A study of self-adjointness, Lie analysis, wave structures, and conservation laws of the completely generalized shallow water equation

Abstract

This article explores the analysis of the completely generalized Hirota–Satsuma–Ito equation through Lie symmetry analysis. The equation under consideration represents a more comprehensive form of the (2+1)-dimensional HSI equation, encom passing four additional second-order derivative terms: 3H , 4H ι, 3H , 4H ι,and 6Hιι,emergingfromtheinclusion of second-order dissipative-type elements. We calculate the infinitesimal generators and determine the symmetry group for each generatorusingtheLiegroupinvariancecondition.EmployingtheconjugacyclassesoftheAbelianalgebra,wetransformtheconsid ered equation into an ordinary differential equation through similarity reduction. Subsequently, we solve these ordinary differential equations to derive closed-form solutions for the completely generalized Hirota–Satsuma–Ito equation under certain conditions. For other scenarios, we utilize the extended direct algebraic method to obtain soliton solutions. Furthermore, we rigorously calculated the conserved quantities corresponding to each symmetry generator, the conservation laws of the model are established using the multiplier approach. Additionally, we present the graphical representation of selected solutions for specific values of the physical parameters of the equation under scrutiny.