Complexiton Solutions of the Toda Lattice Equation

We present a class of solutions of the two-dimensional Toda lattice equation, its fully discrete analogue and its ultra-discrete limit. These solutions demonstrate the existence of soliton resonance and web-like structure in discrete integrable systems such as differential-difference equations, difference equations and cellular automata (ultra-discrete equations).

We consider the Hankel determinant formula of the $\tau$ functions of the Toda equation. We present a relationship between the determinant formula and the auxiliary linear problem, which is characterized by a compact formula for the $\tau$ functions in the framework of the KP theory. Similar phenomena that have been observed for the Painlev\'e II and IV equations are recovered. The case of finite lattice is also discussed.

In this letter, we show that certain Fredholm determinant $D(\lambda;t)$, introduced by Zamolodchikov in his study of 2D polymers, is a continuum limit of soliton solution for the Toda lattice hierarchy with 2-periodic reduction condition.

Determinant formulas for the general solutions of the Toda and discrete Toda equations are presented. Application to the $\tau$ functions for the Painlev\'e equations is also discussed.

We introduce and study a class of analytic difference operators admitting reflectionless eigenfunctions. Our construction of the class is patterned after the Inverse Scattering Transform for the reflectionless self-adjoint Schr\"odinger and Jacobi operators corresponding to KdV and Toda lattice solitons.

Originally a model for wave propagation on the line, the Toda lattice is a wonderful case study in mechanics and symplectic geometry. In Flaschka's variables, it becomes an evolution given by a Lax pair on the vector space of real, symmetric, tridiagonal matrices. Its very special asymptotic behavior was studied by Moser by introducing norming constants, which play the role of discrete inverse variables in analogy to the solution by inverse scattering of KdV. It is a complete...

This paper can be an overview on solutions in Wronskian/Casoratian form to soliton equations with KdV-type bilinear forms. We first investigate properties of matrices commuting with a Jordan block, by which we derive explicit general solutions to equations satisfied by Wronskian/Casoratian entry vectors, which we call condition equations. These solutions are given according to the coefficient matrix in the condition equations taking diagonal or Jordan block form. Limit relati...

The article considers lattices of the two-dimensional Toda type, which can be interpreted as dressing chains for spatially two-dimensional generalizations of equations of the class of nonlinear Schr\"odinger equations. The well-known example of this kind of generalization is the Davey-Stewartson equation. It turns out that the finite-field reductions of these lattices, obtained by imposing cutoff boundary conditions of an appropriate type, are Darboux integrable, i.e., they h...

The notion of Laplace invariants is transferred to the lattices and discrete equations which are difference analogs of hyperbolic PDE's with two independent variables. The sequence of Laplace invariants satisfy the discrete analog of twodimensional Toda lattice. The terminating of this sequence by zeroes is proved to be the necessary condition for existence of the integrals of the equation under consideration. The formulae are presented for the higher symmetries of the equati...

We study a small piece of two dimensional Toda lattice as a complex dynamical system. In particular the Julia set, which appears when the piece is deformed, is shown analytically how it disappears as the system approaches to the integrable limit.