Dynamics of various types of ocean wave-solutions and orbital stability in the generalized geophysical Gardner equation

The Korteweg-de Vries (KdV) equation and its modified version (mKdV) have extensive applications in science and engineering. Incorporating the KdV with the Coriolis effect parameter, known as the geophysical KdV (gKdV), describes the effect of earth’s rotation on ocean water. Both gKdV and its modified version (mgKdV) arise in the application of dispersive tsunami waves in the ocean. In this article, we investigate the geophysical Gardner equation, which combines the gKdV and mgKdV models, to reveal additional physical features of tsunami waves. We adapt some effective analytical schemes to extract water-wave solutions to the proposed model. In particular, we implement the modified tanh-coth expansion and some rational form solutions in terms of trigonometric and hyperbolic functions. Additionally, we examine the influence of the embedded nonlinearity, dispersion, and perturbation parameters on the physical characteristics of the model’s wave propagation. Moreover, by implementing the Grillakis-Shatah-Strauss theory, we investigate the orbital stability state for some obtained solutions. These findings are expected to be highly beneficial for further research into nonlinear problems admitting water-wave solutions.