ELSEVIER Physica D 119 (1998) 115-124

PHYSICA ®

Dynamics of breathers in discrete nonlinear Schr6dinger models Magnus Johansson a,b,., Serge Aubry b, Yuri B. Gaididei c,

Peter L. Christiansen a, K.f3. Rasmussen a a Department of Mathematical Modelling, The Technical University of Denmark, DK-2800 Lyngby, Denmark

b Laboratoire l_don Brillouin (CEA-CNRS), CE Saclay, 91191 Gif-Sur- Yvette Cedex, France c Institute for Theoretical Physics, Metrologicheskaya Street 14B, 252 143 Kiev 143, Ukraine

Received 5 June 1997; received in revised form 2 August 1997

Abstract

We review some recent results conceming the existence and stability of spatially localized and temporally quasiperiodic (non-stationary) excitations in discrete nonlinear Schr6dinger (DNLS) models. In two dimensions, we show the existence of linearly stable, stationary and non-stationary localized vortex-like solutions. We also show that stationary on-site localized excitations can have internal 'breathing' modes which are spatially localized and symmetric. The excitation of these modes leads to slowly decaying, quasiperiodic oscillations. Finally, we show that for some generalizations of the DNLS equation where bistability occurs, a controlled switching between stable states is possible by exciting an internal breathing mode above a threshold value. © 1998 Elsevier Science B.V.

PACS: 03.20+i; 63.20.Pw; 63.20.Ry; 71.38.+i

1. Introduction

The discrete nonlinear Schr6dinger (DNLS) equa-

tion occurs as a first approximation in many different

physical contexts where discreteness, dispersion and

self-interaction are present. Applications include e.g.

polaron formation in electron-lattice coupled systems

[1], localization of vibrational energy in proteins [2],

localization of optical beams in nonlinear waveguide

arrays [3,4], and exciton dynamics in Scheibe aggre-

gates in molecular thin films [5]. In one dimension and

with a general power nonlinearity, the DNLS equation

reads:

i~n q -C(~n+l q- ~n-1) -q- ]aPnl2~n = 0 , (1)

* Corresponding author. E-mail: mjn@ifm.liu.se.

where rr > 0. The ordinary (cubic) DNLS equation

is obtained for cr = 1, while larger values of cr has

frequently been used to simulate higher-dimensional effects in a one-dimensional model. As is well-known

[6-18], the DNLS equation (with arbitrary cr and

in any dimension) has infinitely many exact time-

periodic solutions (stationary states) of the form

~n (t) = Cn eiwt, (2)

where Cn is time independent, so that these solutions

are static in a frame rotating with frequency w. In par-

ticular, there exist exponentially localized stationary

states with a single maximum at a lattice site. These

solutions are the single-site breathers which can be

obtained by continuing the single-site solution at the

anticontinuous (uncoupled) limit C = 0 to non-zero

values of C [12]. Apart from the single-site breathers,

0167-2789/98/$19.00 © 1998 Elsevier Science B.V. All rights reserved PII S0167-2789(98)00070-0

116 M. Johansson et al./Physica D 119 (1998) 115-124

there is also an infinite number of localized or non-

localized stationary solutions, which can be obtained

by continuation from the anticontinuous limit of multi-

site solutions where the frequencies of the excited sites

are integer multiples of 09. However, in a number of different numerical ex-

periments, localized solutions have been observed for

which I~Pn 12 oscillated periodically, and ~n in general

quasiperiodically, with time. These solutions were ob-

tained e.g. when starting with a single-site initial con-

dition [19], in collisions between broad, soliton-like

objects [17], or as final states in the 'quasicollapse'

process [13,14,20], which is the discrete analogue to

the collapse of an initially extended excitation occur-

ring in the continuum limit of Eq. (1) and its multi-

dimensional extensions when t rD > 2, where D is the

number of spatial dimensions [21]. One of the ques-

tions, which we will address here, is then whether

these observed 'quasiperiodic breathers' could be ex-

act solutions to the DNLS equation, or only approx-

imate solutions which would radiate and relax into a

stationary state as t ~ cx~. We will review recent re-

suits [22] proving the existence of exact, spatially lo-

calized and temporally quasiperiodic solutions to the

DNLS equation (1). These solutions are obtained as

multi-site breather solutions (multibreathers) by con-

tinuation from the anticontinuous limit C = 0 of so-

lutions where the excited sites can be divided into two

groups oscillating with mutually incommensurate fre-

quencies. The simplest of these solutions is a two-site

breather, which is also shown to be linearly stable for

small C. (We remark that earlier studies [3,23,24] have

shown the existence of exact, quasiperiodic solutions

in terms of elliptic functions only for small DNLS sys-

tems with two or three sites.) We will also explicitly

discuss an example in two dimensions, which shows

the existence of localized, stationary as well as non-

stationary, solutions of vortex-type.

Comparing the region in parameter space where

exact, quasiperiodic solutions exist with the param-

eters used in the numerical experiments, we argue

that the numerically observed oscillations probably are

not exact solutions, but rather due to the excitation of a localized internal, spatially symmetric 'breath-

ing' eigenmode of the linearized equations around the

single-site breather. We investigate the existence of

these modes numerically, and show that for values of cr

where bistability of stationary states occurs [ 13,16,25],

switching between the stable stationary states occurs if

the internal modes are excited above a threshold value.

2. Construction of quasiperiodic breathers from the anticontinuous limit

In this section, we will summarize some of the re-

sults obtained in [22], to which we refer for more de-

tails. First, we note that at the anticontinuous limit

C = 0, we have the following general solution to the

DNLS equation (1):

1/(2~r ) _i@on t - a n ) ~Zn(t) = 09n e , (3)

where the frequencies 09n and the phases an can be

chosen arbitrarily and independently at each site. The

simplest, time-periodic, breather solution is then ob-

tained from the solution which at the anticontinuous

limit is zero at all sites except one, where it is oscil-

lating at a frequency 09. As was shown in [12], after a

transformation into a frame rotating with an arbitrary

frequency 090, the DNLS equation fulfills the usual an-

harmonicity and non-resonance conditions, so that the

implicit function theorem can be applied to continue

the solution at the anticontinuous limit to finite val-

ues of C for all 09. The resulting solution, a single-site

breather of the form (2), is a stationary (i.e. I~Pn 12 does

not change with time), exponentially localized solu-

tion, which can be continued up to C = ½09, where the

breather frequency enters the band of linear oscilla-

tions and the size of the breather diverges (for tr = 1,

the breather approaches the NLS soliton in this limit).

Quasiperiodic solutions are trivially obtained at the

anticontinuous limit as solutions where different sites

oscillate with incommensurate frequencies 09n. How- ever, for generic nonlinear lattice equations, one ex-

pects that it should not be possible to continue these

solutions to localized breather solutions for non-zero

coupling, since the harmonics of the breather frequen-

cies should fill densely the real axis making it impos- sible to avoid resonances with the linear band [26,27]. The DNLS equation (1) is however exceptional in this

M. Johansson et a l . /Phys i ca D 119 (1998) 115-124

respect, since one frequency always can be made zero by transforming into a rotating frame. Thus, we will be able to construct quasiperiodic breathers with two

(but not more) incommensurate frequencies. The simplest example of a two-frequency breather is

obtained by exciting two sites (not necessarily neigh- bouring) with different frequencies coo and col at the anticontinuous limit, i.e.,

I / (2cr )_io)ot ~ o ( t ) = coo ~ ,

7*J (t) = col/(2a) e i°°lt, (4)

g % ( t ) = 0 , n ~ 0 , 1 .

(The phases c~0 and a l can without loss of generality always be chosen to zero when coo and col are incom- mensurate just by choosing an appropriate origin of time.) We then make the transformation into the frame where site 0 is static, i.e., ~n (t) = q~n (t)e i°J°t, so that

the DNLS equation (1) is transformed into

i~,, + C(rbn+l + 4~n-1)

"-}- IqSn r2a ~pn - - CO0(O n -m- O. (5)

In the new variables 4~n the solution at the anticontin- uous limit will then be periodic, since only site 1 is

oscillating with frequency COo = COl - COo. Since the zero-amplitude frequency CO(0) of the unexcited sites is CO(0) = -COo in the variables 4~n, the standard non-

resonance condition n l COb 5 ~ CO(0) for all integer n l [12] becomes

COl/CO0 5 ~ (h i - - l ) / n l . (6)

Thus, we can directly apply the theory from [12,26] to prove the existence of a unique continuation of the an- ticontinuous solution to non-zero coupling when this condition is fulfilled. In particular, this means that two- frequency breathers exist for all possible incommen-

surate values of (w0, COa ), since all resonances occur at rational values of COl/COo.

To determine the linear stability properties of the quasiperiodic DNLS breathers, we consider the time evolution of a small perturbation {en(0)} at t = 0 of a periodic breather solution {q~°)(t)} to Eq. (5), i.e., we have q~n(t) = 4~°)(t) + en(t). Linearizing around {en = 0} gives the linear tangent equations

117

ien + C(en+l + 6n-1) -I- (1 + a)kb~°)12° Sn /.~.(0) 2 .t(0) 2 a - 2 * + a ~ n ) Iq% I e n - w o e n = 0 . (7)

Then, the solution {~b~°)(t)} is linearly stable if the perturbation {en (t)} as calculated from the linearized equations (7) remains bounded for all times. Although linear stability in general does not guarantee that per-

turbations of the initial conditions of the nonlinear sys-

tem (5) will remain small for all times, it implies that the growth rate of the perturbations necessarily will be slower than exponential, and so linearly stable so-

lutions can be expected to have a significantly longer lifetime than those which are linearly unstable.

For time-periodic solutions {~bn ~°) (t)}, the linear sta- bility analysis can be conveniently performed using

Floquet analysis. Writing en = ~n + ion, where ~n(t) and On (t) are real functions, the Floquet matrix To is defined by the relation

({~n( tb )} ) ( {~n(0)} ) (8) {0n(tb)} = TO {~n(0)} '

t-h~ °) -t where tb = 2zr/cob is the period of L,e ( )], and the left-hand side is calculated by integrating the tangent equations (7) over one period of time. Then the so- lution {~b~°)(t)} will be linearly stable if and only if

the Floquet matrix To has no eigenvalues with modu- lus larger than one. Since (7) is a symplectic map, To will be a symplectic 2N x 2N matrix for a system of

N sites, and as a consequence linear stability requires

that all eigenvalues of To are complex conjugated pairs on the unit circle.

At the anticontinuous limit C = 0, the Floquet ma-

trix will for a general configuration with P excited sites have P degenerate pairs of eigenvalues equal to 1 (corresponding to phase rotations of the excited sites), and N - P pairs equal to e ±i°° (corresponding to lin-

ear oscillations), where 69o = 2Jr(co0/cob) [26]. As C is increased from zero, the complex eigenvalues will raise their degeneracy, but they will stay on the unit circle for small C provided that 6)o is not equal to 0 or zr (this is a consequence of the Krein theory, which implies that only eigenvalues whose imaginary parts have opposite signs at C = 0 can collide and leave the unit circle [26]). Also, for a two-frequency breather, two pairs of eigenvalues will always stay at 1 when

118

C is increased, since {en = q~(0)} and {en = iq~n (°) } are

time-periodic solutions to Eq. (7) with period tb for

all C corresponding to two independent rotations of

the global phase of the breather (for a one-frequency

breather there is only one global phase rotation, and

the two solutions become linearly dependent). Thus,

for the two-site, two-frequency breather all eigenval-

ues stay on the unit circle for small C if the condition

n109b ~ 2090 is fulfilled for all integers n l, so that the breather will be linearly stable.

To prove linear stability for breathers where more

sites are excited at C = 0 is more complicated, since

some additional condition on the phases {~xn } in Eq. (3)

has to be imposed in order to guarantee that none of

the P pairs of eigenvalues at 1 leave the unit circle as

soon as C is increased from zero. However, a general

sufficient condition for linear stability for small C has

been obtained [26] in terms of the nature of the ex-

trema of an effective action which is a function of the

relative phases of the excited sites. For systems with hard (soft) anharmonicity, the local maxima (minima)

of the effective action correspond to linearly stable

phase configurations. Calculating the effective action

for the DNLS equation (1) perturbatively to second

order in C, some explicit multibreather configurations

were proven to be linearly stable for small C in [22].

Regions of existence and stability can be conve-

niently investigated by performing the continuation

from the anticontinuous limit numerically using a

Newton iterative scheme as described in [28]. A

rather detailed numerical investigation of the two-

site two-frequency breather for the DNLS equation

with cr = 1 was performed in [22], but here we will

only make some qualitative observations. First, we

note that no localized breather solution can exist if

any multiple of the breather frequency penetrates the

linear 'phonon' band, i.e., if

n] (091 - 090) = -090 + 2C cos(k), (9)

for any wave number k and integer n 1. This condition

gives for each o9o and 091 a maximal value of C for the possible existence of a localized breather as illus- trated in Fig. 1. The numerical analysis shows, how- ever, that for values of 091/090 where the first phonon resonance occurs at k = 0, the continuation cannot be

M. Johansson et aL/Physica D 119 (1998) 115-124

0.5

0.4

0.3

0.2

0.1

/

/ /

1.5 2 2.5 (O 1

0 1 3 3.5

Fig. 1. Max imum existence region according to Eq. (9) for a two-site quasiperiodic DNLS-breather in units such that w0 = 1, assuming wl > w0 without loss of generality. Resonance with the linear band occurs in the region above the lines. Horizontal line for ~Ol > 3 corresponds to n I = 0 and k = 0; lines with negative (positive) slope correspond to k = 0(k = Jr) and nl ---- - 1 , - 2 , - 3 . . . . from fight to left in the figure.

performed all the way up to the maximum value of

C determined by this resonance. The reason is that a

rather complicated bifurcation, associated with local-

ized eigenmodes of the linearized equations and de-

scribed in detail in [22], will occur prior to the phonon

resonance. In particular, for wl > 3090 when the reso-

nance with the linear band is due to the fundamental

frequency 090 and associated with a size divergence of

the single-site breather with this frequency, the two-

site breather, denoted as (. • • eee01 eee . • .), bifurcates

with the three-site breather (.. • • • • 001 • • • . . . ) (0

and 1 denote sites which are excited with frequencies

090 resp. 091, and the dots denote sites which are unex- cited at the anticontinuous limit). Slightly before this

bifurcation an instability develops, which is connected

with the single-site breather of frequency 090 gaining

mobility. Thus, applying a small random perturbation

of the initial condition in the instability regime results

in a splitting of the breather in two, one pinned and

one moving ('breather fission' [29]). Since the mov-

ing breather is not an exact solution, its velocity will

decrease slowly during the motion as radiation is cre- ated, and it will finally be pinned to the lattice some site far away from its initial site. If the distance be- tween the two sites excited at the anticontinuous limit

M. Johansson et al./Physica

is increased, we find that the values of C where the

instability and the bifurcation occur both approach the value C = ½o90 as expected when the interaction between the breathers becomes negligible.

On the other hand, for some values of ogl/coo such that the first resonance occurs when a higher breather harmonic enters the linear band at k = ~r, we find

that the breather can exist as a linearly stable, lo- calized solution all the way until this resonance oc- curs. Moreover, the continuation can for such values

of ogl/coo in general also be performed into the linear band, and the result is a localized breather supported

by an extended tail (i.e., a phonobrea ther [26,28] or nanopteron [27,30]).

3. An example of a multibreather in 2D: A non-stationary 'vortex-breather'

Since the method of continuation from the anticon- tinuous limit can be used in any dimension, the results

from the previous section can be trivially extended to the 2D or 3D DNLS equation. However, as was shown in [31 ], in general the extension to higher dimensions allows also for a new class of localized multibreather solutions of vortex-type having a phase torsion be-

tween the individual breathers. For the particular case of the 2D DNLS equation,

i~m.n q- C(~m., ,+l q- l/.fm,n-I -~- l[fm+l,n "~- 1/fm-l,n) 2¢r -~-[1/:m,n[ 1/.f m . n = O, (10)

these 'vortex-breathers' can be of two different kinds. The simplest type is a stationary vortex-breather carry-

ing a phase current, but for which I~Pm,~l 2 is constant in time. This solution can be obtained by continua- tion of an anticontinuous solution where a cluster of

sites constituting a closed loop is excited according to Eq. (3) with identical frequencies ogm,n ~- coo but

with uniformly varying phases Ogm,n, such that the to- tal phase twist around the loop is a multiple of 2zr. Since only a finite number of sites is excited at the anticontinuous limit, the resulting solution will be ex- ponentially localized. Also, it will be linearly stable for small C if the phase configuration is such that

D 119 (1998) 115-124 119

the effective action, as defined in [22,26], has a local minimum.

The second type of vortex-breather is the non- stationary, quasiperiodic solution which can be ob-

tained from an anticontinuous solution involving two different frequencies. A simple example of a non-stationary vortex-breather is illustrated in Fig. 2.

Here, we have excited the central site (m0, no) with

a frequency o91, and its four nearest neighbours with frequency coo < ogr and phases tim. n = O, l Yr, Jr,

and ~jr,3 respectively (the phase of the central site can

be chosen arbitrarily as in Section 2). This results in an exponentially localized solution for which the intensity I1//m.n (t)f2 is constant in time for the central

site, while it varies periodically in time for all other sites. The variation is such that within each group of

sites with a fixed distance to the central site (m0, no),

the maximum intensity rotates around (m0, no). The condition for the existence of this solution for small

C is the same non-resonance condition (6) as for the two-site breather. Also, we find that this configura- tion corresponds to a local minimum of the effective action, so that the solution in general will be linearly

stable for small C. Investigating the existence and stability regions of

the non-stationary vortex-breathers by numerical con- tinuation and Floquet analysis, we find that they gen-

erally exist only for a small effective coupling (or, equivalently, a large effective nonlinearity), and that

the stability region is even smaller than the existence region. Defining the effective coupling as

Ceff = C/./V', (1 l)

where

N = Z IOm'n(t)12 (12) m , n

is the norm of the solution (which is a conserved quan- tity for all DNLS-type equations), the existence and

stability regions depend only on the ratio o91/o9o. We find that the largest value of Ceff for which a linearly stable vortex-breather of the type illustrated in Fig. 2 can exist is approximately 0.024, and occurs when ogl/o9o is slightly larger than 3. The nature of the in- stabilities occurring for larger Ceff has not yet been

120 M. Johansson et al./Physica D 119 (1998) 115-124

(a) t, t~

D

i , ; i i i

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(too,no) (b) (mo_l,no)

/too,no+l) ~mo+l ,no) (mo,no-1)

50

45

40

35

30

25

20

15

10

5

0

ta t4

' ' 1 l . l . . . . . . . . . 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 - t

0.5

Fig. 2. Example of a non-stationary vortex-breather. (a) The central spot indicates the site excited with frequency o)1; surrounding arrows indicate sites oscillating with frequency too when C = 0. The direction of the arrows illustrates the phases otto,n, while the length of the arrows schematically illustrates the magnitude of laPm,n(t)[ 2 when C > 0 at four consecutive time instants tl < t2 < t3 < t4. (b) The time evolution of pp,n.n(t)l 2 for the five excited sites when ~r = 1, w l / o ~ ~ 3.7, .A/" ~ 99.5 and C = 2.0.

investigated in detail, but they are different from those

discussed in Section 2 since they origin from collisions

of Floquet eigenvalues on the unit circle out of the

real axis ( 'Krein crunches') as eigenvalues originating

from 1 meet eigenvalues originating from e +iO)° .

4. Internal modes of single-site breathers

The previous sections have shown that the DNLS

equation has many exact localized two-frequency

solutions, which all give rise to a time-periodic oscil-

lation in l aPn (t)l: as observed in the numerical exper-

iments discussed in Section 1. However, a common

feature of all the solutions we have investigated is that

they only exist for a rather small effective coupling

(Ceff < O. 1 for the two-site breather, and even smaller

for other configurations), while the oscillatory states

found in the above-mentioned numerical simulations

typically correspond to Ceff '~" 0.25-0.35. More- over, since the initial conditions for most of these

simulations were spatially symmetric, the resulting oscillatory structures were also spatially symmetric. However, we have found that all the spatially sym-

metric two-frequency breathers which we have inves-

tigated for DNLS equations (1) with tr > 1 have been

linearly unstable with respect to symmetry-breaking

perturbations for all C. On this basis, we believe that

the oscillatory structures observed in the experiments

are not exact quasiperiodic solutions, but rather ex-

tremely long-lived approximate solutions which are

connected with the excitation of a localized, spatially

symmetric internal mode ('breathing mode ' ) of the

single-site breather. The importance of internal modes, which can be found as localized eigenvectors of the

Floquet matrix To, for the dynamics of nonlinear

localized excitations both in discrete [32,33] and con-

tinuum [34] systems has recently been emphasized.

The internal modes are easily found numerically as

Floquet eigenvectors corresponding to isolated eigen-

values outside the continuous spectrum, which can be determined analytically from the linear dispersion re-

lation. (In fact, since the single-site breather is time independent in a rotating frame, no time integration

of the linearized equations is needed, and the Floquet

analysis can be replaced by the simpler, but equivalent, technique described in [8].) For the ordinary 1D DNLS equation (1) with tr = 1 and C = 1, we find that

M. Johansson et al./Physica

single-site breathers of the form (2) possess a localized

breathing mode when 2 < 09< 3.71, or, equivalently,

when.A/" < 3.76 (A/" is a monotonously increasing func-

tion of co for the 1D cubic DNLS equation with .A/" --+

0 as o9 ~ 2 [13]). It is interesting to note that a

breathing mode exists also close to the continuum

limit (09 ~ 2) although it is known [34] that the NLS

soliton has no such mode. However, the localization

length of the eigenvector diverges when approaching

the continuum limit. Thus, our results support the con-

clusions from [34] that the appearance of a breathing

mode is a general phenomenon occurring when the

NLS equation is perturbed away from integrability,

whether the perturbation is due to higher-order nonlin-

earities or discreteness effects. Exciting the breathing

mode by applying a small spatially symmetric and lo-

calized perturbation to the single-site breather at t = 0,

we find that the resulting time-periodic oscillations of

IT t, (t)12 can survive for several hundred periods with-

out significant decrease in amplitude. There will be a

slow, radiative decay, and the state will finally relax

into the stationary single-site breather as t ~ co.

Apart from the breathing mode, the single-site

DNLS breather also possesses a spatially antisym-

metric 'pinning' eigenmode when the width of the

breather is sufficiently large. (In 1D with cr = 1 and

C = 1, the pinning mode exists when 2 < 09 < 3.1 1,

or, equivalently, when .A/'< 3.22) As was shown in

[32], the appearance of a pinning mode generally

implies that the breather gains mobility. As 09 ap-

proaches 2C the Floquet eigenvalue corresponding to

the pinning mode comes very close to +1, implying

a high mobility for broad, 'continuum-like' excita-

tions. (In the continuum limit the Floquet eigenvalue

is exactly 1, since a stationary solution can always be

boosted to an arbitrary velocity due to the Galilean

invariance of the NLS-type equations.)

5. Switching between bistable stationary states through internal breathing modes

The phenomenon of multistability, implying the

coexistence of several stationary localized solutions of type (2) with different frequencies w having the

D 119 (1998) 115-124 121

same value of the norm Af, appears in several dif-

ferent generalizations of the cubic DNLS equation

[13,16,18,25,35]. Typically, in these cases there ex-

ist o91 and co2 such that the function .Af(w) is an

increasing function for 09 < o91 and o9 > o92, but

decreasing in the interval ogi < w < co2. Since for

DNLS-type equations a necessary and sufficient con-

dition for linear stability of stationary states with a

single maximum at a lattice site is dAf/do9 > 0 [13],

the stationary states for ~o < o)1 and o9 > o92 are

stable (often termed 'continuum-like' and 'discrete'

states, respectively) while the intermediate state with

CO l < O9 < 602 i s unstable. For the 1D DNLS equation

(1) with arbitrary power nonlinearity, multistability

occurs when 1.4<~r < 2 [13] (when a > 2 no

stable solution exists in the continuum-limit, where

collapse occurs [21]. In [35] it was recently shown

that when the cubic DNLS equation is attributed

with a long-range dispersive coupling, it is possi-

ble to switch between the stable stationary states by

exciting an internal breathing mode above some well-

defined threshold value. Here, we will illustrate that

the same mechanism for switching can be used also

for the nearest-neighbour DNLS equation (1) in the

bistability domain 1.4 < cr < 2.

In Fig. 3 we illustrate switching from a broad,

continuum-like state to a narrow, discrete state ((a)

and (b)), and vice versa ((c) and (d)) for the case

cr = 1.5. For this value of 0r and C = 1, the stationary

state has a breathing mode when 2 < co < 3.83 (cor-

responding to Af ~< 2.59). However, in constrast to the

case when a < 1.4, where the pair of Floquet eigen-

values corresponding to the breathing mode always

stays on the unit circle, for ~r > 1.4 they collide at +1

when 09 = o92 and go out on the real axis for co < o92,

thereby creating the instability. At co = ogl the eigen-

values return to the unit circle and remain there for

o9 < col if cr < 2 so that a stable state reappears;

when cr ~ 2, wi --+ 2C, and for cr > 2 the eigenval-

ues remain real for all 09 < 092. For the case in Fig. 3 with ~r = 1.5, we find that ogl ~ 2.30 and o92 ~ 2.67,

corresponding to .Af(wl) ~ 2.38 and H(092) ~ 2.29,

so that bistability occurs when .M lies in this interval. To excite the breathing mode, we apply a spatially

symmetric, localized perturbation which we choose to

122 M. Johansson et al./Physica D 119 (1998) 115-124

(a)

2

1.5

1

0.5

0

o=0.18

t~nol2

E30

2

1.8

1.6

1.4

1.2

1

0.8

0.6

0.4 0

(b) (x=O. 17 ...... ct=0.18 - -

, - , ,

~ L I i h i I I

20 40 60 80 100 120 140 160 180 200 t

(C) ~t=-0.15

I~n 12

iL 1 , ,

0.5 ~ 7 0 8 0

- o -

I~no 12

2

1.8

1.6

1.4

1.2

1

0.8

0.6

0.4 0

(d) ~=-0.14 ...... ~ = - 0 . 1 5 - -

i I i I i i i i i

20 40 60 80 100 120 140 160 180 200 t

Fig. 3. (a), (b): Switching from continuum-like state to discrete state, and (c), (d): switching from discrete state to continuum-like state, caused by applying a phase twist to the central sites of the stationary states as described in text. In (a) and (b) the initial state has frequency o~ = 2.22 (A/'~ 2.35) and the threshold value for switching is Ctth ~ 0.18; in (c) and (d) the initial state has frequency o~ = 3.0 (.A/'~ 2.33) and the threshold value for switching is Otth ~ 0.15. (a) and (c) show time evolution of I~Pn(t)l 2 when Ic~l is slightly larger than Ceth (only a small part of a larger system is shown); (b) and (d) show time evolution of I~Pn0(t)l 2 when lal is close to Otth.

be norm conserving in order not to change the effec-

tive nonlinearity of the system. The exact choice of

perturbation is not important for the qualitative fea-

tures of the switching, as long as there is a substan-

tial overlap between the perturbation and the internal

breathing mode. In the simulations shown in Fig. 3 the

perturbation is realized by applying a phase twist ot to

a few sites {n r} around the central site no of the sta-

t ionary state (2), i.e., ap{n,} (0) = c{n,je ia for these sites,

while ~n (0) = cn for all other sites. This can be seen

to be equivalent to applying an external kick at these

sites at t = 0 by adding a term )-~-{n'} °l~(t)t~n,n'l~n

to the left-hand side of Eq. (1). Also, it can be seen

from the linearized equation (7) that a > 0 (or < 0)

implies that d(laPn, 12)/dt > 0 (< 0) at t = 0 for the

kicked sites {n~}, so that we choose a positive when

switching from the continuum-like state to the discrete

state, and ot negative for switching in the opposite di-

rection. In Fig. 3(a) and (b), where the initial state

is continuum-like, the perturbation is applied to the

M. Johansson et al./Physica

three central sites, while in (c) and (d), where the ini-

tial state is discrete, the perturbation is applied only to the central site. The different choices are due to the different nature of the breathing modes for these two cases: for the discrete state I~Pn0(t)l 2 always oscillates in antiphase with I~Pn (t)l 2 for the other sites, while for the continuum-like state used as initial condition in (a) and (b) I~P,(t)[ 2 for the three central sites oscillate in

phase. As is seen from Fig. 3, there is in both cases a clear threshold value otth (depending on the frequency

0) of the initial state) such that when Iotl < ~th slowly decaying breather oscillations around the initial state

will occur, while for Iotl > ~th the state switches into the other stable stationary state, around which breather

oscillations develop. Due to the small radiative losses in the switching process, the switching only occurs

once when loll is close to Otth. The value of Otth is re- lated to the location on the unit circle of the Floquet eigenvalues corresponding to the breathing mode, and approaches zero as the eigenvalues approach + 1, i.e.,

when 0) --~ 0)1 or 0) 2.

Although, as is seen from Fig. 3, the spatial ex-

tent of the bistable continuum-like and discrete states

studied here do not differ as significantly as e.g. for the case when the bistability is due to long-range dispersion [25,35], there is an important qualita-

tive distinction between the two states. Namely, the continuum-like state always possesses a pinning mode, and can therefore be made mobile by applying an appropriate spatially antisymmetric perturbation [32], while the discrete state never has a pinning

mode, and therefore this state always remains pinned at its site. Generally, the pinning mode bifurcates from the continuous band in the instability regime

0)I < 0) < o~2; for the case in Fig. 3 with a = 1.5 this occurs at 0) ~ 2.40 (A/" ~ 2.35).

6. Conclusions

D 119 (1998) 115-124 123

higher dimensions, internal breathing modes of sta- tionary single-site breathers, and bistability of pinned and mobile localized stationary states. In view of the wide applicability of DNLS models, we believe that it should be possible to detect these phenomena experi-

mentally. In particular, the switching between pinned and mobile states could be important for controlling energy storage and transport in molecular systems.

An interesting problem is to what extent these features are present also in models outside the non- linear Schr6dinger class. Exact, quasiperiodic local- ized solutions almost certainly exist only in models

with particular symmetries (for DNLS models, the

existence of two-frequency solutions is connected with the fact that there are two conserved quantities,

norm and Hamiltonian), making it possible to avoid phonon resonances. On the other hand, approximate quasiperiodic breathers arising from internal breath- ing modes of periodic breathers appears to be a very general phenomenon [27]. Concerning multistability and the possibility of achieving a controlled switch-

ing between bistable states, we believe that these phenomena extend beyond the DNLS models, and in

particular in models with long-range dispersion where the bistability arises due to the competition between different length scales. However, this is a topic that

deserves further investigations.

Acknowledgements

We thank T. Cretegny, G. Kalosakas, V.K. Mezentsev, J. Juul Rasmussen for discussions. MJ

acknowledges financial support from the Swedish Foundation for International Cooperation in Research

and Higher Education. This work was supported by the Human Capital and Mobility network of the European Union under grant no. CHRX-CT93-0331.

In summary, we have given some examples illus- trating that the breather dynamics in discrete non- linear Schr6dinger models is very rich, including quasiperiodic breathers as exact solutions, localized stationary and non-stationary vortex-like solutions in

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