Toda Molecule Research Articles

On the inverse scattering transform of the 2 + 1 Toda equation

The method of solution to the 2 + 1 dimensional Toda equation is described in some detail. This equation reduces directly to the well know Toda lattice in 1 + 1 dimension and, by an appropriate asymptotic reduction, to the Kadomtsev-Petviashvili equation in a continuous limit. The solution exhibits a number of interesting aspects depending on certain choices of signs, of which there are four, in the equation. For two choices of sign the equation is well posed and linearly stable/unstable. For the other choices of sign the equation is linearly illposed. In these cases we can relate the solution to a boundary value problem and give a formal construction of the solution. For one choice of signs in the illposed case an analogue of the Sommerfeld radiation condition is developed in order to identify a unique solution. In general the method of solution of the “Toda molecule” equation requires an implementation of the dbar technique to cases where the associated eigenfunctions possess both smooth regions of nonholomorphicity and a discontinuity across a curve, which in this problem is the unit circle. Special lump type solutions and solutions depending on suitable arbitrary functions are presented.

Toda Molecule Equation and Quotient-Difference Method

The numerical algorithm for computing eigenvalues of a given matrix using the Toda molecule equation, suggested recently by Hirota, Tsujimoto and Imai, is shown to be equivalent to the quotient-difference method. This relation, convergence of the algorithm and extension to a much wider range of matrices are described.

Kac-Moody construction of Toda type field theories

Abstract Using the coadjoint orbit method we derive a geometric WZWN action based on the extended two-loop Kac-Moody algebra. We show that under a hamiltonian reduction procedure, which respects conformal invariance, we obtain a hierarchy of Toda type field theories, which contain as submodels the Toda molecule and periodic Toda lattice theories. We also discuss the classical r -matrix and integrability properties.

Explicit One-Soliton Solutions of the 1+1 Dimensional Toda Molecule Equation and Their Large Molecule Limit of the Infinite Toda Lattice

We have obtained the explicit one-soliton solutions of the 1+1 dimensional Toda molecule equation, (∂/∂ t ) 2 log V ( n , t )- V ( n +1, t )+2 V ( n , t )- V ( n -1, t )=0, consisting of arbitrary number of particles which satisfy the molecule boundary condition V ( n , t )=0 at n =- M ′ and M ′′ ( M ′ , M ′′ ≡arbitrary positive integer constants). It has been shown that for the properly chosen values of arbitrary constant parameters and the large molecule limit, the present Toda molecule one-soliton solutions actually reproduce the well known infinite Toda lattice one-soliton solutions.

Solutions to higher Hamiltonians in the Toda hierarchies

A method is presented for constructing the general solution to higher Hamiltonians (nonquadratic in the momenta) of the Toda hierarchies of integrable models associated with a simple Lie group G. The method is representation independent and is based on a modified version of the Lax operator. It constitutes a generalization of the method used to construct the solutions of the Toda molecule models. The SL(3) and SL(4) cases are discussed in detail.

General Cylindrical Soliton Solutions of the Toda Molecule

We consider the 2+1 dimensional Toda Molecule equation Δlog V n - V n +1 +2 V n - V n -1 =0 where Δ≡(∂/∂ x ) 2 +(∂/∂ y ) 2 . The boundary condition is molecule type, which is V n =0 at finite n corresponding to both ends of the molecule. Recently we reported the existence of the Toda Molecule cylindrical solitons which have smooth connection to the Toda Lattice cylindrical solitons in the large molecule limit and are expressed by the associated Legendre functions. We have found that such cylindrical solitons can be generalized further by the use of Jacobi polynomials.

Cylindrical Multi-Soliton Solutions of the Toda Molecule Equation and Their Large Molecule Limit of the Toda Lattice

We consider the 2+1 dimensional Toda Molecule equation Δlog V n - V n +1 +2 V n - V n -1 =0 where Δ≡(∂/∂ x ) 2 +(∂/∂ y ) 2 . The boundary condition is molecule type, which is V n =0 at finite n corresponding to both ends of the molecule. By using the Hirota bilinear method, we derive the cylindrical multi-soliton solutions of this system which are written explicitly by the associated Legendre functions P M n . In the large molecule limit, the present Toda Molecule cylindrical solitons precisely reproduce the (previously known) Toda Lattice cylindrical solitons. Brief study is also made for the cylindrical solitons of the semi-infinite Toda lattice.

The 3+1 Dimensional Toda Molecule Equation and Its Multiple Soliton Solutions

We consider the 3+1 dimensional Toda equation Δ log V n - V n +1 +2 V n - V n -1 =0 where Δ ≡(∂/∂ x ) 2 +(∂/∂ y ) 2 +(∂/∂ z ) 2 . The boundary condition is molecule type which is V n =0 at finite n corresponding to both ends of the finite chain. We take the form of the solution to be V n ( x , y , z )= V n ( r , θ, φ)= r -2 V n (θ,φ) where r , θ and φ are usual spherical coordinates. We have found explicit multiple soliton solutions for V n (θ, φ), which correspond to the superpositions of arbitrary number of axially symmetric solutions with each symmetry axis directing to arbitrary different directions.

Exact Solutions of the Spherical Toda Molecule Equation

It is found that the 3+1 dimensional Toda molecule equation in the spherical coordinates \begin{aligned} \varDelta\log V_{n}=V_{n+1}-2V_{n}+V_{n-1}, \end{aligned} where \begin{aligned} \varDelta=\partial^{2}/\partial r^{2}+(2/r)(\partial/\partial r)+r^{-2}[\partial^{2}/\partial\theta^{2}+\cot\theta(\partial/\partial\theta)+(1/\sin^{2}\theta)\partial^{2}/\partial\varphi^{2}], \end{aligned} which has been introduced by Nakamura, is transformed into the two-dimensional Toda molecule equation under the assumption that V n ( r ,θ,ϕ)= r -2 V n (θ,ϕ). General solutions including Nakamura's are obtained.

Quantum Statistics of the Toda Oscillator in the Wigner Function Formalism

AbstractClassical and quantum mechanical Toda systems (Toda molecules, Toda lattices, Toda quantum fields) recently found growing interest as nonlinear systems showing solitons and chaos. In this paper the statistical thermodynamics of a system of quantum mechanical Toda oscillators characterized by a potential energy V(q) = Vo cos h q is treated within the Wigner function formalism (phase space formalism of quantum statistics). The partition function is given as a Wigner‐ Kirkwood series expansion in terms of powers of h2 (semiclassical expansion). The partition function and all thermodynamic functions are written, with considerable exactness, as simple closed expressions containing only the modified Hankel functions Ko and K1 of the purely imaginary argument iζ with ζ = Vo/kT.

Wronskian Structures of Solutions for Soliton Equations

Solutions for the equations of the Kadomtsev-Petviashvili hierarchy are given in terms of Wronskian. The two-dimensional Toda lattice equation and the two-dimensional Toda molecule equation are investigated and their solutions are expressed in the form of two­ directional Wronskian and double Wronskian. These solutions have common structure so that we can construct systems of these equations and their Wronskian solutions. lattice. There are several ways to get soliton solutions for these equations. For example in the method of the inverse scattering transformation, soliton solutions are associat­ ed with the discrete spectra of the scattering operator and may be written in deter­ minant form. In the direct method, they are obtained by a simple perturbational technique and represented as polynomials in exponentials. The B~cklund transfor­ mation (BT) gives a way of constructing N + 1-soliton solution from N-soliton solu­ tion. A simple and useful expression of these solutions may be the one in terms of Wronskian. The Wronskian representation of the solutions was first given for the KdV and modified KdV equations. 1> This technique has been developed by Freeman and Nimmo2H> for other soliton equations such as the KP, Boussinesq, NLS and Davey-Stewartson (DS) equations. In this expression of the solutions, these non­ linear equations are derived from the Laplace expansion of the determinants which are equal to zero. The general theory of r function has been developed by Sato6> and it has been

Exact Solutions of the Cylindrical Toda Molecule Equation

We consider the cylindrical Toda molecule equation, (∂ 2 /∂ r 2 + r -1 ∂/∂ r ) (log V n )- V n +1 +2 V n - V n -1 =0, with the boundary condition V 0 = V N +1 =0. We have constructed exact solutions to the equation by applying the bilinear method.

Generalizations of the Toda molecule

Abstract Finite-energy monopole solutions are constructed for the self-dual equations with spherical symmetry in an arbitrary integer graded Lie algebra. The constraint of spherical symmetry in a complex noncoordinate basis leads to a dimensional reduction. The resulting two-dimensional (r, t) equations are of second order and furnish new generalizations of the Toda molecule equations. These are then solved by a technique which is due to Leznov and Saveliev. For time-independent solutions a further reduction is made, leading to an ansatz for all SU(2) embeddings of the Lie algebra. The regularity condition at the origin for the solutions, needed to ensure finite energy, is also solved for a special class of nonmaximal embeddings. Explicit solutions are given for the groups SU(2), SO(4), Sp(4) and SU(4).

Non-compact symmetric spaces and the Toda molecule equations

It has been shown by Olshanetsky and Perelomov that the Toda molecule equations associated with any Lie groupG describe special geodesic motions on the Riemannian non-compact symmetric space which is the quotient of the normal real form ofG, GN, by its maximal compact subgroup. This is explained in more detail and it is shown that the “fundamental Poisson bracket relation” involving the Lax operatorA and leading to the Yang-Baxter equation and integrability properties is a direct consequence of the fact that the Iwasawa decomposition forGN endows the symmetric space with a hidden group theoretic structure.

Algebraic structure of Toda systems

Following the Leningrad school an operator P is constructed which guarantees the classical complete integrability of the Toda molecule and Toda lattice equations. This quantity depends in a uniform way upon the root system of the underlying algebra, respectively the simple Lie algebras and the affine euclidean Kac-Moody algebras.

One-Dimensional Toda Molecule. I: General Solution

A mechanism is presented for producing explicitly the general solutions to the one-dimensional Toda molecule equations associated wih the classical complex Lie algebras of rank N : sl (N+1,C), so (2N+1,C), sp (N,C) and so (2N,C). The method itself is applicable, however, to any semi-simple Lie algebra. A particular feature of our solutions is their description in terms of the “initial data”, which is consistent with the fact that the one-dimensional Toda molecule equation is a Newtonian equation of motion. The solutions for the algebras associated with SU(2), SU(3) and Sp(2) are given explicitly.

Self-dual monopoles and toda molecules

Stable static solutions to a gauge field theory with a Higgs field in the adjoint representation and with vanishing self-coupling are self-dual in the sense of Bogomolny. Leznov and Saveliev showed that a specific form of spherical symmetry reduces these equations to a modified form of the Toda molecule equations associated with the overall gauge symmetry G. Values of the constants of integration are found in terms of the distant Higgs field, guaranteeing regularity of the solution at the origin. The expressions hold for any simple Lie group G, depending on G via its root system.