Inverted solutions of KdV-type and Gardner equations

TL;DR

This paper demonstrates that for KdV-type and Gardner equations, inverted wave solutions with opposite signs are also valid solutions, revealing a symmetry property across various nonlinear wave equations.

Contribution

It establishes a general symmetry property where inverted wave solutions are valid for a range of KdV-type and Gardner equations, extending previous understanding.

Findings

Inverted solutions exist for KdV, extended KdV, and Gardner equations.

The symmetry holds for equations with uneven bottom topography.

Inversion corresponds to changing the sign of a key parameter.

Abstract

In most of the studies concerning nonlinear wave equations of Korteweg-de Vries type, the authors focus on waves of elevation. Such waves have general form ~$u_{\text{u}}(x,t)=A f(x-vt)$, where ~$A>0$. In this communication we show that if ~$u_{\text{up}}(x,t)=A f(x-vt)$ is the solution of a given nonlinear equation, then $u_{\text{down}}(x,t)=-A f(x-vt)$, that is, an inverted wave is the solution of the same equation, but with changed sign of the parameter ~$\alpha$. This property is common for KdV, extended KdV, fifth-order KdV, Gardner equations, and generalizations for cases with an uneven bottom.