Analytical and symmetry approaches to the Gardner-Kawahara equation: solitary wave solutions and conservation laws

In this paper, we investigate invariant solutions of the Gardner–Kawahara (GK) equation, an extended form of the Korteweg–de Vries (KdV) equation that models solitary wave propagation in plasmas, shallow water with surface tension, and magneto acoustic media using the Lie symmetry method. The study is divided into two main parts. In the first, we systematically derive infinitesimal generators and perform symmetry reductions to obtain nonlinear ordinary differential equations (ODEs), which are solved using the tanh and power series methods. Exact traveling wave solutions are constructed and illustrated graphically to reveal the influence of key parameters on wave dynamics. Conservation laws are also derived using the multiplier homotopy method. In the second part, we examine the modulational instability (MI) of uniform wave trains through linear stability analysis. The instability condition highlights the contrasting effects of nonlinearity and higher-order dispersion. Numerical simulations demonstrate unstable modulation bands. The present work illuminates wave breaking, energy localization, and pattern formation in nonlinear dispersive media. The results enhance our knowledge of intricate wave dynamics governed by the GK equation, and can be used in various domains such as fluid dynamics, plasma physics, and nonlinear optics.