1-Multisoliton and other invariant solutions of combined KdV - nKdV equation by using symmetry approach

TL;DR

This paper uses Lie symmetry methods to analyze the combined KdV-nKdV equation, deriving invariant solutions including multisoliton, wave, and singular solutions, thereby expanding understanding of its integrability and solution structure.

Contribution

It introduces a symmetry-based approach to find explicit invariant solutions of the combined KdV-nKdV equation, including multisoliton solutions expressed via Weierstrass Zeta functions.

Findings

Derived eight invariant solutions, including multisoliton and wave solutions.

Identified two solutions as progressive waves.

Found five solutions as singular solutions.

Abstract

Lie symmetry method is applied to investigate symmetries of the combined KdV-nKdV equation, that is a new integrable equation by combining the KdV equation and negative order KdV equation. Symmetries which are obtained in this article, are further helpful for reducing the combined KdV-nKdV equation into ordinary differential equation. Moreover, a set of eight invariant solutions for combined KdV-nKdV equation is obtained by using proposed method. Out of the eight solutions so obtained in which two solutions generate progressive wave solutions, five are singular solutions and one multisoliton solutions which is in terms of WeierstrassZeta function.