Weierstrass type projective Riccati equation expansion method and solutions of KdV equation

TL;DR

This paper introduces a novel method using Weierstrass elliptic functions to find explicit solutions of the KdV equation, including solitary and periodic waves, by transforming Riccati equations.

Contribution

It develops the Weierstrass type projective Riccati equation expansion method, providing new elliptic solutions and conversion formulas for the KdV equation.

Findings

New Weierstrass elliptic solutions for Riccati equations

Explicit solitary and periodic wave solutions of the KdV equation

Conversion formulas linking elliptic, hyperbolic, and trigonometric functions

Abstract

This paper presents two new Weierstrass elliptic function solutions of the projective Riccati equations and four conversion formulas for converting the Weierstrass elliptic functions to the hyperbolic and trigonometric functions. The Weierstrass elliptic function solutions to the projective Riccati equations and the conversion formulas are used to propose the called Weierstrass type projective Riccati equation expansion method. The Weierstrass elliptic function solutions, the solitary wave and the periodic wave solutions of the KdV equation are constructed by using the proposed method. The solitary wave like and the periodic wave solutions of the KdV equation are shown through some figures.