On the Lie Symmetry Analysis of the Seventh-Order Nonlinear Kaup–Kuperschmidt Equation

Source abstract

This paper presents a rigorous Lie symmetry analysis of the seventh-order Kaup–Kuperschmidt equation given by ut+2016u3ux+630ux3+2268uuxuxx+504u2uxxx+252uxxuxxx+147uxuxxxx+42uuxxxxx+uxxxxxxx=0 which arises as a higher-order nonlinear evolution equation describing nonlinear wave propagation, dispersive media, and integrable physical systems. The study employs the classical Lie group approach to systematically determine the admitted infinitesimal symmetries of the equation. The total derivative operators and seventh prolongation of the infinitesimal generator are utilized in the invariance criterion to derive the determining system governing the infinitesimal coefficients. The admitted Lie algebra and corresponding one-parameter transformation groups are obtained explicitly. Furthermore, similarity reductions are employed to transform the governing partial differential equation into ordinary differential equations, from which exact invariant solutions are constructed. In addition, exact power series solutions are determined to characterize local analytic behavior of solutions. Stability remarks and qualitative physical interpretations of the resulting wave structures are discussed. The results reveal the underlying geometric structure of the seventh-order Kaup–Kuperschmidt equation and provide exact analytical tools for understanding nonlinear dispersive dynamics.