Optimal system, invariance analysis of fourth-Order nonlinear ablowitz-Kaup-Newell-Segur water wave dynamical equation using lie symmetry approach

Abstract

Nonlinear science is very common and important natural phenomenon in our surroundings. It is quite possible to find a large number of phenomena, which become a cause of the formulation of a non-linear partial differential equation. Due to the presence of non-linear partial differential equations in each branch of science, these equations have become a useful tool to deal with complex natural phenomena. It is interesting to investigate any complex non-linear partial differential equations for different exact solutions and examine the behavior of the solutions. Many effective approaches are developed to obtain the explicit exact solutions of the NLPDEs. Lie symmetry analysis is also one of the significant approaches to investigate the NLPDEs. Based on the Lie group analysis, we investigate a very famous and important equation, which is named as fourth-order Ablowitz-Kaup-Newell-Segur water wave dynamical equation. The symmetry groups, Commutator Tables and Adjoint of infinitesimals are constructed for this equation. Further, using the adjoint table, the optimal system is obtained. According to the optimal system, we tried to find the possible exact solution using symmetry reduction and presented a brief study of the properties of different solutions. We found some new exact solutions described with graphical representation showing solution wave structure, contour plot and wave propagation of the solution profile. The results are often helpful for studying the interaction of waves in many new localized structures and high-dimensional models.