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

Bilinearization and new soliton solutions of Whitham-Broer-Kaup equations with time-dependent coefficients

“…In the second example, we select γ 1 (t) = 1, γ 2 (t) = 1, γ 3 (t) = β, a 0 = −2 α + β 2 , k 1 = 4, l 1 = −2 and set the integration constants of (3.16) and (3.17) as −θ 13 /6 √ 2, then the one-soliton solutions (3.18) and (3.19) give the known solutions [27] where A and B satisfy the constant-coefficient AKNS equations In view of (2.11) and (3.1), we reduce (3.2) and (3.3) as In the procedure of extending Hirota's bilinear method to (1.1) and (1.2), one of the key steps is to reduce (1.1) and (1.2) to the bilinear forms (2.9) and (2.10) by the transformations (2.7), (2.8) and (2.11). It is graphically shown that the dynamical evolutions of one-soliton solutions (3.18) and (3.19), two-soliton solutions (3.20) and (3.21), three-soliton solutions (3.22) and (3.23) possess time-varying amplitudes as Serkin et al [37][38][39] reported in the process of propagations.…”

“…Arshad et al [2] obtained traveling wave solutions in the form of solitons, bell and anti-bell periodic, bright and dark solitary wave by applying a modified extended direct algebraic method. Lin et al [27] obtained multi-soliton solutions by means of Wronskian technique and symbolic computation.…”

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

“…It should be noted that eqs. (1) and (2) are more general than the following known WBK model for the dispersive long waves in shallow water [17][18][19]:…”

“…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…”

<i>N</i>-Fold Darboux Transformation of the Jaulent-Miodek Equation

“…Figure 3). This phenomenon has been described in the Whitham-Broer-Kaup shallow-water-wave model [16]. It seems to be new for the JM Equation.…”