4 Asymptotic of N-soliton solutions in long time:proof of Theorem 1.1

4 Asymptotic of N-soliton solutions in long time:proof of Theorem 1.1

We present in this section the proof of Theorem 1.1,which,in view of Proposition 1.2, focuses on the analysis of solutions constructed in Section 2.Keeping the notations introduced in Section 2,we already proved that(I+T)F=Cand that

Hence,we have

Consequently,we have

Before discussing the general case ofN-solitons,we discuss the two simple cases whereN=1 andN=2.ForN=1,a simple calculation proves that

Then the 1-soliton solution corresponding to data{λ,μ(0)}is

The 2?soliton solution,u2,corresponding to choicesis

Taking into account the frame(in translation)with speed 2λ2defined by the variablewe have

i.e.the solutionu2behaves(asymptotically whent→±∞)as a progressive wave of speed 2λ2.

Now let us come back to the general case(N∈N),for which we can not establish an explicit formula for the solutionu(t,x).To this end,we rewrite

Hence,we need to study the behavior of the sumTo this end rewrite the system (4.1)as follows

where

The system(4.4)can be rewritten as follows

which implies

Letk0∈{1,...,N}.We define the frame(in translation)with speed 2λk0by the variable

We will analyze the asymptotic in long time.Note that

The system(4.5)can be transformed into

We take the limit whent→?∞for fixedη.This leads,denotingto

We introduce the matrices

where theKlkrefer to the cofactors ofK.Fork=k0,we also have

Summing the last equation onk,we obtain

Thus,we finally get

Proposition 4.1.There existsuch that

where U is the function defined in(4.2)which corresponds toμk0(0)and λk0.

Proof.Recall first that

In what concerns(i),remark thatand the fact thatfollows from the following lemma,since

Lemma 4.1.Let M∈MN(C)and JNthe matrix whose coefficients are ones.Denote K=M+XJ. Then

where the Kijrefer to the cofactors of K.

Proof.LetU,V∈CNbe two column vectors.Then

where〈,〉is the usual scalar product in CN.In fact,it suffices to expand the operatorT=I+U.Vtin the basis composed ofVand the basis of the orthogonal toV.Assume now that the matrixK=M+U.Vtis invertible.Then we can write

whereCof(K)is the cofactor matrix ofK.The conclusion therefore follows by takingU=(1,1,...,1)tandV=(X,X,...,X)t.Finally,the caseKnon-invertible is obtained by continuity.

For equation(ii),it suffices to prove that

Subtracting columnk0from columnjin the matrixwe get

which implies??We have used the identitythat

A similar argument proves that

Combining(4.9)and(4.10)we complete the proof of(ii).

We focus now on assertion(iii).Subtracting the linek0from each of the previous lines, we find thatand using(ii),we obtain

Finally,we get

A parallel argument allows,for the limitt→+∞,to obtain

This completes the proof of Theorem 1.1.

[1]Coste C.,Nonlinear Schr¨odinger equation and superuid hydrodynamics.Eur.Phys.J.B Condens.Matter Phys.1(2)(1998),245-253.

[2]Jones C.A.,Roberts P.H.,Motions in a bose condensate v.stability of solitary wave solutions of non-linear Schr¨odinger equations in two and three dimentions.J.Phys.A,Math.Gen.19(15)(1986),2991-3011.

[3]Kevrekidis P.G.,Malomed B.A.,Cuevas J.,Frantzeskakis D.J.,Solitons in quasionedimentional Bose-Einstein condensates with competing dipolar and local interactions.Phys.Rev.A,79(5)(2009),1-11.

[4]Shabat A.B.,Zakharov V.E.,Exact theory of two-dimensional self-focusing and onedimensional self-modulation of waves in nonlinear media.Sov.Phys.JETP34(1972),62-69.

[5]Shabat A.B.,Zakharov V.E.,Interaction between solitons in a stable medium.Sov.Phys. JETP37(1973),823-828.

[6]Kruskal M.,Miura R.,Gardner C.and Greene J.,Korteweg-de-Vries equation and generalization.Comm.Pure Appl.Math.27(1974),97-133.

[7]G′erard P.,Zhang Z.,Orbital stability of traveling waves for the one-dimensional Gross-Pitaevskii equation.Math.Pures Appl.91(2)(2009),178-210.

[8]Eckhaus W.,Harten A.V.,The Inverse Scattering Transformation and the Theory of Solitons,50North-Holland publishing company,Amsterdam,1981.