An integrable generalization of the sine-Gordon equation on the half-line

TL;DR

This paper extends the sine-Gordon equation to a more general form on the half-line and employs the Fokas transform method to solve the initial-boundary value problem through a Riemann-Hilbert problem.

Contribution

It introduces a new integrable generalization of the sine-Gordon equation and applies the Fokas transform method to construct solutions from initial and boundary data.

Findings

Solution constructed via a 2x2 matrix Riemann-Hilbert problem

Method applicable to initial-boundary value problems for integrable PDEs

Provides a framework for analyzing generalized sine-Gordon equations

Abstract

We analyze a generalization of the sine-Gordon equation in laboratory coordinates on the half-line. Using the Fokas transform method for the analysis of initial-boundary value problems for integrable PDEs, we show that the solution $u(x,t)$ can be constructed from the initial and boundary values via the solution of a $2\times 2$-matrix Riemann-Hilbert problem.