Some closed-form solutions, conservation laws, and various solitary waves to the (2 + 1)-D potential B-K equation via Lie symmetry approach

Abstract

The main objective of our work is to obtain some exact closed-form solutions for the (2 + 1)-D potential Bogoyavlensky–Konopelchenko (B-K) equation, which describes the interaction between the Riemann wave propagating and the long-wave propagation along the y-axis and x-axis, by utilizing two effective methods namely: Lie symmetry method and generalized Kudryashov method (GKM). First, we obtained infinitesimals and vector fields of the B-K equation. After that, we used linear combinations of vectors to obtain various reductions of the governing equation. Four ordinary differential equations are obtained; two of them are solved using a series form, while the other two are solved using GKM. Many solitary wave solutions are obtained. The behavior of solutions includes solitary wave solutions, kink wave solutions, anti-kink wave solutions and singular wave solutions. By taking advantage of symbolic computation, the physical phenomena of the demonstrated results are visualized graphically by means of 3D and 2D plots. Using the new conservation theorem and Noether operators, conservation laws and nonlinear self-adjointness for the potential B-K equation are also constructed. It is clearly evident that the obtained results are very useful in studying interactions and processes in optical fibers, mathematical physics, fluid dynamics, engineering and many other areas of science.