A RIEMANN-HILBERT APPROACH TO THE INITIAL-BOUNDARY PROBLEM FOR DERIVATIVE NONLINEAR SCHRÖDINGER EQUATION

doi:10.1016/S0252-9602(14)60063-1

Abstract

Abstract:

We use the Fokas method to analyze the derivative nonlinear Schr¨odinger (DNLS) equation iq_t(x, t) = −q_xx(x, t)+(rq²)_x on the interval [0, L]. Assuming that the solution q(x, t) exists, we show that it can be represented in terms of the solution of a matrix Riemann-Hilbert problem formulated in the plane of the complex spectral parameter ξ. This problem has explicit (x, t) dependence, and it has jumps across {ξ ∈ C|Imξ⁴ = 0}. The relevant jump matrices are explicitely given in terms of the spectral functions {a(ξ), b(ξ)}, {A(ξ), B(ξ)}, and {A(ξ), B(ξ)}, which in turn are defined in terms of the initial data q₀(x) = q(x, 0), the boundary data g₀(t) = q(0, t), g₁(t) = q_x(0, t), and another boundary values f₀(t) = q(L, t), f₁(t) =q_x(L, t). The spectral functions are not independent, but related by a compatibility condition, the so-called global relation.

Key words: Riemann-Hilbert problem, DNLS equation, global relation, finite interval, initial-boundary value problem

CLC Number:

35G31

XU Jian, FAN En-Gui. A RIEMANN-HILBERT APPROACH TO THE INITIAL-BOUNDARY PROBLEM FOR DERIVATIVE NONLINEAR SCHRÖDINGER EQUATION[J].Acta mathematica scientia,Series B, 2014, 34(4): 973-994.

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