A Regularized Shallow‐Water System

The shallow‐water system is a standard model for long waves in shallow water. The system is hyperbolic, and, for a large class of initial data, solutions develop steep gradients, leading to shock formation in finite time. Since such singularities violate the long‐wave assumptions underlying the model, their appearance limits the regime of validity of the equations. In this work, we introduce a regularized shallow‐water system in which the nonlinear terms are modified by a bounded operator. This regularization removes the standard derivative‐steepening mechanism responsible for shock formation in the classical system while remaining consistent with the long‐wave regime. We establish local well‐posedness and small‐data global well‐posedness in Sobolev spaces that exclude singularity formation. In addition, numerical simulations indicate that the system admits solitary‐wave solutions.