On the Marchenko System and the Long-time Behavior of Multi-soliton Solutions of the One-dimensional Gross-Pitaevskii Equation

MOHAMAD Haidar

Departement de Mathematiques,Universite Paris-Sud 11,F-91405 Orsay Cedex.

On the Marchenko System and the Long-time Behavior of Multi-soliton Solutions of the One-dimensional Gross-Pitaevskii Equation

MOHAMAD Haidar?

Departement de Mathematiques,Universite Paris-Sud 11,F-91405 Orsay Cedex.

.We establish a rigorous well-posedness results for the Marchenko system associated to the scattering theory of the one dimensional Gross-Pitaevskii equation (GP).Under some assumptions on the scattering data,these well-posedness results provide regular solutions for(GP).We also construct particular solutions,calledN-soliton solutions as an approximate superposition of traveling waves.A study for the asymptotic behaviors of such solutions whent→±∞is also made.

Non-linear Schr¨odinger equation;Gross-Pitaevskii equation.

1 Introduction

The nonlinear Schr¨odinger equation(NLS)is given by

wheref:R+→R is some smooth function andN∈N.We often complete this equation by the boundary condition at infinity

where the constantρ0≥0 is such thatf(ρ0)=0.Equation(NLS)is called focusing whenf′(ρ0)＞0 and defocusing whenf′(ρ0)＜0.

These equations are widely used as models in many domains ofmathematical physics, especially in non-linear optic,superfluidity,superconductivity and condensation of Bose-Einstein(see for example[1–3]).In non-linear optic case,the quantity|u|2corresponds to a light intensity,in that of the Bose-Einstein condensation,it corresponds to a density of condensed particles.

In what follows,we extend by Gross-Pitaevskii equation the one-dimensional nonlinear Schr¨odinger equation with cubic nonlinearity,namely

and for whichρ0=1.

Multi-soliton solutions

In the early seventies,Shabat and Zakharov discovered an integrable system for the onedimensional cubic Schr¨odinger equation.They also proposed a semi-explicit method of resolution through the theory of inverse scattering.Their results were presented in two short articles,the first one is devoted to the focusing case[4]and the second one is devoted to the defocusing case(i.e.to the Gross-Pitaevskii equation(GP))[5].A similar structure was discovered a few years earlier by,Green,Kruskal and Miura[6]for Korteweg-de Vries equation

These equations have in common special solutions calledsolitons,which play an important role in dynamics in the long time.In the following,we callsolitona progressive wave solution,i.e.a solution of the form

whereUis the wave profile andcits speed.For the Gross-Pitaevskii equation(GP),such solutions(of finite Ginzburg-Landau energy except for trivial constant solutions)exist if and only if the speedcsatisfiesIn this case,for a fixed speed,the profile is unique modulo the invariants of the equation:specifically

whereUcis explicitly given by the formula

andθ0,x0∈R are arbitrary and reflect the invariance by renormalization of the phase and translation.

We will prove the following result:

Theorem 1.1.Letbe arbitrary andfixed.There exists a smooth solution u for the Gross-Pitaevskii equation(GP)on wholeR×Rsuch that

for all x∈Rand k∈{1,···,N},with

In other words,in each coordinate system in translation with speedthe solutionubehaves asymptotically whenlike the solitontranslated appropriately,and the translation factors whengiven bycan be interpreted as a shift of binary collisions between the solitons.A similar result for the focusing equation was described in[4],but the equivalence of Theorem 1.1 has not been proved by[5].

The solutions of Theorem 1.1 are obtained via an integrable system??More exactly,through a family of integrable system depending on two real variables.called Marchenko system.

1.Nreal numbers

2.Nreal strictly negative numbersμ1,...,μN.

Finally,we define the functions of real variable

Proposition 1.1.Let n≥0be such that

There exists a constant ?0＞0,depending on N,n,λiandμi,such that if

then for every x∈Rthe Marchenko system(with parameter x)

has a unique solutionMoreover,the function

belongs to the space

The well-posedness in the previous proposition are obtained by perturbation of the caseβ≡0,which corresponds to the exact multi-solitons solutions,and for which the result is completely of algebraic nature.

We refer the reader to[7]and[8]for more details about the derivation of Marchenko system via the scattering theory associated to(GP).We focus in this paper on the other direction where the link with the Gross-Pitaevskii equation is obtained by the following proposition which introduces an additional time parametertin Marchenko system:

Proposition 1.2.Assume that

where theare assumed to be strictly negative and the function c satisfies assumptions(1.3)above.Then ifthe quantityis small enough,for everythe Marchen-ko system(with fixed parameters x and t)has a unique solutionsuchMoreover,if n≥3,the function

is solution of the Gross-Pitaevskii equation inC1(R2)such that u(t,.)∈C2(R)for all t∈R.Finally,if the νkare pairwise distinct and c is a real-valued function,for t∈Rfixed,we have

where uNis the solution corresponding to(the other coefficients λkandare left equal).

The solutions mentioned in Theorem 1.1 are exactly the above solutionsuN.The link between the speedsckof Theorem 1.1 and the coefficientsλkof Marchenko system is given by the relationshipsck=2λk.

2 N-soliton solutions

The purpose of this section is the resolution of Marchenko system whenβ(λ)=0.We set

1.Nreal numbers

2.Nnegative real numbersμ1,μ2,...,μN.

In this case,the Marchenko system is rewritten as follows

This means that Υ1and Υ2take the forms

and the two Eqs.(2.1)and(2.2)become

Identifying the terms with factors exp(?νky),we get

Assumingxas a parameter,the 2NEqs.(2.3)and(2.4)correspond to linear system with 2Nequations and 2Nunknowns.To analyze its resolution,we set

so that(2.3)and(2.4)can be rewritten in the matrix form:

Proposition 2.1.The matrix(I+T)(x)is invertible for every x∈R.

Proof.The idea of this proof is to demonstrate that?1 is not an eigenvalue forT.To this end,define

hence we have the following equivalence

3 Resolution of Marchenko system:proofs of Propositions 1.1 and 1.2

3.1The case with no time dependence

The solution corresponding towill be obtained by perturbation of theN-soliton solution constructed in the previous section.Denotethe solution of Marchenko system(1.4)corresponding toand the same values of the other scattering coeff icients,i.e.λiandμi.Define

The Marchenko system(1.4)is thus transformed into a system for

Lemma 3.1.The operator?xis positive independently of x.Namely,

Proof.In view of the definition ofF11andF21,we have

Combining(3.4)and(3.5),we obtain

Sinceμkare assumed negative,the sum of the last two terms of the right-hand side of (3.6)is upper-bounded by

which implies thatitis an isomorphism onThis allows us to establish the following corollary:

Corollary 3.1.For every k∈N,the operatoris an isomorphism ofMoreover,we have

Proof.LetWe know that there exists a uniquesuch that

In view of the definition of ?x,we havehence

Eq.(3.1)is equivalent to

Forx∈R,we use the fixed point argument to prove that(3.8)has a unique solutionTo this end,the following lemma will be needed:

Lemma 3.2.Let x∈R.The functionFdefined above is Lipschitz continuous onand there exists a constant C＞0depending on β such that for allwe have

Proof.LetFor everywe have

whereF?1is the inverse Fourier transform.This implies that the left hand side members of previous equations belong toL2(R)and thatApplying the Fourier transform on(3.11)and taking theL2(R)norm,we obtain

A similar argument yields the same inequality from(3.10).This completes the proof.

3.2Variation for Proposition 1.2:addition of time dependence

Assume that

is the unique solution of the corresponding Marchenko system.The dependence ontof the two operators ?xand F does not change the previous results.More specifically,the operator ?xstays non-negative independently of(t,x)∈R2.This allows us to establish a similar corollary to Corollary 3.1.On the other hand,we have

3.3Regularity with respect toxandtof Marchenko system solutions

To give sense to such regularity,the following reformulation

which makes the domain of variables fixed,will be useful.Eq.(3.1)can be rewritten as follows

where Txis the linear part of the operator F.Namely,

Lemma 3.3.Assume that n≥3.Let k be an integer such that k＜n/2.The operatorMx(t)∈L((Hn?2k(R+))2)is contraction independently of t,x.Moreover,The application

is of classCk(R2).

Proof.The contraction property of Mx(t)is a result of the fact thatis contraction independently of(t,x).The two applicationsare of classesrespectively.In fact,we haveThen there existsC＞0 such that for alltheL2(R+)norm of the term

is upper-bounded by

is of class Ck(R2).

In view of Lemma 3.3,K is of class Ck.Finally,for all(t,x)∈R2,we have

3.4Construction of the Gross-Pitaevskii solution

In this section it will be proved that the function

is a solution of Gross-Pitaevskii equation.To this end,the following lemma will be useful

Lemma 3.4.Let(Ψ1,Ψ2)tthe solution of Marchenko system under the assumptions of Proposition1.2with n≥3.Denote

LetWeset

Thenis a solution of

Proof.The proof depends essentially on the well-posedness of Marchenko system and the choice of the function

Sincen≥3,we have the following regularityfor allt∈R.To prove thatψis a solution of(3.13),it suffices to prove that Ψ satisfies the linear system

In fact,replacingQbyQ,Xsatisfies the system(3.13).Hence,we have

A simple calculation proves thatand sincewe haveTo verify that Ψ satisfies(3.15),we set

Then by direct calculation we find thatsatisfies the homogeneous Marchenko system

which has a unique trivial solutionC=0.This proves that Ψ satisfies the linear system (3.15).To prove(3.14),denoteThen(3.14)is equivalent to

Sinceψis a solution of(3.13),we have

Substituting into the expression ofg,we find that

Next,note that the definition ofψimplies that

Hence,Substituting into(3.17),integrating by parts and using(3.15),we obtain

On the other hand,we have

Then to prove(3.16),it suffices to prove that for every

To this end,we verify that for everythe two functionssatisfy the same Marchenko equations in the spaceby using the relationships

which are provided by the definition ofF1andF2.Differentiating(1.4)(in its version depending ont)with respect tot,we have for everyy≥x

On the otherhand,differentiating(1.4)with respect toxand with respect toy,integrating by parts and using(3.15),we obtainConsequently,for everythe two functionssatisfy the same Marchenko equations in the spaceThus,the conclusion follows from the uniqueness.

Sinceψis a solution of(3.13),χis a solution of

On the other hand,we have

Combining the last two equations,we find that

Hence,we have

This proves that

anduis therefor a solution of Gross-Pitaevskii equation.

3.5Behavior of the constructed solution whenx→±∞

It was proved before that the solution Υ of Marchenko system corresponding toβ≡0 is of the form

We have the following lemma

Lemma 3.5.If νkare pairwise distinct,the functions

are bounded onR2.

Proof.DenoteThen developing(2.3)and(2.4),we obtain

We rewrite the previous system in the form of the following two linear systems

It follows from the equivalence of the norms onthat there existsC＞0 such that

Thus,since the functionsakare non-negative,we find,using the CNusual scalar product in(3.20)for everythat

and for every(t,x)∈R2,

This completes the proof.

Denote nowuN(t,x)the solution of(GP)corresponding toβ≡0.We know that

Lemma 3.6.Under the assumptions of Proposition1.2,for fixed t∈R,we have

Proof.For fixedIn fact,we have

We need now to calculate the limit,for fixed(t,x)∈R2,of right hand side member of de (3.22)whenxtends to+∞.We start by the term

The dominated convergence theorem implies thatSimilarly, we find thatFinally,we have

In view of equivalence of norms??It suf fices to take into account the two normsde fined as follows§Sincecis assumed to be realfunction,we havethenF22F12are real-valued functions andis Hermitian.onwe have

where we obtain

Taking the limit in(3.23)and(3.24)prove that

On the other hand,we have

Then we finally get

This completes the proof.

Received 34 March 2015;Accepted 28 May 2015

?Corresponding author.Email address:haidar.mohamad@math.u-psud.fr(H.Mohamad)

AMS Subject Classifications:35Q55

Chinese Library Classifications:O175.29