[Paper Review] Long-time asymptotics for the derivative nonlinear Schrödinger equation on the half-line

Lynnyngs Kelly Arruda, Jonatan Lenells|arXiv (Cornell University)|Feb 7, 2017

Nonlinear Waves and Solitons26 references3 citations

TL;DR

This paper derives long-time asymptotics for the derivative nonlinear Schrödinger equation on the half-line using a nonlinear steepest descent analysis of a Riemann-Hilbert problem. It shows that the solution decays as $ t^{-1/2} $, with the leading-order term $ u_a(x,t) $ vanishing to all orders at $ x=0 $ due to boundary effects encoded in the global relation, which forces the reflection coefficient to vanish at $ \lambda=0 $. This quantifies the boundary's influence on asymptotic decay near the origin, contrasting with the pure initial-value problem where decay is only $ t^{-1/2} $-type without such cancellation.

ABSTRACT

We derive asymptotic formulas for the solution of the derivative nonlinear Schrödinger equation on the half-line under the assumption that the initial and boundary values lie in the Schwartz class. The formulas clearly show the effect of the boundary on the solution. The approach is based on a nonlinear steepest descent analysis of an associated Riemann-Hilbert problem.

Motivation & Objective

• To analyze the long-time behavior of the derivative nonlinear Schrödinger equation (DNLS) on the half-line with initial and boundary data in the Schwartz class.
• To quantify the influence of the boundary on the asymptotic solution structure, particularly in the region near $ x=0 $.
• To extend the nonlinear steepest descent method from initial-value problems to initial-boundary value problems for integrable PDEs.
• To establish that the leading-order asymptotic term $ u_a(x,t) $ vanishes to all orders as $ x/t \to 0 $, due to the global relation linking initial and boundary data.

Proposed method

• Formulates the solution of the DNLS initial-boundary value problem via a matrix Riemann-Hilbert problem with jump matrices defined by four spectral functions derived from initial and boundary data.
• Introduces a new spectral parameter to simplify the structure of the Riemann-Hilbert problem and facilitate asymptotic analysis.
• Applies a nonlinear steepest descent method to deform contours and localize contributions to the jump matrix, focusing on stationary phase points and critical points in the complex plane.
• Constructs local model Riemann-Hilbert problems near essential transition points to approximate the solution in different regions of the $ (x,t) $-plane.
• Uses the global relation—a constraint linking the initial and boundary spectral functions—to show that the reflection coefficient $ r(\lambda) $ vanishes to all orders at $ \lambda=0 $, which is central to the boundary effect.
• Performs detailed asymptotic analysis in the sector $ 0 \leq x \leq Nt $, deriving the leading-order term $ u_a(x,t) $ and error bounds involving $ \ln t / t $.

Experimental results

Research questions

• RQ1 How does the presence of a boundary alter the long-time asymptotic behavior of the derivative nonlinear Schrödinger equation compared to the pure initial-value problem?
• RQ2 What role does the global relation between initial and boundary data play in shaping the asymptotic structure of the solution?
• RQ3 Why does the leading-order asymptotic term $ u_a(x,t) $ vanish to all orders as $ x/t \to 0 $, and how is this related to the boundary conditions?
• RQ4 Can the nonlinear steepest descent method be successfully adapted to initial-boundary value problems for integrable PDEs like DNLS?
• RQ5 What is the precise asymptotic expansion of the solution in the sector $ 0 \leq x \leq Nt $, and what is the rate of decay of the error term?

Key findings

• The solution of the DNLS equation on the half-line satisfies $ u(x,t) = \frac{u_a(x,t)}{\sqrt{t}} + O\left(\frac{\ln t}{t}\right) $ as $ t \to \infty $ in the sector $ 0 \leq x \leq Nt $, with $ u_a(x,t) $ explicitly given in terms of the reflection coefficient $ r(\lambda) $.
• The reflection coefficient $ r(\lambda) $ vanishes to all orders at $ \lambda = 0 $ due to the global relation, which links initial and boundary spectral data.
• As a consequence, $ u_a(x,t) = O\left(\left|\frac{x}{t}\right|^n\right) $ for every $ n \geq 1 $ as $ x/t \to 0 $, indicating enhanced decay near the boundary.
• The leading-order term $ u_a(x,t) $ vanishes at $ x = 0 $, consistent with the assumption of rapidly decaying boundary values and demonstrating the boundary's strong influence on the solution's structure.
• The error term in the asymptotic expansion is $ O(\ln t / t) $, which is smaller than the leading $ t^{-1/2} $ decay, reflecting the effectiveness of the nonlinear steepest descent method.
• In contrast to the pure initial-value problem, where $ r(\lambda) $ does not vanish at $ \lambda=0 $, the half-line problem exhibits a cancellation effect due to the boundary, resulting in faster decay near $ x=0 $.