A New Class of Solutions of Combined KdV-mKdV Equation

TL;DR

This paper derives new exact solutions for the combined KdV-mKdV equation using Jacobi elliptic functions, including special cases with hyperbolic solutions and time-dependent coefficients, expanding the solution space for this nonlinear PDE.

Contribution

It introduces a novel class of solutions for the combined KdV-mKdV equation expressed as superpositions of Jacobi elliptic functions, including cases with time-dependent coefficients.

Findings

Exact solutions expressed as superpositions of Jacobi elliptic functions.

Special hyperbolic solutions when the elliptic modulus m equals one.

Solutions with time-dependent velocity coefficients.

Abstract

In this work, the exact solutions for combined KdV-mKdV generalized equation as a linear superposition of Jacobi elliptic functions, $c_n(\xi,m)$, $d_n(\xi,m)$. When $m$ is set to one, the solution matches with well-known hyperbolic solutions of generalized combined KdV-mKdV equation. Similar solution is also derived for the case for time dependent co-efficient. In the latter case the velocity term become time dependent.