Gauge-Transformed Exponential Integrator for Generalized KdV Equations with Rough Data

In this paper, we propose a new exponential-type integrator for solving the gKdV equation under rough data. By introducing new frequency approximation techniques and a key gauge transform, the proposed scheme is explicit, stable, and efficient in practice. The optimal convergence result of the scheme is established, which says that it is first order accurate in the -norm for the solution in with any . The required regularity is lower than classical approaches. The theoretical results are confirmed by numerical experiments.