Long-time asymptotic for the derivative nonlinear Schrödinger equation with step-like initial value

2013/04/17 by Jian Xu, Xu, Jian, Engui Fan +1 · 1 citation
Physics and Astronomy · #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Physical sciences #nlin.SI

paper · pdf · doi:10.48550/arxiv.1304.4681

43 pages. arXiv admin note: text overlap with arXiv:solv-int/9701001 by other authors

arxiv created 2013/04/17 · arxiv updated 2013/04/18

Abstract

We consider the Cauchy problem for the Gerdjikov-Ivanov(GI) type of the derivative nonlinear Schrödinger (DNLS) equation: iq_t+q_xx-iq²q_x+(1)/(2)|q|⁴q=0. with steplike initial data: q(x,0)=0 for x≤ 0 and q(x,0)=Ae^-2iBx for x>0,where A>0 and B∈ \R are constants.The paper aims at studying the long-time asymptotics of the solution to this problem.We show that there are four regions in the half-plane -∞<x<∞,t>0,where the asymptotics has qualitatively different forms:a slowly decaying self-similar wave of Zakharov-Manakov type for x>-4tB, a plane wave region:x<-4t(B+√(2A²(B+(A²)/(4)))), an elliptic region:-4t(B+√(2A²(B+(A²)/(4))))<x<-4tB. The main tool is the asymptotic analysis of an associated matrix Riemann-Hilbert problem.

Citations

• Leading-order temporal asymptotics of the modified nonlinear Schrödinger equation: solitonless sector

Cited by

• Soliton gas for the derivative nonlinear Schrödinger equation: continuum dbar-problem, genus reduction and asymptotics

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