論文

In this paper, we propose the concept of $(\pm)$-discrete Dirac structures over a manifold, where we define $(\pm)$-discrete two-forms on the manifold and incorporate discrete constraints using $(\pm)$-finite difference maps. Specifically, we develop $(\pm)$-discrete induced Dirac structures as discrete analogues of the induced Dirac structure on the cotangent bundle over a configuration manifold, as described by Yoshimura and Marsden (2006). We demonstrate that $(\pm)$-discrete Lagrange--Dirac systems can be naturally formulated in conjunction with the $(\pm)$-induced Dirac structure on the cotangent bundle. Furthermore, we show that the resulting equations of motion are equivalent to the $(\pm)$-discrete Lagrange--d'Alembert equations proposed in Cort\'es and Mart\'inez (2001) and McLachlan and Perlmutter (2006). We also clarify the variational structures of the discrete Lagrange--Dirac dynamical systems within the framework of the $(\pm)$-discrete Lagrange--d'Alembert--Pontryagin principle. Finally, we validate the proposed discrete Lagrange--Dirac systems with some illustrative examples of nonholonomic systems through numerical tests.

In this paper, we develop discrete versions of Darboux transformations and Crum's theorems for two second order difference equations. The difference equations are discretised versions (using Darboux transformations) of the spectral problems of the KdV quation, and of the modified KdV equation or sine-Gordon equation. Considering the discrete dynamics created by Darboux transformations for the difference equations, one obtains the lattice potential KdV equation, the lattice potential modified KdV equation and the lattice Schwarzian KdV equation, that are prototypes of integrable lattice equations. It turns out that, along the discretisation processes using Darboux transformations, two families of integrable systems (the KdV family, and the modified KdV or sine-Gordon family), including their continuous, semi-discrete and lattice versions, are explicitly constructed. As direct applications of the discrete Crum's theorems, multi-soliton solutions of the lattice equations are obtained.