A singular limit problem for the Ibragimov-Shabat equation

TL;DR

This paper investigates the singular limit of the Ibragimov-Shabat equation with nonlinear dispersion, showing convergence to weak solutions of a scalar conservation law as diffusion vanishes, using a priori estimates and compensated compactness.

Contribution

It establishes the convergence of solutions of the dispersive Ibragimov-Shabat equation to scalar conservation law solutions in the zero-diffusion limit, a novel analysis in this context.

Findings

Solutions converge to discontinuous weak solutions as diffusion tends to zero

Derived new a priori estimates for the dispersive equation

Applied compensated compactness in the L^p setting for the analysis

Abstract

We consider the Ibragimov-Shabat equation, which contains nonlinear dispersive effects. We prove that as the diffusion parameter tends to zero, the solutions of the dispersive equation converge to discontinuous weak solutions of a scalar conservation law. The proof relies on deriving suitable a priori estimates together with an application of the compensated compactness method in the L^p setting