A new class of integrable two-dimensional dilaton gravity theories, in which scalar matter fields satisfy the Toda equations, is proposed. The simplest case of the Toda system is considered in some detail, and on this example we outline how the general solution can be obtained. We also demonstrate how the wave-like solutions of the general Toda systems can be simply derived. In the dilaton grav...

Some techniques of bilinearization of the non-autonomous 1+1 dimensional discrete soliton equations are discussed by taking the discrete KdV equation, the discrete Toda lattice equation, and the discrete Lotka-Volterra equation as examples. Casorati determinant solutions to those equations are also constructed explicitly. §

1.1. The Toda lattice equation introduced by Toda as a Hamilton equation describing the motion of the system of particles on the line with an exponential interaction between closest neighbours gave rise to numerous important generalizations and helped to discover many of the exciting phenomena in the theory of integrable equations. In Flaschka’s variables [F] the finite non-periodic Toda lattic...

A new ansatz is presented for a Lax pair describing systems of particles on the line interacting via (possibly nonsymmetric) pairwise forces. Particular cases of this yield the known Lax pairs for the Calogero-Moser and Toda systems, as well as their relativistic generalisations. The ansatz leads to a system of functional equations. Several new functional equations are described and the general...

this paper studies the existence of solutions for acoupled system of nonlinear fractional differential equations. newexistence and uniqueness results are established using banach fixedpoint theorem. other existence results are obtained using schaeferand krasnoselskii fixed point theorems. some illustrative examplesare also presented.