A Riemann theta function formula with its application to double periodic wave solutions of nonlinear equations

TL;DR

This paper introduces a method using Riemann theta functions and Hirota's bilinear form to explicitly construct double periodic wave solutions for nonlinear differential and difference equations, linking these solutions to solitons.

Contribution

It provides a unified, straightforward approach to derive periodic solutions from bilinear forms using theta functions, applicable to various nonlinear equations.

Findings

Successfully constructs double periodic solutions for multiple nonlinear equations.

Establishes rigorous relations between periodic wave solutions and soliton solutions.

Demonstrates efficiency of the method on water wave, Bogoyavlenskii-Schiff, and differential-difference KdV equations.

Abstract

Based on a Riemann theta function and Hirota's bilinear form, a lucid and straightforward way is presented to explicitly construct double periodic wave solutions for both nonlinear differential and difference equations. Once such a equation is written in a bilinear form, its periodic wave solutions can be directly obtained by using an unified theta function formula. The relations between the periodic wave solutions and soliton solutions are rigorously established. The efficiency of our proposed method can be demonstrated on a class variety of nonlinear equations such as those considered in this paper, shall water wave equation, (2+1)-dimensional Bogoyavlenskii-Schiff equation and differential-difference KdV equation.