Higher Order Solitary Wave Solutions of the Standard KdV Equations

Abstract

Considered under their standard form, the fifth-order KdV equations are a sort of reading table on which new prototypes of higher order solitary waves residing there, have been uncovered and revealed to broad daylight. The mathematical tool that made it possible to explore and analyze this equation is the Bogning-Djeumen Tchaho-Kofané method extended to the new implicit Bogning' functions. The analytical form of the solutions chosen in this manuscript is particular in the sense that it contains within its bosom, a package of solitary waves made up of three solitons, especially, the bright type soliton, the hybrid soliton and the dark type soliton which we estimate capable in their interactions of generating new hybrid or multi-form solitons. Existence conditions of the obtained solitons have been determined. It emerges that, these existence conditions of the chosen ansatz could open the way to other new varieties of fifth-order KdV equations including to which it will be one of the solutions. Some of the obtained solitons are exact solutions. Intense numerical simulations highlighted numerical stability and confirmed the hybrid character of the obtained solutions. These results will help to model new nonlinear wave phenomena, in plasma media and in fluid dynamics, especially, on the shallow water surface.

Dispersive analytical wave solutions of the strain waves equation in microstructured solids and Lax’ fifth-order dynamical systems

“…In the past some decades, novel exact solutions may help to find new phenomena. A variety of powerful techniques, such as inverse scattering scheme, [1,2] Hirota bilinear tranformation [3,4], the tanh-sech method [5][6][7][8], sine-cosine method [9,10], Expfunction method [11][12][13][14] and ¢ G G ( ) expansion methods [15][16][17][18][19][20], the Lie group symmetry method [21], the homotopy analysis scheme [22,23], the first integration technique [24],the theta function method [25,26], the homogeneous balance method [27], the Jacobi elliptic function method [28,29], the Adomian decomposition method [30] and Some new and important developments for searching for analytical solitary wave solutions for NLPDEs as [31][32][33][34][35][36][37][38][39][40][41] were used to develop nonlinear dispersive and dissipative nonlinear wave problems.…”