Long-time Asymptotic Behavior for the Derivative Schrödinger Equation with Finite Density Type Initial Data  

作　　者：Yiling YANG Engui FAN 

机构地区：[1]School of Mathematical Sciences,Fudan University,Shanghai 200433,China

出　　处：《Chinese Annals of Mathematics,Series B》2022年第6期893-948,共56页数学年刊（B辑英文版）

基　　金：supported by the National Natural Science Foundation of China(Nos.51879045,1202624,118013233,11671095)。

摘　　要：In this paper,the authors apply■steepest descent method to study the Cauchy problem for the derivative nonlinear Schrödinger equation with finite density type initial data iqt+qxx+1(lq|^(2)q)_(x)=0,q(x,0)=q0(x),where lim x→±∞ qo(x)=g0(x)=q±and|q±|=1.Based on the spectral analysis of the Lax pair,they express the solution of the derivative Schrödinger equation in terms of solutions of a Riemann-Hilbert problem.They compute the long time asymptotic expansion of the solution in differeit space-time regions.For the regionζ=x/t with|ζ+2|<1,the long time asymptotic is given by q(x,t)=T(∞)^(-2)q^(r)Λ(x,t)+O(t^(-3/4)),in which the leading term is N(I)solitons,the second term is a residual error from a■equation.For the regionζ+2|>1,the long time asymptotic is given by q(x,t)=t(∞)^(-2)q^(r)Λ(x,t)-t^(-1/2)if11+O(t^(-3/4)) in which the leading term is N(I)solitons,the second t^(-1/2)order term is soliton-radiation interactions and the third term is a residual error from a■equation.These results are verification of the soliton resolution conjectuore for the derivative Schrödinger equation.In their case of finite density type initial data,the phase functionθ(z)is more complicated that in finite mass initial data.Moreover,two triangular decompositions of the jump matrix are used to open jump lines on the whole real axis and imaginary axis,respectively.

关 键 词：Derivative Schrödinger equation Riemann-Hilbert problem ■steepest descent method Long-time asymptotics Soliton resolution Asymptotic stability 

分 类 号：O175.29[理学—数学]