Comment on: “Spherical Kadomtsev–Petviashvili equation and nebulons for dust ion-acoustic waves with symbolic computation” [Phys. Lett. A 340 (2005) 243]

Comment on “Solitons, Bäcklund transformation, and Lax pair for the (2 + 1)-dimensional Boiti-Leon- Pempinelli equation for the water waves” [J. Math. Phys. 51, 093519 (2010)]

“…For u(x, y,t), the horizontal velocity of the water wave, and w(x, y,t), the elevation of the water wave, variable-coefficient-dependent auto-Bäcklund transformation (5)- (7) with (10)- (15) has been constructed out, under variable-coefficient constraints (8) and (9). Variable-coefficient-dependent shock-wave-type solutions (19) and (20) have also been obtained, under variable-coefficient constraints (9), (17), and (18), with respect to the water waves.…”

“…15 for other (2 + 1) 014101 (2015) evolution, nonuniform boundaries, and/or inhomogeneous media, the variable-coefficient partial differential equations can model many dynamical processes more realistically than their constantcoefficient counterparts do. [17][18][19] We hereby propose to investigate a variable-coefficient generalization of System (1), which can be written as…”

Odd-Soliton-Like Solutions for the Variable-Coefficient Variant Boussinesq Model in the Long Gravity Waves

“…Painlevé property for System (2) has been reported under some constraints [19], and the N-solitonic solutions in terms of the Wronskian determinant has been presented by the Hirota technique [19]. Lax pair and Darboux transformations (DTs) have been constructed for System (2), by which the one-and two-solitonic solutions have been given [20].…”

“…To better understand the dynamics of the water waves in the nonuniform backgrounds and provide useful information for the coastal and civil engineers to apply the nonlinear water wave models in a harbour and coastal design, it is valuable to seek for more solutions of System (2). In this paper, we will focus on the odd-soliton-like solutions in terms of the Vandermonde-like determinant via the N-fold DT method and give the analysis on the dynamics for System (2), including the three-parallel solitonic waves, head-on collisions, double structures, and inelastic interactions.…”

“…In this paper, we will focus on the odd-soliton-like solutions in terms of the Vandermonde-like determinant via the N-fold DT method and give the analysis on the dynamics for System (2), including the three-parallel solitonic waves, head-on collisions, double structures, and inelastic interactions. To our knowledge, such results have not been reported in the existing literatures as yet.…”