Exact solutions to the variable-coefficient Bogoyavlensky-Konopelchenko equation by using two newly extended methods

Abstract

 This study presents two new methods for solving nonlinear evolution equations with variable coefficients. The expanded method has the advantage of being able to solve nonlinear evolution equations with both constant and variable coefficients, while the basic version of the methods under consideration can only be used for problems with constant coefficients. The proposed methods are used to solve the Bogoyavlensky–Konopelchenko (BK) equation with variable coefficients, which represent the interaction of a Riemann wave propagating along the y-axis and a long wave propagating along the x-axis in a fluid. Further, the BK equation with variable coefficients is applied for internal waves with layers, shallow-water waves, ion-acoustic waves, and propagation of water in a liquid. The BK equation with constant coefficients is the subject of the majority of research in the literature. This may result in a lack of comprehension of the physical phenomena that the model reveals. The BK equation is modified to include terms with time-varying coefficients to get around this restriction. The model becomes closer to the real problem and the physical phenomenon with the inclusion of these terms. The main focus point of this study is to develop two analytical methods with variable coefficients; the expansion of two variants of Kudryashov’s method. The proposed methods are applied to the variable-coefficient BK equation, showcasing distinctions from existing literature. A range of analytical solutions, including solitary wave solutions and their corresponding parameter constraints, are obtained. The results demonstrate effectiveness of the proposed methods in capturing the intricate dynamics of nonlinear systems, providing valuable insights into the qualitative behavior of solutions under varying conditions. In addition, the effects of time-varying coefficients on solitons and their interactions with each other in the generated traveling wave solutions are analyzed in detail under certain restrictive conditions. The results shed light on the physical behavior of the BK equation with variable coefficients and contribute to a better understanding of similar models. The proposed methods open new possibilities for the study of nonlinear evolution equations with variable coefficients and provide avenues for analytical investigation of their solutions. This work not only contributes to the theoretical understanding of NLPDEs but also offers practical implications for fields such as fluid dynamics and mathematical physics.