Group invariant solutions and conservation Laws of certain nonlinear evolution equations In mathematical physics

Abstract

This research project aims to study some nonlinear partial differential equations that arise in many branches of physics such as particle physics, fluid dynamics, plasma astrophysics, ocean dynamics, atmospheric science, computational fluid mechanics, cosmology, condensed matter physics, statistical physics, nonlinear acoustics, vehicular traffic, electronic transport, etc. Exact solutions, conservation laws and solution solutions are derived for such equations using various methods. The nonlinear partial differential equations that are studied in this research work are two (3 + 1)-dimensional Kadomtsev-Petviashvili (KP) like equations, a generalized (2 + 1)-dimensional Bogoyavlenskii-Kadomtsev-Petviashvili (BKP) equation, a generalized dispersive water waves system and an extended (2 + 1)-dimensional coupled Burgers system in fluid mechanics. The classical symmetry approach will be employed to search for exact solutions of a first (3+1)-dimensional KP like equation. Thereafter, we will derive the admitted conserved vectors of the aforementioned equation. We employ some ansatz methods to derive topological solutions of a second (3 + 1)-dimensional KP like equation. Furthermore, mixed solutions consisting of singular and periodic solutions and others are derived. Moreover, other analytical solutions based on modern group analysis are obtained. In addition, low-order conservation laws are constructed. We further, determine novel exact solutions of a generalized (2 + 1)-dimensional Bogoyavlenskii-Kadomtsev-Petviashvili (BKP) equation by utilizing the multiple exp-function algorithm and the modern group analysis method. Then, we compute conserved currents using the invariance and multiplier technique. Symmetry analysis is performed for a generalized dispersive water waves system. This symmetry analysis will lead to similarity reductions and new exact solutions

with the aid of the simplest equation method. The solutions obtained include the solitary waves and the traveling wave solutions. In addition, conservation laws are derived using the multiplier approach. Finally, we determine novel exact solutions of an extended (2 + 1)-dimensional coupled Burgers system in fluid mechanics by the Lie symmetry method in conjunction with the Kurdyshov method. Conservation laws of the above-mentioned system are generated.