Convergence result for the short pulse equation

Abstract In this paper, we consider a non-local elliptic-hyperbolic system related to the short pulse equation. It is a model which describes the evolution of the electrical field of linearly polarized continuum spectrum pulses in optical waveguides, including fused-silica telecommunication-type or photonic-crystal fibers, as well as hollow capillaries filled with transparent gases. We prove that the solution of a non-local elliptic-hyperbolic system related to the short pulse equation converges to the unique entropy one of the short pulse equation. The proof relies on deriving suitable a priori estimates together with an application of the compensated compactness method in the $$L^p$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:math> setting.