS equation (1) isregarded as a critical NLS equation. For the critical NLSequation, as we have seen, scaling transformations representan extra symmetry. As a consequence, in particular, thedispersion �� I1� and nonlinear �� I2� terms in H behavesimilarly under such transformations. For the NLS equationwith the power-law nonlinearity in (46), the critical behavioris associated with s � 1� 2=D. For smaller s, the Hamilto-nian is bounded from below for a fixed number of particles,and the soliton solution, which corresponds to the minimumof H, turns out to be stable.

6.4 Collapse in a boundary layerThe critical behavior does not occur solely for the NLSequation. For example, the two-dimensional model

ut � qqx

k̂uÿ 6uux � qqx

dHdu

�87�


H ��

1

2uk̂uÿ u 3

�dr � 1

2I1 ÿ I2

also belongs to the class of critical models. Here, k̂ is theintegral operator whose Fourier transform is the modulusjkj � �k 2

x � k 2y �1=2. Equation (87) describes low-frequency

oscillations in a boundary layer for large Reynolds numbers,Re4 l. The mean profile of velocity (parallel to the x axis)v � U�z� �04 z <1� is assumed to be a monotonicallyincreasing function that tends to a constant as z!1. Thedimensionless amplitude u is related to velocity fluctuationsalong the flow as

dvx � ÿ6huU 0�z� ; �88�

where h � U�0�=U 0�0� is the boundary layer thickness.Equation (81), first derived by Shrira [69], is a two-dimensional generalization of the Benjamin±Ono equationdescribing long waves in stratified fluids. In one dimension,Eqn (87) was derived in Refs [70, 71] with account for (small)viscosity.

Scaling transformations preserving the x-component ofthe momentum Px � 1=2

�u 2 dr, similarly to those in (52),

u�r� ! 1

aD=2u

�r

a

�;

where D � 1; 2, lead to the following dependence of theHamiltonian H on the scale parameter a:

H�a� � 1

2

I1aÿ I2

aD=2:

It then follows that this model becomes critical forD � 2. ForD � 1, the Hamiltonian is bounded from below, and itsminimum corresponds to the Benjamin±Ono soliton. Thissolution, having the form u � us�xÿ Vt; y�, can be foundexactly:

us � 2V

3��xÿ Vt�2V 2 � 1

� ; V > 0 : �89�

However, this soliton turns out to be unstable with respect totwo-dimensional perturbations [72, 73]. In the long-wave


limit, the increment of this instability is [72]

G � kyV��2p :

Figure 2 shows the evolution of u�x; y; t� illustrating thisinstability. The initial condition is the perturbed periodicwave

uini�x; y� � up�x��1� 0:1 cos

2pyLy

�; �90�

where up�x� is the exact one-dimensional solution of theBenjamin±Ono equation in the form of a periodic wave,

up�x� � k

3

sinh �k=V �cosh �k=V � ÿ cos �kx� ÿ 1 : �91�

As k! 0, solution (91) transforms into soliton solution (89).It can be shown that the solution in the form of a periodicwave can be rewritten as a periodic lattice of solitons with theperiod 2p=k; wave (91) accommodates one soliton per period.

For the chosen variant of initial conditions (90),k � 0:0625, V � 0:4, and the size of the domain Lx coincideswith the wave period in (90).

The instability described above is analogous to theKadomtsev±Petviashvili (KP) instability [74, 75] for acousticsolitons in media with positive dispersion (see also Refs [76,

77]). (See Section 6.5 on the cause of this instability.) In twodimensions, this model has a solution in the form of acylindrically symmetric soliton without zeros. Such a solu-tion, found numerically in Ref. [78], plays the same role as theTownesmode for the two-dimensional NLS equation [23]: forPx < Px; cr, where Px; cr is the value of the x-component ofmomentum in the ground-state (without zeros) solitonsolution, the Hamiltonian turns out to be positive with aminimum that tends to zero as u! 0. In this case, collapse isimpossible, just as in the analogous case of the two-dimensional NLS equation. The Hamiltonian is unboundedfor states with negative values of H. It was numericallyconfirmed in [72, 73] that the collapse in system (88) occursfor H < 0, and the collapsing solution is cylindricallysymmetric and approaches the soliton distribution as t! t0.The initial condition was chosen as

u�r� � 2=3jVj1� �Vxx�2 � �Vyy�2

:

The parameters Vx and Vy were varied so as to satisfy theconditions Px > Px; cr and H < 0. On approaching thesingularity, the peak anisotropy disappeared and the distribu-tion became symmetric. A typical example of collapse isshown in Fig. 3.

To conclude this section, we point to a series of interestingexperiments [79, 80], performed over several years, on theexcitation of coherent structures in a transient boundary layerover a plate. The structures were excited with the help of avibrating mechanical system placed at the front edge of theplate. According to experimental data, one-dimensionalsolitons developed at the initial stage, and later, far down-stream from the leading edge, the solitons lost stability, givingway to so-called thornsÐ three-dimensional coherent struc-tures. Further, self-focusing of these structures was observed.At even later stages of thorn development, vortices wereformed and separated. Theoretical arguments and numericalmodeling performed in Refs [72, 73] qualitatively explain theentire sequence of experimental observations, up to theformation of vortices, where Eqn (87) loses applicability. Itis worth noting that a more elaborate analysis [70, 71], basedon one-dimensional model (87), i.e., performed in the frame-work of the one-dimensional Benjamin±Ono equation,showed a rather good agreement between the theory andexperiment.

t � 70

512

512

0

t � 20

512

512

0 x, rel. unitsx, rel. units

t � 60

512

512

0

512

0

t � 15


t � 40

512

512

0

t � 0

512


y,rel.u

nits

y,rel.u

nits

y,rel.u

nits

y,rel.u

nits

y,rel.u

nits

y,rel.u

nits

Figure 2. The evolution of u�x; y; t� (isolines) for initial condition (90),

demonstrating the solution instability and cluster formation.

ÿ6.40

ÿ6.40

ÿ6.40

ÿ2.80

ÿ2.80

ÿ10.0

Figure 3. Isolines u�x; y� for t � 45 in the collapse mode. The contours

correspond to values of u�x; y� in the range from ÿ10:0 to 0.8 with the

interval 0.90000.


A question that naturally arises is why instability and self-focusing in such prominent phenomena as one-dimensionalsolitons have not been observed in other experiments. In ouropinion, the answer to this question is related to the characterof collapse in boundary layersÐ the collapse is strong, i.e., itsappearance requires a finite pulse energy. If the pulseamplitude is insufficient, the phenomenon does not occur.

6.5 Weak collapseIn three dimensions �D � 3�, virial relation (84) for the NLSequation takes the form

d2

dt 2

�r 2jcj2 dr � 2Hÿ 1

2I2 : �92�

Accordingly, because I2 is positive, equality (92) can bereplaced by the inequality

Nhr 2i < Ht 2 � C1t� C2 ; �93�

where C1 and C2 are integration constants determined fromthe initial conditions. Relation (93) provides the samecollapse criterion as for D � 2: H < 0 [17].

However, the criterion H < 0 for the NLS equation inthree dimensions is rather rough. According to Refs [81, 82],this criterion can be improved. A more precise criterion isgiven by two conditions:

H < Hs ; I1 > I1s : �94�

Criterion (94) indicates once again that the soliton corre-sponds to a saddle point.When the system passes through thissaddle point, collapse becomes possible.

A more precise inequality, which follows from virialrelation (92) for D � 3, has the form

d2

dt 2

�r 2jcj2 dr4 3�HÿHs� : �95�

Integration of (95) gives

Nhr 2i4 3

2�HÿHs�t 2 � C1t� C2 ;

whereHs is the value ofH in the ground-state (without zeros)soliton solution.

We next consider a self-similar substitution for Eqn (1),

c � 1

�t0 ÿ t�1=2�iaw�x� : �96�

Here, the self-similarity variable x � r�t0 ÿ t�ÿ1=2 corre-sponds to Eqn (83) and the function w�x� is assumed to bespherically symmetric. Inserting c into the NLS equationresults in the equation for the function w [39]

i

��1

2� ia

�w� 1

2xwx

�� 1

2wxx �

1

xwx � jwj2w � 0 ; �97�

where a plays the role of a spectral parameter. We areinterested only in regular solutions of Eqn (97) that decay atinfinity. Obviously, as x!1, w satisfies the linear equation�

1

2� ia

�w� 1

2xwx � 0 : �98�

It defines the asymptotic form of w:

w! C

x 1�2ia ; �99�

where C is some constant, which can be taken real and,moreover, positive without loss of generality. The require-ment that the solution be regular eliminates the ambiguity inthe choice of a and C. In fact, we are dealing with a nonlinearspectral problem for Eqn (97), whose numerical solution gives

a � 0:545 ; C � 1:01 : �100�

We now discuss the characteristics of the self-similarsolution. First, for any fixed point of physical space with acoordinate r, the corresponding self-similar coordinate xtends to infinity as t! t0. The self-similar solution in thiscase passes to its asymptotic form (99), acquiring a singularity

c! C

r 1�2ia�101�

in the physical variable r, which is independent of time t.Second, the self-similar solution cannot exist simulta-

neously in the entire space: it can be realized only in somedomain with coordinates r < r0, where the size r0 must beconstrained by the quantity N=�4pC 2�.

Singularity (101), which `sprouts' in the center of thisdomain as r! 0, is integrable. At a first glance, self-similaransatz (96) results in the nonconservation of the integral

N ��jcj2 dr � �t0 ÿ t�1=2

�jwxj2 dx �102�

because of the appearance of the factor �t0 ÿ t�1=2. On theother hand, the conservation of N implies that integral (102)must be infinite. This is the case when the self-similar solutionis considered in the entire space. But in any finite domainr < r0, the integral N remains finite. Indeed, upon substitut-ing (96) in Eqn (102), for r < r0, we obtain

N � �t0 ÿ t�1=24p� x�

0

x 2��w�x��2 dx ; x� � r0�t0 ÿ t�ÿ1=2 :

�103�With the asymptotic form in (99), the integral in Eqn (103)increases linearly as the upper limit increases as t! t0. Wesuppose that the size r0 is sufficiently large. Then the value ofthe integralN in the domain r < r0 must approach its value atthe instant of collapse t � t0. In other words, the relation�1

0

�jwj2 ÿ C 2

x 2

�x 2 dx � 0 �104�

must hold. It was verified numerically that relation (104)holds with high accuracy for w�x� and C found by integratingEqn (97).

The solution constructed here corresponds to a weakcollapse. In this state, speaking formally, zero energy istrapped in the singularity at r � 0 [39]. In fact, this impliesthat ifc0 is a characteristic amplitude at which the absorptionof energy occurs [Eqn (1) is then not valid], then the amountof energy lost to dissipation in a single act is of the order of

DN � c 20 r

30 �

1

c0

;


where r0 � 1=c0 is the characteristic scale of the dissipationdomain.

The self-similar solution related to the weak collapsemode describes the most rapid collapse in which radiationplays the primary role; in contrast to the semiclassical collapsemode (82), it essentially accelerates the process of singularityformation; as a result, only a small fraction of the energyreaches the singularity.

We note that the weak collapse mode was first simulatedin numerical experiments in [83] and then in [84].

6.6 Black-hole regimeAs we have seen in Section 6.5, the time-independentsingularity in (101) forms in the regime of weak collapse,which corresponds to the attracting potential

U � ÿC 2

r 2; �105�

with a constant C 2 > 1. As is known from quantummechanics (see, e.g., Ref. [85]), a particle can fall on thecenter in this case. Moreover, as the center r � 0 isapproached, such a fall in quantum mechanics becomescloser and closer to the semiclassical one.

For all waves surrounding the singularity, potential (105)serves as an attractor, i.e., the singularity plays the role of afunnel for these waves. Because the energy dissipating in thesingularity for weak collapse is infinitely small in physicalterms, the process of pulling particles into the singularityfrom its peripheryÐ the postcollapseÐcan be consideredquasistationary. The postcollapse is characterized by a finiteflux of the number of particles into the singularity. As wasfirst shown in Refs [86, 87] (see also Refs [88±90]), the densityjcj2 behaves in the vicinity of the singularity in this mode as

jcj2 � 1

2r 2j log rj :

The incessant flow into the funnel, mediated by wavestrapped in the singularity from the periphery, ensures theexistence of long-lived `burning' pointsÐ locations where allthe energy supplied from outside is `burnt'. Such a regime isalso called the black-hole regime.

7. The role of dispersion in collapse

Dispersion affects collapse quite significantly. In stationaryself-focusing light, wave dispersion is unimportant: theappearance of a singularityÐ the focusÐ stems from theconcurring nonlinearity from the Kerr effect and diffrac-tion. In this case, the two-dimensional NLS equation isapplicable, with the role of time played by the coordinate zalong the beam propagation. In three dimensions, however,everything depends on the sign of dispersion, positive ornegative; in optics, in particular, it depends on whether thedispersion is anomalous or normal. Three-dimensional NLSequation (1) is applicable for anomalous dispersion. In thecase of normal dispersion, the Laplace operator in Eqn (1)must be replaced with the hyperbolic operator D?ÿ q2=qz 2,where D? � qxx � qyy. This is why the NLS equation of theform

iqcqt� D?cÿ czz � jcj2c � 0 �106�

is often called the hyperbolic nonlinear SchroÈ dinger equation.The different sign of the second-order operator stems fromthe sign of the frequency derivative of the group velocity,qVgr=qo. The derivative qVgr=qo is negative for normaldispersion.

The hyperbolic operator in Eqn (106) essentially modifiesthe character of nonlinear interaction. For the anomalousdispersion, all `directions' are equivalent, but for the normaldispersion, we have repulsion instead of attraction in Eqn (1)along the z direction, whereas attraction is preserved inthe lateral direction. Correspondingly, quasiparticles areattracted to each other in the lateral direction, which leads tobeam narrowing in the plane perpendicular to the z axis. Putdifferently, in NLS equation (106), the mass of particles in thelongitudinal direction is negative, and therefore the non-linearity promotes an increase in the beam longitudinalscale. In this respect, the main question is whether the lateralbeam compression is capable of producing a singularity inspite of the longitudinal beam expansion.

We demonstrate that in the framework of hyperbolicSchroÈ dinger equation (106) in three dimensions, the collapseof a wave packet as a whole is impossible at the stage mostfavorable for collapse, when a pulse is compressed in all threedirections. The idea of the proof is to use integral estimates forthe virial equations.

Similarly to Eqn (1), Eqn (106) is Hamiltonian with

H ��jH?cj2 drÿ

�jczj2 drÿ

�1

2jcj4 dr � I? ÿ Iz ÿ I2 :

�107�The different signs of the first and second integrals corre-spond to the difference in the behavior in the lateral andlongitudinal directions.

We consider the behavior of mean square sizes hz 2i andhr 2?i, along and transverse to the z axis. Calculations similarto those in Eqn (84) give

Nd2

dt 2hr 2?i � 4

�2

�jH?cj2 drÿ

�jcj4 dr

�; �108�

Nd2

dt 2hz 2i � 8

�jczj2 dr� 2

�jcj4 dr : �109�

From Eqn (108) for hr 2?i and I?, just as from Eqn (109) forhz 2i and Iz, we can obtain the uncertainty relations

I?hr 2?i5N ; Izhz 2i5 N

4: �110�

Using relations (110) and the definition ofH in (107), we canestimate the right-hand sides of Eqns (108) and (109):

Nd2

dt 2hr 2?i � 8H� 8Iz 5 ÿ 4H� 2

N

hz 2i ; �111�

Nd2

dt 2hz 2i � ÿ4H� 4Iz � 4I? > ÿ4H� 4

N

hr 2?i: �112�

We now consider the contraction mode in all directions,the most preferential from the standpoint of collapse, when

d

dthr 2?i < 0 ;

d

dthz 2i < 0 ;

and demonstrate that the collapse, understood as thecontraction to zero of the mean square transverse andlongitudinal sizes �hr 2?i ! 0, hz 2i ! 0�, is impossible in thiscase.


First, we prove that the mean square longitudinal scale z 2

of the wave packet cannot vanish if dhz 2i=dt < 0. FromEqn (109), also using Eqn (110), we derive a closed-forminequality for hz 2i:

Nd2hz 2idt 2

5 8Iz 5 2N

hz 2i : �113�

Formula (113) is a second-order differential inequality. Itsfirst integration, analogous to finding the energy integral inmechanics, with dhz 2i=dt < 0, gives the relation

E�t� � 1

2

�dhz 2idt

�2

ÿ 2 log hz 2i4 E�0� ; �114�

where E�0� is the initial value of E�t�. As hz 2i ! 0, the left-hand side of (114) tends to infinity because of the logarithmicterm, which would violate the inequality. Hence, shrinkinghz 2i ! 0 is impossible.

We now show that the collapse in the transverse direction(when hr 2?i ! 0) is also impossible. For this, we multiplyinequality (111) by dhz 2i=dt < 0, and inequality (112) bydhr 2?i=dt < 0, add the results, and integrate the resultingequation over time from zero to t. This yields

E�t� � Ndhr 2?idt

dhz 2idtÿ 8Hhz 2i � 4Hhr 2?i

ÿ 2N log hz 2i ÿ 4N log hr 2?i4E�0� ; �115�

where E�0� is the value of E�t� at the initial instant. Frominequality (115), it immediately follows that collapse isimpossible at the stage of uniform contraction because thefirst term in the right-hand side of (115) is positive bydefinition, the terms proportional to H are finite, and thelogarithmic term becomes infinitely large as hr 2?i ! 0. Thisdoes not agree with the fact that the function E�t� is boundedfrom above by the initial value E�0�. Hence, the collapse of athree-dimensional wave package as a whole is impossible atthe most `dangerous' stage of shrinking in all directions [91](see also Ref. [92]).

Does this imply that the collapse in such a system iscompletely impossible? Strictly speaking, no, because, first,the Vlasov±Petrishchev±Talanov criterion is only a sufficientone and, second, the collapse, if it is possible, should besought among states that correspond to the longitudinaldispersion of the gas of quasiparticles, which leads to anincrease in the longitudinal scale. On the other hand, long-itudinal stretching results in a decrease in the particle linedensity. For the two-dimensional SchroÈ dinger equation, aswe have seen, collapse is prohibited for N < Ncr. It is worthnoting that in spite of keen interest in this problem (see, e.g.,Refs [93±95]), it is still awaiting its full solution.

8. Collapse in Kadomtsev±Petviashvili equations

Finally, we discuss the question of collapse in Kadomtsev±Petviashvili (KP) systems. We begin with the classic KPequation with positive dispersion

qqx�ut � uxxx � 6uux� � 3D?u ; �116�

which describes the propagation of a beam of weaklynonlinear acoustic waves in a medium with weak disper-

sion. The second term in the left-hand side of Eqn (116) isresponsible for dispersion effects, the third term describesthe effects of sound wave steepening, and the term in theright-hand side describes the diffraction �D? � q2y � q2z�.Equation (116) is written in the coordinate frame movingat the speed of sound cs along the x axis. All the termsresponsible for nonlinearity, dispersion, and diffraction aresmall compared to the `fast' propagation at the speed ofsound, and therefore the KP equation has an additionalintegral, the adiabatic invariant P � �1=2� � u 2 dr, whichcoincides with the x-projection of the momentum. It isalso important that P is the total energy up to a constantfactor.

It is pertinent to say a few words about sound waves withpositive dispersion. For example, they include fast magneto-acoustic waves in plasmas with a small parameter b (the ratioof the thermal plasma pressure to the magnetic fieldpressure), b5 1, for propagation directions that are notclose to the longitudinal and lateral ones (with respect to themagnetic field). One more example of positive dispersion isoffered, under certain conditions, by sound waves in liquidhelium [96]. Finally, we mention surface waves in shallowwater when the (positive) dispersion due to capillarityexceeds that due to a finite depth (see, e.g., Ref. [97]).

Equation (116), which is Hamiltonian, can be written inthe form

ut � qqx

dHdu

�117�


H � 1

2

�u 2x dr�

3

2

��H?w�2 drÿ

�u 3 dr � I1

2� 3I2

2ÿ I3 :

�118�Here, w plays the role of the hydrodynamic potential, u � wx.

Solutions in the form of a multi-dimensional solitonpropagating along the x axis, u � us�xÿ Vt; r?�, are foundby integrating the equation

q2

qx 2�ÿVu� uxx � 3u 2� � 3D?u : �119�

The linear operator in Eqn (119),

q2

qx 2

�ÿV� q2

qx 2

�ÿ 3D? ;

is positive definite if the soliton velocity is positive, V > 0. Aswas first noted by Petviashvili [98], this requirement is in factthe existence condition for soliton solutions of the KPequation. Physically, this condition implies the absence of aCherenkov resonance between the moving soliton and small-amplitude waves [50, 99].

In one dimension, the solution of Eqn (119) is the KdVsoliton

us � 2k 2

cosh2 k�xÿ 4k 2t� ;

and in two dimensions, it is the lump solution [100]

us � 4V 01� 4V 0 2y 2 ÿ V 0�xÿ Vt�21� 4V 0 2y 2 � V 0�xÿ Vt�2 :


In three dimensions, solutions of Eqn (119) can only be foundnumerically.

The soliton solution of Eqn (119) corresponds to astationary point of Hamiltonian (118) with the momentumfixed:

d�H� VP� � 0 :

Using the explicit expression for the Hamiltonian, Eqn (118),together with integral estimates [77] (see also Refs [3, 51]), itis possible to show that in one and two dimensions, theHamiltonian for the KP equation is a functional boundedfrom below if the momentum P is fixed. These solitonsolutions, however, turn out to be unstable under three-dimensional perturbations, which was discovered for thefirst time for one-dimensional solitons in the long-wavelimit by Kadomtsev and Petviashvili [74]. Later, with thehelp of the inverse scattering method, Zakharov found theexact expression for the growth rate of this instability [76].The cause of the instability is that the speed of a soliton inphysical variables decreases as its amplitude increases. As aconsequence, if such a soliton is modulated in the lateraldirection, the regions with larger amplitude take over thosewith a smaller amplitude. This leads to a focusing-typeinstability [75].

For D � 3, Hamiltonian (118) is unbounded from below,which, notably, follows by applying the scaling transforma-tions [57, 101]

u! aÿ1=2bÿ1u�x

a;r?b

�;

which preserve P, to Hamiltonian (118),

H�a; b� � I12a 2� 3I2a 2

2b 2ÿ I3

a 1=2b; �120�

which, as a function of a and b, has a fixed point at a � b � 1,corresponding to a three-dimensional soliton. It is clear thatat the fixed point, the conditions

qHqa� qH

qb� 0 ;

are satisfied; they are equivalent to the following relationsbetween the integrals:

I1 � 3I32; I2 � I3

3:

Equipped with these relations, it is easy to see that this fixedpoint is a saddle. This fact indicates a possible instability ofthe three-dimensional soliton under small perturbations. Asregards the instability under finite perturbations, it followsfrom the unboundedness of the Hamiltonian as a; b! 0.Indeed, if we consider parabolas b / a 2, the first two terms inH behave as aÿ2, while the cubic term is proportional toaÿ5=2. This indicates that H is unbounded (from below) atsmall scales owing to the nonlinear term. The unboundednessof H, as we have seen in other examples, is a criterion for thecollapse to occur. Under scaling transformation along theparabolas b / a 2, the exponents in the dispersion term (equalto ÿ2) and in the cubic term �ÿ5=2� do not coincide, andhence the possible collapse is not critical and should be weak.

The self-similar substitution

u � tÿ2=3u�

x

t 1=3;r?t 2=3

�

corresponds to this collapse regime, with t � t0 ÿ t and t0 thecollapse time.

Numerical integration of three-dimensional KP equation(116) for cylindrically symmetric distributions [57, 101] hasshown that a singularity forms for initial conditions with anegative value of the Hamiltonian.

Figure 4 shows the profile of u on the axis r? � 0 at fourtime instants. The spatial distribution of u�x; r?� for thesesame instants is given in Fig. 5. The evolving distribution hasa horseshoe shape, which corresponds to the linear stage ofthe KP instability. Numerical experiments have corroboratedthat this collapse is a weak one. Zero boundary conditionswere chosen for u�x; r?� in numerical experiments; because ofthe KP equation nonlocality, they did not ensure that theHamiltonian is preserved, since radiation from the integra-tion domain is possible. Figure 6 plots theHamiltonianH, themomentum Px computed over the integration domain, andthe maximum amplitude as functions of time. As can be seen,

u�x;0�

0

t � 0

0x

10ÿ10

10

20

x

u�x;0�

t � 0.1

0 10ÿ10

10

20

0

u�x;0�

t � 0.2

0x

10ÿ10

10

20

0

u�x;0�

t � 0.325

0x10ÿ10

15

0

30

Figure 4.The dependence of u on the axis r? � 0 for four moments of time.

r?

t � 0

x10ÿ10 0

t � 0.1

r?

x10ÿ10 0

r?t � 0.32

ÿ10x

0 10

r?t � 0.2

x10ÿ10 0

Figure 5. Isolines of u�x; r?� for the same time instants as in Fig. 4.


a decrease of Px and jH j with time is accompanied by asimultaneous increase in maxr u, which points precisely to theweak nature of the collapse.

The question of the sufficient criterion of collapse and,accordingly, the finiteness of the singularity formation timefor three-dimensional KP equation (116) is presently open,although the possibility of a collapse is indicated by the factthat the Hamiltonian H is unbounded from below, if themomentum P � �1=2� � u 2 dr is fixed, and also by the aboveresults of numerical simulations of the collapse.

8.1 Virial inequalitiesWe demonstrate how the Vlasov±Petrishchev±Talanov cri-terion can be used in studies of collapses for the generalizedthree-dimensional KP equation

qqx

ÿut � uxxx � n�nÿ 1�unÿ2ux

� � 3D?u : �121�

The case n � 3 in Eqn (121) corresponds to the classic KPequation (116). The KP equation with n � 4 emerges in thedescription of acoustic waves in antiferromagnets for certainpropagation angles [102].

We begin from Eqn (116) assuming n � 4. We considerthe quantity

hr 2?i �1

2P

�r 2?u

2 dr ;

which, by virtue of the conservation of P � � u 2 dr, can beinterpreted as the mean beam cross section. As was firstshown in Ref. [102], an analog of virial theorem (92) can beformulated for hr 2?i as

d2

dt 2

�r 2?u

2 dr � 48Hÿ 8

�u 2x dr ; �122�

whereH is a conserved quantity, the Hamiltonian, written forn � 4 as

H ��

u 2x

2� 3

H?w 2

2ÿ u 4

�dr :

This immediately leads to the estimate

d2

dt 2

�r 2?u

2 dr < 48H : �123�

Hence, for H < 0, the beam collapses [102], formally shrink-ing to hr 2?i � 0. It can be shown that the collapse criterionH < 0 pertains to all integer values n > 4, but, unfortunately,the case n � 3, which corresponds to the classic KP equationwith quadratic nonlinearity, escapes this proof.

We dwell on one more example involving the generalizedKP equation. This is the model proposed in Ref. [103] for thedescription of self-focusing of ultrashort pulses, which doesnot rely on averaging over the frequency of a quasimonochro-matic wave. The pulse spectrum is assumed to be broad and tolie in the nonresonance domain. However, the nearestresonance (the transition frequency o0) and the plasmaresonance (at the plasma frequency op) are incorporated inthis model. An interesting feature is that the dispersiondependence contains intervals of both anomalous andnormal dispersion. Other limitations involve the Kerr non-linearity and small-angle approximation, which is character-istic of the KP equation,

2n0c

q2Eqz qt

� 4pdw 3

�co0�2q2E 3

qt 2ÿ 4pw

�co0�2q4Eqt 4� o2

p

c 2E � D?E : �124�

Equation (124) is written for the electric field amplitude E inthe case of a linearly polarized wave. Here, e0 � n 2

0 � 1� 4pwis the static dielectric permittivity, c is the speed of light,t � tÿ zn0=c is the time in the comoving frame of reference,and 4pdw 3oÿ20 is the Kerr constant, which is assumedpositive. Compared with Eqn (5) in Ref. [103], Eqn (124)neglects linear wave damping. Using dimensionless variables,Eqn (124) can be rewritten in the form

qqt

�quqz� 3u 3 qu

qtÿ b

q3uqt 3

�� au � D?u ; �125�

with positive constants a and b. Equation (125), which is thegeneralizedKP equation, is Hamiltonian, like the original KPequation (116),

quqz� q

qtdHdu

;


H � 1

2

��ÿbu 2

t � aw 2 � �H?w�2 ÿ 1

2u 4

�dt dr? ;

where u � wt, i.e., w plays the role of the hydrodynamicpotential. The expression for H indicates that its quadraticpart is not necessarily positive definite in the general case andbecomes so only in the low-frequency domain, where the firstterm bu 2

t can be neglected as being small compared with thesecond one. It is also obvious that the quadratic form ispositive definite for b � 0. Just in these cases a sufficientcriterion for the occurrence of collapse can be proposed,which is analogous to (123):

d2

dz 2

�r 2?u

2 dt dr? � 4Hÿ 8

�aw 2 dt dr?4 4H :

Integration of the last inequality results in a sufficientcondition for the occurrence of collapse: H < 0 [103].Numerical integration of Eqn (125) demonstrated thetendency to beam self-focusing in the transverse direction.A rather interesting effectÐ the buildup of sharp gradientson the beam axis prior to self-focusingÐwas also discov-ered in [103].

0.1ÿ40

ÿ4:5

0 0.2 0.3 t4

34

umax

umax

Px

HH

2.8

8.8

Px

Figure 6. The dependences of the HamiltonianH, the momentum Px, and

the maximum amplitude umax on time.


9. Conclusions

We have presented two methods for exploring the stability ofsolitons, one linked to the Vakhitov±Kolokolov criterion andits generalizations to vector systems of the NLS type, and theother based on the analysis of stability in the Lyapunov sense.The combination of these two methods furnishes an effectiveapproach to exploring soliton stability.

Another key element of the present review is theapplication of embedding theorems, which play an essentialrole in proving the soliton stability. In particular, the use ofsuch an approach helped establish soliton stability for thethree-wave system describing the coupling of electromagneticwaves in w 2-media and for three-dimensional solitons ofmagnetized ion-acoustic waves described in terms of theanisotropic KdV equation [14].

We note that by the w 2-media, we first and foremost meancrystals without a center of symmetry; only in that case doesthe three-wave matrix element differ from zero. Electromag-netic waves propagating in such crystals have anisotropicdispersion laws, which, notably, implies that the tensors oab

cannot be simultaneously reduced to a diagonal form forevery wave package in general. However, the methodpresented in this review requires no diagonalization of thedispersion tensors oab. The only essential condition is thatthey have the same sign definiteness. Only in this case dosolitons exist. The sign definiteness of the dispersion tensorsallows introducing the appropriate Sobolev spaces andobtaining necessary integral estimates for the Hamiltonians.It is also important that solitons realizing the minimum of theHamiltonian are stable with respect to not only small but alsofinite perturbations. In this sense, the Lyapunov stabilitycriterion is equivalent to the energy principle.

When the Hamiltonian is an unbounded functional, wavecollapse leading to a singularity stands out as the mostprobable scenario for nonlinear system behavior. Collapsein this case is similar to the process of particle fall in a self-consistent potential (see, e.g., Ref. [39]).

As the space dimension increases, the role of nonlineareffects also increases. Stable solitons are therefore observed atlow dimensions, whereas collapses are typical for higher-dimensional systems. In collapsing systems, solitons repre-sent a separatrix separating collapsing and noncollapsingdistributions. Such (unstable) solitons define the thresholdof wave collapse.

AcknowledgmentsThis work was supported by a grant from the government ofRussia for the state support of scientific research, conductedunder the supervision of leading scientists in Russianeducational institutions in the system of high professionaleducation (Contract 11.G34.31.0035 from 25November 2010between theMinistry for Education and Science, NovosibirskState University, and the leading scientist). E.A.K. thanksLPCE CNRS (OrleÂ ans, France), where the work was carriedout under the support of grant of the RAS and CNRS project``Nonlinear Processes in Anisotropic Plasma Turbulence.''The work was also supported by a grant from the RFBR, theProgram of the RAS Presidium ``Fundamental Problems ofNonlinear Dynamics in Mathematical and PhysicalSciences,'' and a grant from leading scientific schools ofRussia, NSh 6170.2012.2.

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Solitons and collapses: two evolution scenarios of nonlinear...

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