Inverted solutions of KdV-type and Gardner equations

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 ~$u_{\text{u}}(x,t)=Af(x-vt)$, where ~$A>0$. In this communication we show that if ~$u_{\text{up}}(x,t)=Af(x-vt)$ is the solution of a given nonlinear equation, then $u_{\text{down}}(x,t)=-Af(x-vt)$, 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.

8 pages

References in corpus (1)

• Superposition of Elliptic Functions as Solutions For a Large Number of Nonlinear Equations