New exact superposition solutions to KdV2 equation

TL;DR

This paper derives new exact periodic solutions for the KdV2 equation, an extended model of shallow water waves, expanding the set of known solutions beyond solitonic and single elliptic function forms.

Contribution

The paper introduces novel exact solutions to the KdV2 equation expressed as combinations of Jacobi elliptic functions, enriching the solution space for this extended wave model.

Findings

New periodic solutions involving elliptic functions derived

Solutions complement existing solitonic and elliptic function solutions

Enhances understanding of wave behaviors in shallow water models

Abstract

New exact solutions to the KdV2 equation (known also as the extended KdV equation) are constructed. The KdV2 equation is a second order approximation of the set of Boussinesq's equations for shallow water waves which in first order approximation yields KdV. The exact solutions ~$\frac{A}{2}\left(\dn^2[B(x-vt),m]\pm \sqrt{m}\,\cn [B(x-vt),m]\dn [B(x-vt),m]\right)+D$~ in the form of periodic functions found in the paper complement other forms of exact solutions to KdV2 obtained earlier, i.e., the solitonic ones and periodic ones given by a single $\cn^2$ or $\dn^2$ Jacobi elliptic functions.