The existence and uniqueness of weak solutions for a highly nonlinear shallow-water model with Coriolis effect.

This article focuses on the analysis of a highly nonlinear shallow water wave model incorporating the Coriolis effect, described by a complex partial differential equation (Eq. 1.1) derived under specific scaling regimes relevant to equatorial ocean waves. The authors establish the existence and uniqueness of weak solutions in lower order Sobolev spaces \( H^s(\mathbb{R}) \) for \( 1 < s \leq \frac{3}{2} \), and prove local well-posedness of strong solutions in higher order Sobolev spaces \( H^s(\mathbb{R}) \) with \( s > \frac{3}{2} \) using a pseudoparabolic regularization technique. The work extends known results for related models such as the Camassa-Holm and rotation-Camassa-Holm equations, addressing challenges posed by the equation's combined quadratic and cubic nonlinearities. The paper provides rigorous functional analytic estimates, convergence arguments, and uniqueness proofs within the Sobolev space framework, contributing to the mathematical understanding of nonlinear wave propagation influenced by rotational effects.