Casorati Determinant Solution for the Relativistic Toda Lattice Equation

The relativistic Toda lattice equation is decomposed into three Toda systems, the Toda lattice itself, B\"acklund transformation of Toda lattice and discrete time Toda lattice. It is shown that the solutions of the equation are given in terms of the Casorati determinant. By using the Casoratian technique, the bilinear equations of Toda systems are reduced to the Laplace expansion form for determinants. The $N$-soliton solution is explicitly constructed in the form of the Casorati determinant.Comment: 19 pages in plain Te

Artà al Bt. Ramon Llull

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Third-order integrable difference equations generated by a pair of second-order equations

We show that the third-order difference equations proposed by Hirota, Kimura and Yahagi are generated by a pair of second-order difference equations. In some cases, the pair of the second-order equations are equivalent to the Quispel-Robert-Thomson(QRT) system, but in the other cases, they are irrelevant to the QRT system. We also discuss an ultradiscretization of the equations.Comment: 15 pages, 3 figures; Accepted for Publication in J. Phys.