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 u (x, t) = Af (x−vt), where A > 0. In this communication we show that if u up (x, t) = Af (x − vt) is the solution of a given nonlinear equation, then u 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 α. This property is common for KdV, extended KdV, fifth-order KdV, Gardner equations, and generalizations for cases with an uneven bottom.