The unified method for the three-wave equation on the half-line

TL;DR

This paper develops a Riemann-Hilbert problem framework to solve the initial-boundary value problem for the three-wave equation on the half-line, providing a unified approach to analyze this integrable system.

Contribution

It introduces a novel Riemann-Hilbert formalism specifically tailored for the three-wave equation with boundary conditions on the half-line.

Findings

Provides a systematic method to solve the initial-boundary value problem.

Establishes a unified analytical framework for the three-wave equation.

Facilitates future analytical and numerical studies of the system.

Abstract

We present a Riemann-Hilbert problem formalism for the initial-boundary value problem for the three-wave equation: \[p_{ij,t}-\frac{b_i-b_j}{a_i-a_j}p_{ij,x}+\sum_k(\frac{b_k-b_j}{a_k-a_j}-\frac{b_i-b_k}{a_i-a_k})p_{ik}p_{kj}=0,\quad i,j,k=1,2,3.\] on the half-line.