Yiling YANG Engui FAN
Abstract In this paper,the authors applysteepest descent method to study the Cauchy problem for the derivative nonlinear Schrödinger equation with finite density type initial data
Keywords Derivative Schrödinger equation,Riemann-Hilbert problem,steepest descent method,Long-time asymptotics,Soliton resolution,Asymptotic stability
The study on the long-time behavior of nonlinear wave equations which is solvable by the inverse scattering method was first carried out by Manakov in 1974(see[1]).By using this method,Zakharov and Manakov gave the first result for large-time asymptotic of solutions for the NLS equation with decaying initial data(see[2]).The inverse scattering method also worked for long-time behavior of integrable systems such as KdV,Landau-Lifshitz and the reduced Maxwell-Bloch system(see[3—5]).In 1993,Deift and Zhou developed a nonlinear steepest descent method to rigorously analyze the long-time asymptotic behavior of the solution for the mKdV equation by deforming the original Riemann-Hilbert(RH for short)problem to a model one whose solution is calculated in terms of parabolic cylinder functions(see[6]).Since then this method has been widely applied to the focusing NLS equation,KdV equation,Fokas-Lenells equation,short-pulse equation and Camassa-Holm equation etc.(see[7—12]).
In recent years,McLaughlin and Miller further presented asteepest descent method which combines steepest descent with-problem rather than the asymptotic analysis of singular integrals on contours to analyze asymptotic of orthogonal polynomials with non-analytic weights(see[13—14]).When it was applied to integrable systems,thesteepest descent method also has displayed some advantages,such as avoiding delicate estimates involving Lpestimates of the Cauchy projection operators,and leading the non-analyticity in the RH problem reductions to a-problem in some sectors of the complex plane which can be solved by being recast into an integral equation and by using Neumann series.Dieng and McLaughin used it to study the defocusing NLS equation under essentially minimal regularity assumptions on finite mass initial data(see[15]).Thissteepest descent method was also successfully applied to prove asymptotic stability of N-soliton solutions to focusing NLS equation(see[16]).Jenkins et al.studied soliton resolution for the derivative NLS equation for generic initial data in a weighted Sobolev space(see[17]).Their work provided the soliton resolution property for derivative NLS equation,which decomposes the solution into the sum of a finite number of separated solitons and a radiative parts when t→ ∞.Its dispersive part contains two components,one coming from the continuous spectrum and another from the interaction of the discrete and continuous spectrum.For finite density initial data,Cuccagna and Jenkins studied the defocusing NLS equation(see[18]).Recently,we further extended this method to obtain the long-time asymptotics and the soliton resolution conjecture for some integrable systems(see[19—23]).
In this paper,we study the long time asymptotic behavior for the derivative nonlinear Schrödinger(DNLS for short)equation with finite density initial data
Much work was done on the N-soliton solutions for the DNLS equation with zero/nonzero boundary conditions on discrete spectrum by using inverse scattering transform(see[36—41]).Tsutsumi and Fukuda established the local existence of the DNLS equation for initial value q0∈Hs(R),s>3 by using a parabolic regularization(see[42]).Later,they used the first five conserved quantities of the DNLS equation to establish the global existence of solutions for q0∈H2(R)with small initial data in H1(R)(see[43]).Hayashi proved local and global existence of solutions to the DNLS equation for q0∈H1(R)with small initial data in L2(R)(see[44]).For Schwartz initial value q0(x)∈S(R),we first used Deift-Zhou steepest descent method to derive the long-time asymptotic for the DNLS equation(1.1)in soliton-free region(see[45])
Later we further investigated the long-time asymptotic for the DNLS equation(1.1)with steplike initial data(see[46]).Pelinovsky and Shimabukuro studied the existence of global solutions to the DNLS equation with the inverse scattering transform method(see[47]).Recently,generic initial data in a weighted Sobolev space defined by
where qsol(x,t;DI)is the soliton solutions of(1.1)with modulating reflectionless scattering data(1.1)(see[17,48]).
In our present paper,for finite density initial data q0(x)−q±∈H2,2(R),we applysteepest descent method to obtain the following long-time asymptotic of the DNLS equation(1.1).
where meanings of the notations(x,t),T(z)and f11are shown in Proposition 3.1,Corollary 6.2 and(8.14),respectively.Our work is different from those[17,45]in the following three aspects.Firstly,for our case with finite density initial data,the corresponding phase function and its phase points are more complicated.On the jump contour iR and R,there does not always exist phase point.And in the case that phase point absences on iR(or R),unlike usual steepest descent method to open jump contour at phase points,we open the jump contour iR(or R)at z=0.And under this method,jump contour will decay to zero,and its non-analytical component is transformed into aequation.So we do not need to consider usual paraboliccylinder model.Secondly,from characteristics of two triangular decompositions of jump matrix in the case of non-zero boundary conditions,one decomposition is used to open jump line on the whole real axis,another is used to open jump line on the whole imaginary axis.Thirdly,in the case of non-vanishing initial data,to avoid multi-valued function,we need to introduce uniformization variables.This also results in extra singularities on two branch cut points±i,which leads to some adjustments in the structure of standard matrix factorizations.
This paper is arranged as follows.In Section 2,we recall some main results on the construction process of the RH problem with respect to the initial problem of the DNLS equation(1.1)obtained in[38,41],which will be used to analyze long-time asymptotics of the DNLS equation in our paper.In Section 3,we introduce a function T(z)to define a new RH problem for M(1)(z),which admits a regular discrete spectrum and two triangular decompositions of the jump matrix.In Section 4,by introducing a matrix-valued function R(z),we obtain a mixed-RH problem for M(2)(z)by continuous extension of M(1)(z).In Section 5,we decompose M(2)(z)into a model RH problem for MR(z)and a pure-problem for M(3)(z).The MR(z)can be obtained via a modified reflectionless RH problem M(r)(see Section 6),local RH problem Mlo(z)(see Section 7)and error function E(z)(see Section 8).In Section 9,we analyze the-problem for M(3)(z).Finally,in Section 10,based on the result obtained above,a relation formula is found
from which we then obtain the long-time asymptotic behavior for the DNLS equation(1.1)via a reconstruction formula.
with ζn∈ D+and∈ D−.The distribution of Z on the z-plane is shown in Figure 2.As shown in[41],the zero zngives the breather solution of the DNLS equation with nonzero boundary conditions(NZBCs),while the zero wmgives the soliton solution.As shown in[38],there exists a constant bnsuch that
Figure 1 The domains D−,D+and boundary Σ =R∪ iR{0}.
Figure 2 Distribution of the discrete spectrum Z.The red curve is unit circle.
Figure 3 In these figure we take ξ= −4,−3,−2.6,−1.5,−1,0,respectively to show all type of Imθ.The green curve is unit circle.In the red region,Imθ>0 while Imθ=0 on the red curve.And Imθ<0 in the white region.
Figure 4 The blue curve,including R,iR and the small circles constitute Σ(1).For ζn ∈ Z Λ,we change it to jump on ∂D(ζn,ρ).In this figure,we take wmas the pole point which satisfies Imθ(wm)=0 as an example,while take znas the pole point which satisfies Imθ(zn)/=0 as an example.
Figure 5 The yellow and blue region is Ω.The red circles constitute Σ(2)together.Similarly in this figure we suppose that Imθ(wm)=0 while Imθ(zn)=0.
Figure 6 Figure(a)and(b)are corresponding to the ξ> −1 and ξ< −3,respectively.There are four stationary phase points ξ1,···,ξ4with ξ1= −ξ4==−.
The long-time asymptotic of RHP 0 is affected by the growth and decay of the exponential function e±2itθappearing in both the jump relation and the residue conditions.Therefore,in this section,we introduce a new transform M(z)→M(1)(z),which make that the M(1)(z)is well behaved as t→∞along any characteristic line.
In this section,we make continuous extension for the jump matrix V(1)to remove the jump from Σ.Besides,the new problem is hoped to take advantage of the decay/growth of e2itθ(z)for zΣ.For this purpose,we introduce some new regions and contours relayed on ξ.
4.1 For the region ξ ∈ (−3,−1)
4.2 For the region|ξ+2|>1
These intervals are shown in Figure 6.Then Σjk,and Ijkcommon constitute the region Ωjkas boundary.And Σktogether with iR constitute the region Ωkas boundary when ξ> −1 while Σktogether with R constitute the region Ωkas boundary when ξ< −3.These contours separate complex plane C into sectors.In addition,let
Figure 7 The yellow and blue region is Ω.The red circle around the poles and Σ11constitute Σ(2)together.
Figure 8 Jump contours Σ(0)of Mlo(z).The figures(a)and(b)are corresponding to the cases−1< ξ and ξ< −3,respectively.
Figure 9 The contour Σpcin case ξ> −1 and ξ< −3,respectively.
Figure 10 The jump contour Σ(E)for the E(z;ξ).The blue circles are U(ξ).
Remark 6.2 The N(Λ)-solution is not N(Λ)-soliton solution.Because when the discrete spectrum ζnis not on unit circle,it corresponds to breather solution,while when the discrete spectrum ζnis on unit circle,it corresponds to soliton solution.Suppose that the discrete spectrum only distributes on unit circle,then it corresponds to pure soliton solution.We will show that under this assumption,through Beal-Cofiman theorem,the N-soliton can be expressed asymptotically as a sum(adjusted for boundary conditions)of N simple solitons.
6.1 Error estimate between M(r)(z)and (z)
Now we consider the asymptotics behavior of M(3)(z).The∂-problem 4 of M(3)(z)is equivalent to the integral equation
The long time asymptotic expansion(10.8)—(10.9)shows the soliton resolution of for the initial value problem of the derivative nonlinear schrödinger equation,which can be characterized with an N(Λ)-solution whose parameters are modulated by a sum of localized soliton-soliton interactions.Our results also show that the poles on curve soliton solutions of the derivative Schrdinger equation has dominant contribution to the solution as t→ ∞.