Rasha Al Jamal

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.

The generalized Korteweg–de Vries–Burgers–Kuramoto–Sivashinsky equation (GKdVBKS) is a nonlinear partial differential equation that models propagation of waves in a thick elastic tube filled with a viscous fluid. First, the dynamics of the GKdVBKS equation that depend on the physical parameters and , where is a positive parameter and is a real number, are presented. We show that the set of constant equilibrium solutions is unstable when , Lyapunov stable when , and globally exponentially stable when . Second, we show that the controlled GKdVBKS equation has a unique strong solution. Then, we synthesize a single bounded input‐feedback controller for the GKdVBKS equation through its linearized system avoiding any spillover. The approach used is valid due to the Fréchet differentiability of the nonlinear ‐semigroup generated by the nonlinear GKdVBKS equation with its derivative being the linear ‐semigroup generated by the linearized system. Finally, numerical results for different values of the parameters of the equation are presented to illustrate the developed theory.