Bilinear B\"acklund transformations and Lax pair for the Boussinesq equation

TL;DR

This paper derives the bilinear Bäcklund transformation and Lax pair for the Boussinesq equation, demonstrating its integrability using Hirota's method, which facilitates solution construction for soliton systems.

Contribution

The paper introduces the bilinear Bäcklund transformation and Lax pair for the Boussinesq equation, advancing the understanding of its integrability and solution methods.

Findings

Derived bilinear Bäcklund transformation for the Boussinesq equation

Obtained Lax pair from the Bäcklund transformation

Confirmed the integrability of the Boussinesq equation

Abstract

Hirota's bilinear approach is a very effective method to construct solutions for soliton systems. In terms of this method, the nonlinear equations can be transformed into linear equations, and can be solved by using perturbation method. In this paper, we study the bilinear Boussinesq equation and obtain its bilinear B\"{a}cklund transformation. Starting from this bilinear B\"{a}cklund transformation, we also derive its Lax pair and test its integrability.