Solitonic interactions and double-Wronskian-type solutions for a variable-coefficient variant Boussinesq model in the long gravity water waves

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.…”

“…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] …”

The elastic-fusion-coupled interaction for the Boussinesq equation and new soliton solutions of the KP equation

“…the fusion behavior coexists with the elastic collision. Although elastic collision and solitary waves fusion have been investigated in [5][6][7][8][9][10] by many authors recently, to the authors' knowledge, the elastic-fusion-coupled interaction is a new phenomenon for the Boussinesq equation (1). The solutions with this property could be useful to describe the run-up of a tsunami wave on a vertical wall [11].…”

Binary Bell polynomials, Hirota bilinear approach to Levi equation

“…So both the bilinear forms and bilinear Bäcklund transformations can be obtained directly. Lots of studies have devoted to the application and extension of this method [11][12][13][14][15][16][17][18][19][20][21].…”