Exact solutions to the modified Korteweg-de Vries equation

TL;DR

This paper derives explicit exact solutions for the modified Korteweg-de Vries (mKdV) equation using inverse scattering transform and matrix equations, providing new analytical tools for understanding this nonlinear PDE.

Contribution

The paper introduces a novel explicit solution formula for the mKdV equation based on matrix exponentials and solutions to Sylvester and Lyapunov equations, advancing analytical solution methods.

Findings

Explicit solutions expressed via matrix exponentials and algebraic equations.

Solutions constructed using the unique solutions to Sylvester and Lyapunov equations.

Examples demonstrating the application of the derived solutions.

Abstract

A formula for certain exact solutions to the modified Korteweg-de Vries (mKdV) equation is obtained via the inverse scattering transform method. The kernel of the relevant Marchenko integral equation is written with the help of matrix exponentials as $$\Omega(x+y;t)=Ce^{-(x+y)A}e^{8A^3 t}B,$$ where the real matrix triplet $(A,B,C)$ consists of a constant $p\times p$ matrix $A$ with eigenvalues having positive real parts, a constant $p\times 1$ matrix $B$, and a constant $1\times p$ matrix $C$ for a positive integer $p$. Using separation of variables, the Marchenko integral equation is explicitly solved yielding exact solutions to the mKdV equation. These solutions are constructed in terms of the unique solution $P$ to the Sylvester equation $AP+PA=BC$ or in terms of the unique solutions $Q$ and $N$ to the respective Lyapunov equations $A^\dagger Q+QA=C^\dagger C$ and $AN+NA^\dagger=BB^\dagger$, where the $\dagger$ denotes the matrix conjugate transpose. Two interesting examples are provided.