Inverted Solutions of KdV-Type and Gardner Equations

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 paper 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), i.e. the inverted wave is the solution of the same equation, but with the changed sign of the parameter alpha. This property is common for Korteweg-de Vries equation, extended Korteweg-de Vries equation, fifth-order Korteweg-de Vries equation, Gardner equation, and their generalizations for cases with an uneven bottom.