The differential-q-difference 2D Toda equation: bilinear form and soliton solutions (original) (raw)
Rational solutions of the Toda lattice equation in Casoratian form
Chaos, Solitons & Fractals, 2004
A recursive procedure is presented for constructing rational solutions to the Toda lattice equation through the Casoratian formulation. It allows us to compute a broad class of rational solutions directly, without computing long wave limits in soliton solutions. All rational solutions arising from the Taylor expansions of the generating functions of soliton solutions are special ones of the general class, but only a Taylor expansion containing even or odd powers leads to non-constant rational solutions. A few rational solutions of lower order are worked out.
q-discrete Toda molecule equation
Physics Letters A, 1993
A q-discrete version of the two-dimensional Toda molecule equation is proposed through the direct method. Its solution, Bäcklund transformation and Lax pair are discussed. The reduction to the q-discrete cylindrical Toda molecule equation is also discussed.
Generalized Casorati Determinant and Positon–Negaton-Type Solutions of the Toda Lattice Equation
Journal of the Physical Society of Japan, 2004
A set of conditions is presented for Casorati determinants to give solutions to the Toda lattice equation. It is used to establish a relation between the Casorati determinant solutions and the generalized Casorati determinant solutions. Positons, negatons and their interaction solutions of the Toda lattice equation are constructed through the generalized Casorati determinant technique. A careful analysis is also made for general positons and negatons, the resulting positons and negatons of order one being explicitly computed. The generalized Casorati determinant formulation for the two dimensional Toda lattice (2dTL) equation is presented. It is shown that positon, negaton and complexiton type solutions in the 2dTL equation exist and these solutions reduce to positon, negaton and complexiton type solutions in the Toda lattice equation by the standard reduction procedure.
Complexiton solutions of the Toda lattice equation
Physica A: Statistical Mechanics and its Applications, 2004
A set of coupled conditions consisting of differential-difference equations is presented for Casorati determinants to solve the Toda lattice equation. One class of the resulting conditions leads to an approach for constructing complexiton solutions to the Toda lattice equation through the Casoratian formulation. An analysis is made for solving the resulting system of differential-difference equations, thereby providing the general solution yielding eigenfunctions required for forming complexitons. Moreover, a feasible way is presented to compute the required eigenfunctions, along with examples of real complexitons of lower order.
Casorati determinant solution for the relativistic Toda lattice equation
Journal of Mathematical Physics, 1993
The relativistic Toda lattice equation is decomposed into three Toda systems, the Toda lattice itself, Bäcklund 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.
The modified Lax and two-dimensional Toda lattice equations associated with simple Lie algebras
Ergodic Theory and Dynamical Systems, 1981
We associate to each complex simple Lie algebra g a hierarchy of evolution equations; in the simplest case g = sl(2) they are the modified KdV equations. These new equations are related to the two-dimensional Toda lattice equations associated with g in the same way that the modified KdV equations are related to the sinh-Gordon equation.
2021
This paper investigates a relativistic Toda lattice system with an arbitrary parameter that is a very remarkable generalization of the usual Toda lattice system, which may describe the motions of particles in lattices. Firstly, we study some integrable properties for this system such as Hamiltonian structures, Liouville integrability and conservation laws. Secondly, we construct a discrete generalized (m, 2Nm)-fold Darboux transformation based on its known Lax pair. Thirdly, we obtain some exact solutions including soliton, rational and semi-rational solutions with arbitrary controllable parameters and hybrid solutions by using the resulting Darboux transformation. Finally, in order to understand the properties of such solutions, we investigate the limit states of the diverse exact solutions by using graphic and asymptotic analysis. In particular, we discuss the asymptotic states of rational solutions and exponential-and-rational hybrid solutions graphically for the first time, which might be useful for understanding the motions of particles in lattices. Numerical simulations are used to discuss the dynamics of some soliton solutions. The results and properties provided in this paper may enrich the understanding of nonlinear lattice dynamics.
A modified Toda spectral problem and its hierarchy of bi-Hamiltonian lattice equations
Journal of Physics A: Mathematical and General, 2004
Starting from a modified Toda spectral problem, a hierarchy of generalized Toda lattice equations with two arbitrary constants is constructed through discrete zero curvature equations. It is shown that the hierarchy possesses a bi-Hamiltonian structure and a hereditary recursion operator, which implies that there exist infinitely many common commuting symmetries and infinitely many common commuting conserved functionals. Two cases of the involved constants present two specific integrable sub-hierarchies, one of which is exactly the Toda lattice hierarchy.
The nonabelian Toda lattice: Discrete analogue of the matrix Schrödinger spectral problem
Journal of Mathematical Physics, 1980
We investigate the discrete analog of the matrix Schrodinger spectral problem and derive the simplest nonlinear differential-difference equation associated to such problem solvable by the inverse spectral transform. We also display the one and two soliton solution for this equation and tersely discuss their main features.
Combined Wronskian solutions to the 2D Toda molecule equation
Physics Letters A, 2011
By combining two pieces of bi-directional Wronskian solutions, molecule solutions in Wronskian form are presented for the finite, semi-infinite and infinite bilinear 2D Toda molecule equations. In the cases of finite and semi-infinite lattices, separated-variable boundary conditions are imposed. The Jacobi identities for determinants are the key tool employed in the solution formulations.
Solvable systems of wave equations and non-Abelian Toda lattices
Journal of Physics A: Mathematical and General, 1992
ABSTRACT This paper relates equivalence classes of coupled systems of N linear wave equations to motions of an N*N matrix dynamical system, the two-dimensional non-Abelian Toda lattice. In particular, the correspondence is shown to relate those coupled systems of wave equations with progressing-wave general solutions to motions of the finite non-Abelian Toda lattice with free ends, generalizing a known result for the N = 1 case. Some non-trivial motions of such Toda lattices are found, and the corresponding coupled wave equations and their progressing wave general solutions are given. Other consequences of the correspondence and possible application of the progressing waves are discussed.
On the Integrable Generalization of the 1D Toda Lattice
2010
A generalized Toda Lattice equation is considered. The associated linear problem (Lax representation) is found. For simple case N = 3 the τ -function Hirota form is presented that allows to construct an exast solutions of the equations of the 1DGTL. The corresponding hierarchy and its relations with the nonlinear Schrodinger equation and Hersenberg ferromagnetic equation are discussed.
Physics Letters A, 1995
We present an unifying description of the graded SL(p, q) KP-KdV hierarchies, including the Wronskian construction of their tau-functions as well as the coefficients of the pertinent Lax operators, obtained via successive applications of special Darboux-Bäcklund transformations. The emerging Darboux-Bäcklund structure is identified as a constrained generalized Toda lattice system relevant for the two-matrix string model. It allows simple derivation of the n-soliton solutions of the unconstrained KP system. Also, the exact Wronskian solution for the two-matrix model partition function is found.