Similarity solutions of a generalized inhomogeneous-nonautonomous (2 + 1)-dimensional Konopelchenko – Dubrovsky equation. Stability analysis

Abstract

The (2 + 1) dimensional Konopelchenko–Dubrovsky equation (2D-KDE) is an integro differential equation which describes two-layer fluid in shallow water near ocean shores and stratified atmosphere. In this respect, the solutions of (2D-KDE) describe two dual functions which represent two-layer fluid. Here, we are concerned with deriving self-similar (similarity) solutions of the inhomogeneous-nonautonomous (2D-KDE). In this context, the model equations are nonlinear partial differential equations (NLPEs) with space and time dependent coefficients. It is worth noticing to mention that, the (2D-KDE) with space and time dependent coefficients was not considered in the literature up to date. So that, the present work is novel. In fact, only the study of (NLPDEs) with space dependent or time dependent coefficients was carried. This was done by using Lie symmetry. Here, we use a different technique together with introducing similarity transformations. The similarity solution are obtained via the extended unified method (EUM), which is an alternative technique to the use of Lie group symmetries of (NLPDEs). It is worthy to mention that the (EUM) is of low cost time in symbolic computations and it provides a wide class of solutions. The solutions presented here, are classified to hyperbolic and elliptic functions.The results are obtained via symbolic computations and they are evaluated numerically and displayed in graphs. Multiple solutions structures are observed. Among them dromian pattern, bridge shape, lumps vector and interaction between longitudinal and lateral complex waves. These results are completely new. A unified approach for analyzing the stability of the steady state solution is established. The stability of the is determined against varying the relevant parameters. It is found that stability holds against the dispersion coefficient, while instability holds against the nonlinearity coefficient and the phase velocity.