Hybrid solitary wave solutions of the Camassa–Holm equation

Abstract

The Camassa–Holm equation governs the dynamics of shallow water waves or in its reduced form models nonlinear dispersive waves in hyperelastic rods. By using the straightforward Bogning-Djeumen Tchaho-Kofané method, explicit expressions of many solitary wave solutions with different profiles not previously derived in the literature are constructed and classified. Geometric characterizations of the solutions in terms of three new mappings are presented. Intensive numerical simulations carried confirm the stability of the solutions even with relatively high critical velocities and reveal that solitary waves with large widths are more stable than the ones with small widths.

Hybridization of Solitary Wave Solutions in (2+1)-dimentional Complex Ginzburg-Landau Equation

“…In eq. ( 9), the respective coefficient terms A1 and A2 are hybrid solitary waves [22][23][24] while the respective coefficient terms B1 and B2 are the well known classic solitons under the names: Kink and Dark. Thereafter, the consideration of eq.…”

“…Faced with these complications, a good number of effective direct methods [14] to obtain the exact solutions of these models have been proposed, in particular the homoclinic test technique [15][16][17], Exp-function method [18], the F-expansion method [19], the extended tanhfunction method [20; 21], and so on. Hybrid solutions [22][23][24] formed by an assembly of analytical sequences of the classical solitary waves are very difficult to obtain. These last two decades have seen the birth of other techniques for solving some of the proposed models, which, apart from exact solutions, also offer approximate or forced solutions with several solitons, such as the Bogning-Djeumen Tchaho-Kofan é method (BDKm) [25][26][27][28][29][30] extended to the implicit Bogning functions(iB-functions) [29][30][31][32].…”