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)]

Abstract

Recent studies on the water waves have been impressive. Of current interest in fluid physics, Jiang et al. [J. Math. Phys. 51, 093519 (2010)] have reported certain soliton interactions along with binary-Bell-polynomial-type Bäcklund transformation and Lax pair for the (2 + 1)-dimensional Boiti-Leon-Pempinelli system for water waves. However, the story introduced by that paper can be made more complete, since in fluid physics and other fields, the variable-coefficient models can describe many physical processes more realistically than their constant-coefficient counterparts. Hereby, on a (2 + 1)-dimensional variable-coefficient Boiti-Leon-Pempinelli generalization, water-wave symbolic computation is performed. For the horizontal velocity of the water wave as well as the wave elevation, variable-coefficient-dependent auto-Bäcklund transformation is constructed out, along with some variable-coefficient-dependent shock-wave-type solutions. Relevant variable-coefficient constraints are also given, with respect to water waves.

A system of coupled partial differential equations exhibiting both elevation and depression rogue wave modes

“…• Instead of considering waves on fluids of finite depth, dynamical systems which elucidate motion on shallow water (long waves), e.g. Kadomtsev Petviashvili and Boiti-Leon-Manna-Pempinelli equations also exhibit intriguing features regarding solitons [29,30,31].…”

Analytical multi-soliton solutions of a (2+1)-dimensional breaking soliton equation

“…In recent years, many research have studied soliton and its evolvements in nonlinear equations via kinds of method (Zarmi 2014 ; Chen and Ma 2013 ; Ashorman 2014 ; Zhang and Chen 2016 ; Meng and Gao 2014 ; Mohamed 2016 ; Liu and Liu 2016 ; Jiang and Ma 2012 ; Guo and Hao 2013 ; Dou et al 2007 ). Vertex dynamics in multi-soliton solutions and some new exact solution of breaking equation are studied in Zarmi ( 2014 ) and Chen and Ma ( 2013 ), the methods of Multi-soliton Solutions are given in Ashorman ( 2014 ), Zhang and Chen ( 2016 ), Meng and Gao ( 2014 ), Mohamed ( 2016 ), Liu and Liu ( 2016 ), Jiang and Ma ( 2012 ), Guo and Hao ( 2013 ), Dou et al ( 2007 ), Zuo and Gao ( 2014 ), Huang ( 2013 ), Liu and Luo ( 2013 ), Côtea and Muñoza ( 2014 ), Xu and Chen ( 2014 ), Hua and Chen ( 2014 ) and Zhang and Cai ( 2014 ), complex solutions for the [BLP System are proposed in Ma and Xu ( 2014 )], and these so-called new solutions is identical to the universal formula in Doungmo Goufo ( 2016 ), Atangana and Doungmo Goufo ( 2015 ), Gao ( 2015a , b , c , d ), Xie and Tian ( 2015 ), Sun and Tian ( 2015 ) and Zhen et al ( 2015 ).…”

Multisoliton Solutions and Breathers for the Coupled Nonlinear Schrödinger Equations via the Hirota Method

“…with Δ ± = Δ 1 ± Δ 2 , ± = 1 ± 2 , where 1 is the group velocity coefficient, is the parameter related to nonlinear effects, and Δ ± , ± are related to linear effects. Although the investigations into the CNLS equations have been conducted via the different methods including Darboux transform [17], the Hirota method [31,32], extended Jacobi's elliptic function method [18], Painlevé's analysis [14], and Bäcklund transformation [33][34][35][36] and many soliton solutions have been derived including one-soliton solutions, there are still several points to continue to explore as follows:…”