Solitary wave solutions of the higher order KdV equations available via a direct method

Abstract:
The famous Korteweg–de Vries (KdV) equation arises in many physical contexts as an equation governing weakly nonlinear long waves when nonlinearity and dispersion are in balance at leading order. If higher order nonlinear and dispersive effects are of interest, then the asymptotic expansion can be extended to the higher orders in the wave amplitude which leads to the higher-order KdV equations. In this paper, a modification of the direct method, designed specifically for constructing solitary wave solutions of evolution equations, is developed and applied to a general class of scalar higher-order (fifth-order) KdV equations (defined, for example, in [1]). New types of exact and explicit solitary wave solutions of the higher-order KdV equations found using this method are presented. In particular, a new type of solutions of the standard KdV equation is defined. It describes solitons propagating from the boundary at which a constant value of the variable is maintained.

1. Kichenassamy S. and Olver P.J., Existence and nonexistence of solitary wave solutions to higher-order model evolution equations, SIAM J. Math. Anal. 23 (1992), 1141–1166.