NON-CLASSICAL LIE SYMMETRY ANALYSIS OF KDV, JAULENT–MIODEK, AND JIMBO–MIWA EQUATIONS

Abstract

The Lie symmetry analysis of non-classical is presented in the context of three important nonlinear partial differential equations, the (1+1)-dimensional Kortewegde Vries (KdV) equation, the (1+2)-dimensional Jaulent-Miodek (JM) equation and the (1+3)-dimensional modified Jimbo-Miwa equation. The non-classical symmetry approach, a variation on the classical Lie symmetry analysis, by adding an invariant surface condition, is employed to achieve symmetry reductions and invariant solutions of these equations. The corresponding determining equations are obtained and solved to construct non-classical symmetry generators. These generators are subsequently used to elicit smaller forms and even exact solutions to the invariance.

The results of the analysis are subsequently validated and interpreted by visualizing representative solutions through contour plots, surface plots and line profiles. Graphical Analysis the KdV equation forms localised travelling wave structures, the JM equation gives periodic wave structures in two space dimensions and the modified Jimbo-Miwa equation, demonstrates complex multidimensional interactions of waves. The findings underscore how the non-classical symmetry approach can be efficiently used to find physically significant solutions, especially to higher-dimensional nonlinear PDEs. The study extends the already existing literature on non-classical symmetry analysis to the multidimensional models and provides both analytical and graphical understanding of the behaviour of the solutions involving the multidimensional models.