Inelastic interactions and double Wronskian solutions for the Whitham–Broer–Kaup model in shallow water

Abstract

Under investigation in this paper is the Whitham–Broer–Kaup (WBK) model for the dispersive long wave in shallow water. Connection between the WBK model and a second-order Ablowitz–Kaup–Newell–Segur (AKNS) system is revealed. By means of the Darboux transformation for the second-order AKNS system, the multisoliton solutions in terms of the double Wronskian determinant for the WBK model are derived. Inelastic interactions are graphically discussed. Our results could be helpful for interpreting certain nonlinear wave phenomena in shallow water.

Bright N-Soliton Solutions to the Vector Hirota Equation from Nonlinear Optics with Symbolic Computation

“…We have also analyzed some algebraic properties of the (m +1)-component Wronskians in (16) and (17), and derived the asymptotic formulae in (26) of (21) as Z → ∓∞ for any given set of spectral parameters {λ k } N k=1 . Furthermore, we have revealed some properties for the twoand three-soliton collisions, including the asymptotic patterns of (21) with N = 2 and 3 (see Tables 1 and 2), parametric conditions in (28) and (32) for the preservation of all soliton components in the collision, phaseshift formulae in (29), (33), and (34) induced by the vector-soliton collisions, and generalized LFTs in (30), (35)- (37) for directly describing the state changes undergone by the colliding vector solitons.…”

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

“…where A and B are taken to be the real differentiable functions of x and t. Through a direct substitution to (1), we obtain the equations as follows [12]:…”

“…Our result is novel, different from those in Refs. [6,12,13], which display the complete elastic or inelastic interactions between/among the solitons.…”

“…For (1), people have obtained the solitons [6], traveling wave solutions [7,8], periodic solutions [9], rational solutions [10], and complexitons [10,11]. They have also investigated the following soliton-evolution properties between the solitons of (1) [6,12,13]: the head-on and overtaking interactions for three soliton-kink solutions [6], pure elastic interactions of single solitons [12], and the fission interaction for one soliton-kink solution for u (or one soliton-kink solution for v) [13]. Our purpose in this paper is to search the solitons, rational solutions, and complexitons of (1) by virtue of the matrix extension, new and different from those in the existing papers as far as we know.…”

Generalized (2+1)-dimensional Gardner model: bilinear equations, Bäcklund transformation, Lax representation and interaction mechanisms

“…By virtue of computerized symbolic computation [23,[27][28][29][30][31][32][33][34][36][37][38][39][40][41][42][43][44][45][46], the Painlevé integrability conditions of (3) among the coefficient functions have been derived [23], which reduce all the coefficient functions to be proportional only to γ (t), giving the special Painlevé-integrable case of (3) as [23] …”