Integrable self-adaptive moving mesh schemes for multi-component short pulse type equations with nonzero boundary values

Abstract

In this paper, we construct integrable self-adaptive moving mesh schemes for multi-component modified short pulse and short pulse equations with nonzero boundary values by using the consistency condition with the hodograph transformation. The essential point is that the edge point $x_{0}$ of the hodograph transformation cannot be kept fixed when the boundary flux is nonzero. We derive the evolution equation for $x_{0}$ and incorporate it into the semi-discrete moving mesh scheme. This supplies a moving-edge mechanism that extends the previously fixed-edge schemes and, in particular, allows periodic computations with nonzero boundary values. These schemes automatically adjust the mesh intervals according to the solution profile. We also derive multi-soliton solutions in Pfaffian form for the proposed schemes, which preserve the integrable structure in the discrete scheme. Numerical experiments for one- and two-soliton solutions demonstrate that the proposed schemes achieve high accuracy even in regions with rapid variation, while maintaining stability over long-time simulations, with small relative errors near peak amplitudes.