Special Polynomials and Exact Solutions of the Dispersive Water Wave and Modified Boussinesq Equations

TL;DR

This paper derives exact solutions for dispersive water wave and modified Boussinesq equations using special polynomials linked to Painleve IV rational solutions, introducing generalized solutions with arbitrary constants.

Contribution

It introduces a novel method of obtaining exact solutions via special polynomials related to Painleve IV, extending the solution space with generalized solutions involving arbitrary constants.

Findings

Exact solutions expressed in terms of special polynomials.

Generalized solutions with infinite arbitrary constants.

Connections to rational solutions of Painleve IV equation.

Abstract

Exact solutions of the dispersive and modified equations are expressed in terms of special polynomials associated with rational solutions of the fourth Painleve equation, which arises as generalized scaling reductions of these equations. Generalized solutions that involve an infinite sequence of arbitrary constants are also derived which are analogues of generalized rational solutions for the Korteweg-de Vries, Boussinesq and nonlinear Schrodinger equations