540 research outputs found
Self-adaptive moving mesh schemes for short pulse type equations and their Lax pairs
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.Comment: 13 pages, 6 figures, To be appeared in Journal of Math-for-Industr
On the -functions of the reduced Ostrovsky equation and the two-dimensional Toda system
The reciprocal link between the reduced Ostrovsky equation and the
two-dimensional Toda system is used to construct the -soliton
solution of the reduced Ostrovsky equation. The -soliton solution of the
reduced Ostrovsky equation is presented in the form of pfaffian through a
hodograph (reciprocal) transformation. The bilinear equations and the
-function of the reduced Ostrovsky equation are obtained from the period
3-reduction of the or two-dimensional Toda system,
i.e., the two-dimensional Toda system. One of -functions of
the two-dimensional Toda system becomes the square of a pfaffian
which also become a solution of the reduced Ostrovsky equation. There is
another bilinear equation which is a member of the 3-reduced extended BKP
hierarchy. Using this bilinear equation, we can also construct the same
pfaffian solution.Comment: 16 pages, several typos were correcte
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