  Integrable self-adaptive moving mesh schemes for short pulse type equations
(the short pulse equation, the coupled short pulse equation, and the complex
short pulse equation) are investigated. Two systematic methods, one is based on
bilinear equations and another is based on Lax pairs, are shown. Self-adaptive
moving mesh schemes consist of two semi-discrete equations in which the time is
continuous and the space is discrete. In self-adaptive moving mesh schemes, one
of two equations is an evolution equation of mesh intervals which is deeply
related to a discrete analogue of a reciprocal (hodograph) transformation. An
evolution equations of mesh intervals is a discrete analogue of a conservation
law of an original equation, and a set of mesh intervals corresponds to a
conserved density which play an important role in generation of adaptive moving
mesh. Lax pairs of self-adaptive moving mesh schemes for short pulse type
equations are obtained by discretization of Lax pairs of short pulse type
equations, thus the existence of Lax pairs guarantees the integrability of
self-adaptive moving mesh schemes for short pulse type equations. It is also
shown that self-adaptive moving mesh schemes for short pulse type equations
provide good numerical results by using standard time-marching methods such as
the improved Euler's method.
