A Search Method for Hirota Bilinear Systems of Nonlinear Evolution Equations

TL;DR

This paper introduces a systematic method for deriving Hirota bilinear systems from nonlinear evolution equations, exemplified on the nonlinear Schrödinger equation, facilitating the generation of solution classes like solitons and breathers.

Contribution

The paper presents a novel systematic approach to identify Hirota bilinear systems from solutions, enabling efficient classification and generation of solution families for nonlinear evolution equations.

Findings

Multiple solution-dependent bilinear systems can correspond to the same nonlinear equation.

All solutions within a class share a common bilinear system.

The method is successfully applied to a nonintegrable NLSE with dual nonlinearity and external potential.

Abstract

We present a systematic search method for finding Hirota bilinear systems of nonlinear evolution equations, with emphasis on the nonlinear Schr\"odinger equation (NLSE). Using a known exact solution, couplings between the different terms of the differential equation are identified, which are then used to derive the bilinear system. We show that a nonlinear evolution equation may have many solution-dependent Hirota bilinear systems. Nonetheless, all solution members of a certain solution class are associated with a single bilinear system. This has been demonstrated for the known solutions of the NLSE. For instance, all solution members of the N-bright soliton solutions class lead to the same bilinear system. Similarly, the class of N-dark soliton solutions and the class of breathers solutions are associated with their own distinctive bilinear systems. Identifying the bilinear system of a known seed solution will thus help in finding the rest of the solution members of the same class using the Hirota method. We apply our method to the nonintegrable NLSE with dual nonlinearity and external potential.