Lie symmetries, invariant solutions and phenomena dynamics of Boiti–Leon–Pempinelli system

Abstract

The present work deals with highly nonlinear dispersive water equation named Boiti–Leon–Pempinelli (BLP) system, which is a well known two dimensional generalization of sinh-Gordon equation. The BLP system describes interaction of horizontal wave component propagating in plane in infinite narrow channel with constant depth. The Lie symmetry method is employed to derive infinitesimal generators under invariance conditions of one-parameter transformations. The invariance conditions ensure the reducibility of independent variables and preserve the characteristics of system. The commutative relations of infinitesimal vectors show infinite dimensional algebra of symmetries. Thereafter, similarity variables are derived using infinitesimal generators, which lead to first symmetry reductions. Thus, a repeated process of symmetry reductions provide equivalent system of ODEs. These ODEs are integrated and thus some invariant solutions of BLP system are constructed under parametric constraints. The derived solutions are general than previous researches [J Q Yu et al 2010 Exact solutions and conservation laws of (2 + 1)-dimensional Boiti–Leon–Pempinelli equation, Appl. Math. Comput. 216 2293–2300; M Kumar, R Kumar, 2014 On new similarity solutions of the boiti–leon–pempinelli system, Commun. Theor. Phys. 61 121–126; M Kumar et al 2015 Some more similarity solutions of the (2 + 1)-dimensional BLP system, Comput. Math. Appl. 70 212–221; J Fei et al 2015 Symmetry reduction and explicit solutions of the (2 + 1)-dimensional boiti–leon–pempinelli system, Appl. Math. Comput. 268 432–438; M Kumar et al 2021 Some more invariant solutions of (2 + 1)-water waves, Int. J. Appl. Comput. Math. 7 18] as they contain all four arbitrary functions involved in infinitesimals and several free parameters as well. The deduction of previous results obtained by taking particular choices of arbitrary functions and constants, ensure novelty and importance of these solutions as well as the authenticity and effectiveness of Lie symmetry method. To demonstrate the distinct physical structures of phenomena, these solutions are expanded via numerical simulation. Thus, doubly soliton, line soliton, multi soliton, soliton fusion and fission nature have been analyzed. The graphical representations are seemed useful to understand the dynamics of BLP system and propagation of dispersive water waves in infinite narrow channel.