Extended double Wronskian solutions to the Whitham–Broer–Kaup equations in shallow water

Abstract

Whitham-Broer-Kaup (WBK) equations describing the propagation of shallow-water waves, with a variable transformation, are transformed into a generalized Ablowitz-Kaup-Newell-Segur system, the bilinear forms of which are obtained via the rational transformations. Employing the matrix extension and symbolic computation, we derive types of solutions of the WBK equations through the selection of different canonical matrices, including solitons, rational solutions, and complexitons. Furthermore, dynamic properties of the solutions are discussed graphically and a novel phenomenon is observed, i.e., the coexistence of the elastic-inelastic interactions without disturbing each other.

Numerical Solution of Dispersive Optical Solitons with Schrödinger‐Hirota Equation by Improved Adomian Decomposition Method

“…Nonlinear Schrodinger equations (NLSE) play a significant role in the fiber optics, optical soliton, prorogation of pulses in metamaterials and network engineering applications among others [1,2]. Thus, different forms of NLSE exist and serve for a variety of purposes mostly in communication and networking engineering including, for example, the transcontinental and trans-oceanic data transfer requiring phase modulation [3] and beyond; see [4][5][6][7][8][9][10][11][12][13][14][15][16] for different mathematical studies on these equations. Furthermore, due to their immense application, many analytical methods and few numerical methods have been proposed in the past decades.…”

Elastic–inelastic-interaction coexistence and double Wronskian solutions for the Whitham–Broer–Kaup shallow-water-wave model

“…3(a) and (b) display a novel phenomenon for Model (1) that the elastic and inelastic interactions occur simultaneously during the interaction [44][45][46]. That is to say, in addition to the elastic interaction of the soliton solutions with the increase of t, the fission of the localized coherent structures for the fields u and v can also be found.…”

“…To build a bridge between Model (1) and the Ablowitz-Kaup-Newell-Segur (AKNS) systems, we will employ the following transformations [44,45],…”

“…Hereby, f, g and h are differentiable functions about x and t, and the operators D t , D 2 x and D 2 t are the bilinear derivative operators [44][45][46] …”

New multi-soliton solutions of Whitham-Broer-Kaup shallow-water-wave equations

“…In 1971, Hirota proposed a direct method [7] for constructing multi-soliton solutions of non-linear PDE. Since put forward by Hirota, Hirota's bilinear method has developed to a systematic method [8] for multi-soliton solutions [9][10][11][12][13][14][15][16][17][18][19]. In this paper, we shall extend Hirota's bilinear method to new and more general Whitham-Broer-Kaup (WBK) equations with arbitrary constant coefficients ( 1, 2, , 6) i…”