Research ArticleSelf-Similarity Analysis of the Nonlinear Schrödinger Equationin the Madelung Form

Imre F. Barna ,1,2 Mihály A. Pocsai,1,3 and L. Mátyás4

1Wigner Research Center for Physics of the Hungarian Academy of Sciences, Konkoly-Thege Miklós út 29-33, 1121 Budapest, Hungary2ELI-HU Nonprofit Kft., Dugonics Tér 13, H-6720 Szeged, Hungary3University of Pécs, Institute of Physics, Ifjúság Útja 6, H-7624 Pécs, Hungary4Sapientia University, Department of Bioengineering, Libertătii Sq. 1, 530104 Miercurea Ciuc, Romania

Correspondence should be addressed to Imre F. Barna; barna.imre@wigner.mta.hu

Received 16 July 2018; Accepted 12 September 2018; Published 1 October 2018

Academic Editor: Emmanuel Lorin

Copyright © 2018 Imre F. Barna et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

In the present study a particular case of Gross-Pitaevskii or nonlinear Schrödinger equation is rewritten to a form similar to ahydrodynamic Euler equation using theMadelung transformation.The obtained systemof differential equations is highly nonlinear.Regarding the solutions, a larger coefficient of the nonlinear term yields stronger deviation of the solution from the linear case.

1. Introduction

In 1926, at the advent of the quantum mechanics Madelung[1, 2] gave a hydrodynamical foundation of the Schrödingerequation and raised interpretation questions which areopen till today. Later, people realized that the MadelungAnzatz is just the complex Cole-Hopf transformation [3, 4]which is sometimes applied to linearize nonlinear partial-differential equations (PDEs). The Madelung form of theSchrödinger equation has a remarkable property that Planck’sconstant (representing quantum mechanics) appears onlyonce, namely, as the coefficient of the quantum potential.This quantity also appears in the nonmainstream attemptof quantum mechanics (the de Broglie-Bohm pilot wavetheory [5] (in other context)) which interprets quantummechanics as a deterministic nonlocal theory. TheMadelungtransformation may create a common language betweenscientist of fluid dynamics and quantum mechanics and canbe fruitful to give an additional interpretation of quantummechanics. In the last decades, hydrodynamical descriptionof quantum mechanical systems become a popular technicaltool in numerical simulations. Detailed reviews in this fieldcan be found in a booklet [6] or in Wyatt’s monograph [7].

The application of the original Madelung (or to say Cole-Hopf) transformation grew out the linear quantum theory

and were applied to more sophisticated field theoreticalproblems. Recently, Chavanis and Matos [8] applied theMadelung Ansatz to the Klein-Gordon-Maxwell-Einsteinequation in curved space-time possibly describing self-gravitating Bose-Einstein condensates, coupled to an electro-magnetic field. This study clearly presents the wide applica-bility of the original Madelung Ansatz to refined quantumfield theories with the possible hydrodynamic interpretationpicture.

As an analytical tool of investigation we use the self-similar Ansatz of [9]. It is a powerful method to study theglobal properties of the dispersive solutions of various non-linear partial-differential equations (PDEs) In other words,the intermediate asymptotic of the problem is described.Therefore this solution is valid when the precise initialconditions are no longer important; however, the final steadystate is not yet reached. For real physical systems this regionattracts the most attention. In contrast to the full solutionit is much simpler and easier to understand for the entireparameter space. The second reason for studying them isthat they are solutions of a system of ordinary differentialequations (ODEs) (not a PDE) and thus do not suffer thefull inherent numerical problems of the full PDEs. Most ofthe cases this tool helps to understand some of the globalproperties of the solution, e.g., shockwaves or the existence of

HindawiAdvances in Mathematical PhysicsVolume 2018, Article ID 7087295, 5 pageshttps://doi.org/10.1155/2018/7087295

http://orcid.org/0000-0001-6206-3910https://creativecommons.org/licenses/by/4.0/https://doi.org/10.1155/2018/7087295

2 Advances in Mathematical Physics

compact supports. First, we applied this Ansatz to the mod-ified Cattaneo-Venotte heat-conduction problem [10] andobtained a compact solution with finite propagation speed.Later, we successfully generalized the Ansatz to multispacedimensions and studied numerous viscous fluid equations[11, 12] ending up with a book chapter of [13]. As main resultswe are now able to interpret the birth of Benard cells in heatedfluids [14].

In our former study [15] we investigated the Schrödingerequation in the Madelung form with the self-similar Ansatzand presented analytic solutions with discussion. Unlike toclassical fluids the obtained density of the Madelung fluidhas infinite number of zeroes which is a fingerprint of thequantum mechanical origin. Now we analyze the Madelungform of nonlinear Schrödinger (NLS) equation with the sameAnsatz and compare the obtained results to the previouslinear one. As far as we know, there are no direct clear-cut analytic solutions existing for the Madelung form ofthe NLS equation by now. It is important to mention thatcelebrated standard numerical algorithms exist for solvingthe NLS [16, 17]. We hope that our present study helps toattract attention to this question. Parallel to the well-knownquantummechanical picture there is a looming hydrodynam-ical world hidden behind the Schrödinger equation from thevery beginning which is poorly understood.

2. Theory and Results

In a weakly interacting Bose system, the macroscopic wavefunction Ψ appears as the order parameter of the Bose-Einstein condensation, obeying the Gross-Pitaevskii (GP)equation [18–21]:

𝑖ℏ𝜕Ψ𝜕𝑡 = (−ℏ2∇22𝑚 + 𝑛 |Ψ|2 − 𝜇)Ψ. (1)

Here, 𝑛 = 4𝜋ℏ2𝑎/𝑚 represents the coupling constant cha-racterized by the s-wave scattering length a. The particlemass and the chemical potential are 𝑚 and 𝜇, respectively.Due to the second (nonlinear) term of the right hand sideof this equation, it is definitely different from the linearSchrödinger equation [15]; therefore the following analysiswith the presented results are independent as well.

The role and the properties of the GP system has beeninvestigated from various viewpoints. Without completenesswe mention some of them. The question of collective excita-tions of the trapped Bose gases was analyzed by [22]. Later,it is considered in the study of self-organization of the Bose-Einstein (BE) condensates [23].

Certain generalized versions of (1) have been also appliedto the study of BE condensate of polaritons [24–32].

Considering the Madelung Ansatz for the order param-eter with the amplitude and phase Ψ = √𝜌𝑒𝑖𝜙 where thedensity of the condensate is 𝜌 and the superfluid velocity isk = (ℏ/𝑚)∇𝜙. (The expression “superfluid velocity” is takenfrom Tsubota et al. [33] and means, in this content, that thefluid is inviscid.)

The obtained hydrodynamical equations, the continuityand the Euler, are as follows:

𝜕𝜌𝜕𝑡 + ∇ ⋅ (𝜌k) = 0,𝜕k𝜕𝑡 + k ⋅ ∇k = ℏ

2

2𝑚2∇(√𝜌√𝜌 ) −

∇𝑚𝜌 (𝑛𝜌2

2 ) ,(2)

where we consider free motion (no extra potential energytermU(r) is used) and the chemical potential 𝜇 was set to zeroas well. It is easy to show that an additional chemical potentialterm 𝜇 gives no contribution to the final hydrodynamicalequations. The role of the external potential energy term𝑈(r)will be mentioned later.

The first term of the right hand side of the secondequation is the quantum pressure term. The second termmakes the difference to the original Madelung equations; thisis a kind of second quantum pressure due to the nonlinearorigin of the GP equation. Not, that these are most generalvectorial equation for the velocity field k which means thatone-, two-, or three-dimensional motions can be investigatedaswell. In the following wewill consider the two-dimensionalflow motion k = (𝑢, V) in Cartesian coordinates. Thefunctional form of the three- and one-dimensional equationsystem will be mentioned briefly, too.

We search the solution of (2) in the formof the self-similarAnsatz [9]:

𝜌 (𝑥, 𝑦, 𝑡) = 𝑡−𝛼𝑓(𝑥 + 𝑦𝑡𝛽 ) fl 𝑡−𝛼𝑓 (𝜂) ,𝑢 (𝑥, 𝑦, 𝑡) = 𝑡−𝛿𝑔 (𝜂) ,V (𝑥, 𝑦, 𝑡) = 𝑡−𝜖ℎ (𝜂) ,

(3)

where 𝑓, 𝑔, and ℎ are the shape functions of the densityand the velocity field, respectively. The similarity exponents𝛼, 𝛽, 𝛿, 𝜖 are of primary physical importance since 𝛼, 𝛿, 𝜖 rep-resent the damping of the magnitude of the shape function.The exponent 𝛽 represents the spreading of the solution.Additional technical details about the general features ofthe Ansatz can be found in our former papers [12, 14].Except some extraordinary cases only all positive similar-ity exponents correspond to physically relevant dispersivesolutions which mean decaying features at 𝑥, 𝑦, 𝑡 → ∞.Usually negative exponents describe exploding solutions.Substituting the Ansatz (3) into (2) and executing additionalalgebraicmanipulation the followingODE system canwrittendown for the shape functions:

− 𝛼𝑓 − 𝛼2𝑓𝜂 + 𝑓𝑔 + 𝑓𝑔 + 𝑓ℎ + 𝑓ℎ = 0,− (32𝛼 − 1)𝑔 − 𝛼2𝑔𝜂 + 𝑔𝑔 + ℎ𝑔

= ℏ22𝑚2 ( 𝑓3

2𝑓3 − 𝑓𝑓𝑓2 + 𝑓

2𝑓 ) − 𝑛𝑚𝑓,

Advances in Mathematical Physics 3

− (32𝛼 − 1) ℎ − 𝛼2 ℎ𝜂 + 𝑔ℎ + ℎℎ

= ℏ22𝑚2 ( 𝑓3

2𝑓3 − 𝑓𝑓𝑓2 + 𝑓

2𝑓 ) − 𝑛𝑚𝑓,(4)

where prime means derivation with respect to the newvariable 𝜂. The relations among the similarity exponents arethe following: 𝛼 can be taken arbitrary, 𝛽 = 𝛼/2 and 𝜖 =𝛿 = 3𝛽 − 1 = 3𝛼/2 − 1. At this point we have to emphasize,that a particular form of the external potential 𝑈(𝑥, 𝑦, 𝑡)can be added to (1) without leading to contradiction amongthe similarity exponents. The form in 2D is the following:𝑈(𝑥, 𝑦, 𝑡) = 1/𝑡2𝛽 ⋅V(𝜂)where V(𝜂) can be any kind of functionwith existing first derivative.

(Of course, there is no evidence that the obtained modi-fied (4) could be solved with analytical means.)

Note, that the ODE system of the linear Schrödingerequation [15] has a much simpler form of

− 12𝑓 − 12𝑓𝜂 + 𝑓𝑔 + 𝑓𝑔 + 𝑓ℎ + 𝑓ℎ = 0,− 12𝑔 − 12𝑔𝜂 + 𝑔𝑔 + ℎ𝑔

= ℏ22𝑚2 ( 𝑓3

2𝑓3 − 𝑓𝑓𝑓2 + 𝑓

2𝑓 ) ,− 12ℎ − 12ℎ𝜂 + 𝑔ℎ + ℎℎ

= ℏ22𝑚2 ( 𝑓3

2𝑓3 − 𝑓𝑓𝑓2 + 𝑓

2𝑓 ) ,

(5)

where all the similarity exponents have the fixed value of+1/2. Our experience tells that this is usual for regular heatconduction, diffusion or for noncompressible Navier-Stokesequations [13]. For the linear case after some additional trivialalgebraic steps a decoupled ODE can be derived for the shapefunction of the density:

2𝑓𝑓 − (𝑓)2 + 𝑚2𝜂2𝑓22ℏ2 = 0. (6)For one, two, or three dimensions themultiplicative factor

of ℏ in the last term is different. It is 1, 2, or 3, respectively.With the help of Bessel functions [34] the solution of (6)

can be expressed:

𝑓 (𝜂)= 𝜋𝜂64 [𝑐1𝐽1/4 (

√2𝑚𝜂28 ) − 𝑐2𝑌1/4 (√2𝑚𝜂28 )]

2 , (7)

where 𝑐1 and 𝑐2 are the usual integration constants. All theadditional properties of this solution can be found in ourformer paper [15]. (There is a misprint in the final [15] at Eq.7, the parenthesis is missing in front of the exponent 2.This isthe correct form.)

The most general equation of (4) cannot be fully inte-grated by analytical means. The first ODE of such systems[13] explicitly means mass conservation. Furthermore in themost cases it can be integrated analytically. However, the firstequation of (4) is not a total derivative for any 𝛼.This is highlyunusual among our investigated systems till now. As a secondcase let us consider (3/2𝛼 − 1) = 𝛼/2 this gives 𝛼 = 1,𝛽 = 𝛿 = 𝜖 = 1/2. Now the sum of the second and thirdequation of (5) can be integrated leading to

𝑔22 + 𝑔 (ℎ − 𝜂2) + ℎ2

2 + ℎ𝜂2 + 2𝑛𝑓𝑚− ℏ22𝑚2 (− 𝑓

2

4𝑓2 + 𝑓

2𝑓) + 𝑐1 = 0.(8)

For the sake of simplicity we set 𝑐1 to zero. Note, that thevariables 𝑔, 𝑓 and ℎ are still coupled in this single equation.However, (8) is quadratic in 𝑔(𝜂) and the solutions can beexpressed with the well-know formula of

𝑔1,2 = 𝜂2 − ℎ ± √ 𝜂2

4 − 4𝑛𝑓𝑚 + ℏ2

2𝑚2 (− 𝑓2

4𝑓2 + 𝑓

2𝑓). (9)Even this includes both functions of ℎ and 𝑓. To avoid thisproblem let us fix the discriminant to zero, which meansthat the velocity shape function will be single valued. Theremaining ODE reads as follows:

2𝑓𝑓 − (𝑓)2 + 𝑓2 (𝜂2𝑚22ℏ2 − 8𝑛𝑚𝑓ℏ ) = 0 (10)with the additional restraint that 4𝑓2 ̸= 0 meaning that thephysical density of a fluidmust be positive. Note, that withoutthe last term we get back (6) with the solution of (7). We triednumerous variable transformations unfortunately, we couldnot find any closed solution for the full equation (10). (Notethat if the coefficient of the second term would be twice asthe first term there would be a possible solution for the fullequation including the third order term.) It is also clear from(9) that the velocity shape functions have the trivial form of𝑔 + ℎ = 𝜂/2 = (𝑥 + 𝑦)/2√𝑡 which is the same as for the linearcase.

Figure 1 shows the analytical solution of (7) and numer-ically solution of Eq. (10) for various nonlinear parameters𝑛 with the same initial conditions of 𝑓(0) = 1 and 𝑓(0) =0. The ℏ = 𝑚 = 1 units are used. All solutions have astrong damping with stronger and stronger oscillations atlarge arguments. The functions are positive for all values ofthe argument, (which is physical for a fluid density), but suchoscillatory profiles are completely unknown in regular fluidmechanics [13]. The most interesting feature is the infinitenumber of zero values which cannot be interpreted physicallyfor a classical real fluid. Remembering the restraint 4𝑓2 ̸= 0 ofthe calculation the solution function falls into infinite numberof finite intervals. We think that this is a clear fingerprintthat the obtained Euler equations have a quantummechanicalorigin.

4 Advances in Mathematical Physics

f()

10

1

15

2

3

4

5

5

Figure 1: Various shape functions of the density. The blue curveis the analytic solution of equation (7) with 𝑛 = 0 nonlinearityparameter. Black, red, and green functions are numerical results ofequation (10) for 𝑛 = 0.078, 0.11 and 0.13 nonlinearity parameters.(All curves are for the same initial conditions of 𝑓(0) = 1 and𝑓(0) = 0.) Note that larger parameters mean larger deviation fromthe analytic result.

The major difference to the analytic solution is thefollowing the ∫∞

0𝑓(𝜌)𝑑𝜌 can be evaluated (now of course just

numerically up to an arbitrary 𝜌𝑚𝑎𝑥) giving us a convergentintegral. This clearly shows that the wave function of thecorresponding quantummechanical system (see below) is theelement of the 𝐿2 space. We investigated numerous initialconditions all giving the same result of a finite integral. Ofcourse, this is just an empirical statement and not a rigorousmathematical proof.

The √𝜌 function could be interpreted as the fluidmechanical analogue of the real part of the wave function ofthe free GP quantum mechanical particle. For the linear casethe square root of (7) is a kind of far analogue of the originalfree Gaussian wave packet. Unfortunately, we cannot find anydirect transformation between these two functions.

To obtain the complete original wave function, the imag-inary part has to be evaluated as well. It is simple from 𝜂 =(𝑥 + 𝑦)/𝑡1/2 = 2(𝑔 + ℎ) that

𝑆 = 𝑚ℏ ∫r1

r0kdr = 𝑚ℏ

(𝑥 + 𝑦)24𝑡 . (11)

Now

Ψ(𝑥, 𝑦, 𝑡) = √𝜌 (𝑥, 𝑦, 𝑡)𝑒𝑖𝑆(𝑥,𝑦,𝑡)

= √𝑡−1 ⋅ 𝑓([𝑥 + 𝑦]𝑡1/2 )𝑒(𝑖𝑚/ℏ)((𝑥+𝑦)2/4𝑡),

(12)

where 𝑓(𝜂) is now a numerical function. Figure 2 shows theprojection of the real part of the wave function to the 𝑥, 𝑡subspace. At small times there are clear oscillations, but atlarger times the strong damping is evident. If the time isfixed (𝑡 = 𝑡0 so to say) there are infinite number of spatialoscillations which can be interpreted as a quantum mech-anical heritage of the system.

24

6

0

0

5

1

−1

10

15

−2

t

Re[Ψ]

x

Figure 2: The projection of the real part of the wave function (12)along the x-t plane.

3. Summary and Outlook

We gave a short historical review of the interpretation anddevelopment of the Madelung equation. As second pointwe introduced (the not so well known) self-similar Ansatzwhich is a robust tool to investigate the local and globalproperties of nonlinear PDEs. Finally, the Madelung form ofthe free NLS equation was investigated and the results werecompared to the ones obtained from the linear Schrödingerequation. We think that the highly oscillatory fluid density(with the infinite number of zeroes) are a clear fingerprint ofthe quantummechanical origin of the system.We also believethat the noticeable quadratic argument of the solutions mightget an interpretation in the future. However, the presentedapproach is still evolving and works are in progress. We planto apply our method to the Madelung equations of moregeneral nonlinear Schrödinger equations like [35].

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The ELI-ALPS project (GINOP-2.3.6-15-2015-00001) is sup-ported by the European Union and cofinanced by the Euro-pean Regional Development Fund.

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