Long-time Asymptotic for the Derivative Nonlinear Schrodinger Equation with Step-like Initial Value

被引:60
|
作者
Xu, Jian [1 ]
Fan, Engui [2 ,3 ]
Chen, Yong [4 ]
机构
[1] Fudan Univ, Sch Math Sci, Shanghai 200433, Peoples R China
[2] Fudan Univ, Inst Math, Sch Math Sci, Shanghai 200433, Peoples R China
[3] Fudan Univ, Key Lab Math Nonlinear Sci, Shanghai 200433, Peoples R China
[4] E China Normal Univ, Shanghai Key Lab Trustworthy Comp, Shanghai 200062, Peoples R China
基金
美国国家科学基金会;
关键词
Riemann-Hilbert problem; Nonlinear Schrodinger equation; Long-time asymptotic; Step-like initial value problem; PERIODIC BOUNDARY-CONDITION; HAMILTONIAN-SYSTEMS; INVERSE SCATTERING; SOLITONS;
D O I
10.1007/s11040-013-9132-3
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
We study long-time asymptotics of the solution to the Cauchy problem for the Gerdjikov-Ivanov type derivative nonlinear Schrodinger equation iq(t) + q(xx) - iq(2)(q) over bar (x) + 1/2 vertical bar q vertical bar(4)q = 0 with step-like initial data q(x, 0) = 0 for x <= 0 and q(x, 0) = Ae(-2i Bx) for x > 0, where A > 0 and B is an element of R are constants. We show that there are three regions in the half-plane {(x, t)| - infinity < x < infinity, t > 0}, on which the asymptotics has qualitatively different forms: a slowly decaying self-similar wave of Zakharov-Manakov type for , a plane wave region: x < -4t (B + root 2A(2) (B + A(2)/4)), an elliptic region: -4t (B + root 2A(2) (B + A(2)/4)) < x < -4tB. Our main tools include asymptotic analysis, matrix Riemann-Hilbert problem and Deift-Zhou steepest descent method.
引用
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页码:253 / 288
页数:36
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