Inverted Solutions of KdV-Type and Gardner Equations

A. Karczewskaa, P. Rozmejb aInstitute of Mathematics, University of Zielona Góra, Szafrana 4a, 65-246 Zielona Góra, Poland bInstitute of Physics, University of Zielona Góra, Szafrana 4a, 65-246 Zielona Góra, Poland

Full Text PDF

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 uu(x,t)=Af(x-vt), where A>0. In this paper we show that if uup(x,t)=Af(x-vt) is the solution of a given nonlinear equation, then udown(x,t)=-Af(x-vt), i.e. the %that is, an inverted wave is the solution of the same equation, but with the changed sign of the parameter α. 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.

DOI:10.12693/APhysPolA.140.445 topics: solitons, KdV-type equations, Gardner equation, travelling wave solutions