Bifurcation Theory, Lie Group-Invariant Solutions of Subalgebras and Conservation Laws of a Generalized (2+1)-Dimensional BK Equation Type II in Plasma Physics and Fluid Mechanics

Abstract

The nonlinear phenomena in numbers are modelled in a wide range of fields such as chemical physics, ocean physics, optical fibres, plasma physics, fluid dynamics, solid-state physics, biological physics and marine engineering. This research article systematically investigates a (2+1)-dimensional generalized Bogoyavlensky–Konopelchenko equation. We achieve a five-dimensional Lie algebra of the equation through Lie group analysis. This, in turn, affords us the opportunity to compute an optimal system of fourteen-dimensional Lie subalgebras related to the underlying equation. As a consequence, the various subalgebras are engaged in performing symmetry reductions of the equation leading to many solvable nonlinear ordinary differential equations. Thus, we secure different types of solitary wave solutions including periodic (Weierstrass and elliptic integral), topological kink and anti-kink, complex, trigonometry and hyperbolic functions. Moreover, we utilize the bifurcation theory of dynamical systems to obtain diverse nontrivial travelling wave solutions consisting of both bounded as well as unbounded solution-types to the equation under consideration. Consequently, we generate solutions that are algebraic, periodic, constant and trigonometric in nature. The various results gained in the study are further analyzed through numerical simulation. Finally, we achieve conservation laws of the equation under study by engaging the standard multiplier method with the inclusion of the homotopy integral formula related to the obtained multipliers. In addition, more conserved currents of the equation are secured through Noether’s theorem.

Roadmap of the Multiplier Method for Partial Differential Equations

“…Another equation worked using this method are the BK [42] where Lie group-invariant solutions of subalgebras and conservation laws of a (2 + 1)-dimensional BK equation of type II used in plasma physics and fluid mechanics is also studied using this method.…”