Construction of solutions and asymptotics for the sine-Gordon equation in the quarter plane

TL;DR

This paper constructs solutions to the sine-Gordon equation in the quarter plane using Riemann-Hilbert techniques and derives detailed asymptotic formulas for large space-time, revealing sector-based behaviors including solitons and radiation.

Contribution

It introduces a method to construct solutions from spectral data and provides comprehensive asymptotic analysis in different sectors of the quarter plane.

Findings

Solutions approach integer multiples of 2π at infinity

Asymptotic behavior characterized by four sectors

Explicit formulas for soliton-radiation interactions

Abstract

We consider the sine-Gordon equation in laboratory coordinates in the quarter plane. The first part of the paper considers the construction of solutions via Riemann-Hilbert techniques. In addition to constructing solutions starting from given initial and boundary values, we also construct solutions starting from an independent set of spectral (scattering) data. The second part of the paper establishes asymptotic formulas for the quarter-plane solution $u(x,t)$ as $(x,t) \to \infty$. Assuming that $u(x,0)$ and $u(0,t)$ approach integer multiples of $2\pi$ as $x \to \infty$ and $t \to \infty$, respectively, we show that the asymptotic behavior is described by four asymptotic sectors. In the first sector (characterized by $x/t \geq 1$), the solution approaches a multiple of $2\pi$ as $x \to \infty$. In the third sector (characterized by $0 \leq x/t \leq 1$ and $t|x-t| \to \infty$), the solution asymptotes to a train of solitons superimposed on a radiation background. The second sector (characterized by $0 \leq x/t \leq 1$ and $x/t \to 1$) is a transition region and the fourth sector (characterized by $x/t \to 0$) is a boundary region. We derive precise asymptotic formulas in all sectors. In particular, we describe the interaction between the asymptotic solitons and the radiation background, and derive a formula for the solution's topological charge.