Abstract
Abstract 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.
Highlights
The studies of discrete integrable systems were initiated in the middle of 1970s
We demonstrate how to construct self-adaptive moving mesh schemes for short pulse type equations whose Lax pairs are written in the WKI type form which is transformed into the Ablowitz-KaupNewell-Segur (AKNS) type form by reciprocal transformations
4 Concluding remarks We have proposed two systematic methods for constructing self-adaptive moving mesh schemes for a class of nonlinear wave equations which are transformed into a different class of nonlinear wave equations by reciprocal transformations
Summary
Hirota discretized various soliton equations such as the KdV, the mKdV, and the sine-Gordon equations based on the bilinear equations [16,17,18,19,20], Ablowitz and Ladik proposed a method of integrable discretizations of soliton equations, including the nonlinear Schrödinger equation and the modified KdV (mKdV) equation, based on the Ablowitz-Kaup-NewellSegur (AKNS) form [1,2,3,4,5]. Following the pioneering works of Hirota and Ablowitz-Ladik, the studies of discrete integrable systems have been expanded in diverse areas (see, for example, [6,7,15,40]). Integrable discretization of soliton equations in the WKI class had been regarded as a difficult problem until recently. The method employed in our previous papers was rather technical, it is not easy to extract a fundamental structure of discretizations to apply this method to a broader class of nonlinear wave equations including nonintegrable systems
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