Analytical solutions for the generalized (2+1)–dimensional Konopelchenko–Dubrovsky equation via Lie symmetry analysis

Abstract

In this study, we examine the (2+1)–dimensional Konopelchenko–Dubrovsky (KD) equation, which models the dynamics of a two-layer fluid in shallow water near ocean shores and within a stratified atmosphere. Here, we focus on obtaining similarity solutions to the KD equation by applying Lie symmetry analysis. At first, five–dimensional Lie point symmetries are found by generalizing the invariance criterion for the integro differential equation. Further, two–dimensional optimal classification is performed for the five–dimensional Lie algebra. Moreover, invariant solutions like parabolic type and inverted bell-shaped solutions are obtained for different classes of two–dimensional optimal sets. Additionally, a special solution in terms of Airy functions is obtained for the KD equation, which is derived from the second Painlevé equation $P_{II}$. Finally, physically relevant solutions including traveling wave solutions, notably kink-type solitons, and multi solitons, are derived through traveling wave transformations, and all solutions are represented graphically.