Toda Molecule Research Articles

Integrable quadratic structures in peakon models

We propose realizations of the Poisson structures for the Lax representations of three integrable n-body peakon equations, Camassa-Holm, Degasperis-Procesi and Novikov. The Poisson structures derived from the integrability structures of the continuous equations yield quadratic forms for the r-matrix representation, with the Toda molecule classical r-matrix playing a prominent role. We look for a linear form for the r-matrix representation. Aside from the Camassa-Holm case, where the structure is already known, the two other cases do not allow such a presentation, with the noticeable exception of the Novikov model at n=2. Generalized Hamiltonians obtained from the canonical Sklyanin trace formula for quadratic structures are derived in the three cases.

Min-plus eigenvalue of tridiagonal matrices in terms of the ultradiscrete Toda equation

The discrete Toda molecule equation can be used to compute eigenvalues of tridiagonal matrices over conventional linear algebra, and is the recursion formula of the well-known quotient difference algorithm for tridiagonal eigenvalues. An ultradiscretization of the discrete Toda equation leads to the ultradiscrete Toda (udToda) equation, which describes motions of balls in the box and ball system. In this paper, we associate the udToda equation with eigenvalues of tridiagonal matrices over min-plus algebra, which is a semiring with two operation types: and . We also clarify an interpretation of the udToda variables in weighted and directed graphs consisting of vertices and edges.

Combinatorial expressions of the solutions to initial value problems of the discrete and ultradiscrete Toda molecules

Combinatorial expressions are presented of the solutions to initial value problems of the discrete and ultradiscrete Toda molecules. For the discrete Toda molecule, a subtraction-free expression of the solution is derived in terms of non-intersecting paths, for which two results in combinatorics, Flajolet’s interpretation of continued fractions and Gessel–Viennot’s lemma on determinants, are applied. By ultradiscretizing the subtraction-free expression, the solution to the ultradiscrete Toda molecule is obtained. It is finally shown that the initial value problem of the ultradiscrete Toda molecule is exactly solved in terms of shortest paths on a specific graph. The behavior of the solution is also investigated in comparison with the box–ball system.

Black brane solutions and their solitonic extremal limit in Einstein-scalar gravity

We investigate static, planar, solutions of Einstein-scalar gravity admitting an anti-de Sitter (AdS) vacuum. When the squared mass of the scalar field is positive and the scalar potential can be derived from a superpotential, minimum energy theorems indicate the existence of a scalar soliton. On the other hand, for these models, no-hair theorems forbid the existence of hairy black brane solutions with AdS asymptotics. By considering a specific example (an exact integrable model which has the form of a Toda molecule) and by deriving explicit exact solution, we show that these models allow for hairy black brane solutions with non-AdS domain wall asymptotics, whose extremal limit is a scalar soliton. The soliton smoothly interpolates between a non-AdS domain wall solution at $r=\infty$ and an AdS solution near $r=0$.

Combined Wronskian solutions to the 2D Toda molecule equation

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.

Toda molecule and Tomimatsu–Sato solution—towards the complete proof of Nakamura's conjecture

We discuss the Nakamura's conjecture stating that the Tomimatsu–Sato black hole solution with an integer deformation parameter n is composed of the special solutions of the Toda molecule equation at the nth lattice site. From the previous work, in which the conjecture was partly analytically proved, we go further towards the final full proof by rearranging the rotation parameter. The proof is explicitly performed for the highest and lowest orders. Though the proof for all orders still remains unsolved, the prospect to the full proof becomes transparent and workable by our method.

Solutions to the ultradiscrete Toda molecule equation expressed as minimum weight flows of planar graphs

We define a function by means of the minimum weight flow on a planar graph and prove that this function satisfies the ultradiscrete Toda molecule equation, its Bäcklund transformation and the two-dimensional Toda molecule equation. The method we employ in the proof can be considered as fundamental to the integrability of ultradiscrete soliton equations.

Black holes as generalised Toda molecules

In this note we compare the geodesic formalism for spherically symmetric black hole solutions with the black hole effective potential approach. The geodesic formalism is beneficial for symmetric supergravity theories since the symmetries of the larger target space lead to a complete set of commuting constants of motion that establish the integrability of the geodesic equations of motion, as shown in arXiv:1007.3209. We point out that the integrability lifts straightforwardly to the integrability of the equations of motion with a black hole potential. This construction turns out to be a generalisation of the connection between Toda molecule equations and geodesic motion on symmetric spaces known in the mathematics literature. We describe in some detail how this generalisation of the Toda molecule equations arises.

A q-difference version of the ϵ-algorithm

In this paper, a q-difference version of the ϵ-algorithm is proposed. By using determinant identities the solutions of an initial value problem thus arisen can be expressed as ratios of Hankel determinants. It is shown that in numerical analysis this algorithm can be used to compute the approximation limt→∞f(t), and in the field of integrable systems it could be viewed as the q-difference version of the modified Toda molecule equation.

STRINGS AND MEMBRANES FROM EINSTEIN GRAVITY, MATRIX MODELS AND W∞ GAUGE THEORIES AS PATHS TO QUANTUM GRAVITY

It is shown how w∞, w1+∞ gauge field theory actions in 2D emerge directly from 4D gravity. Strings and membranes actions in 2D and 3D originate as well from 4D Einstein gravity after recurring to the nonlinear connection formalism of Lagrange–Finsler and Hamilton–Cartan spaces. Quantum gravity in 3D can be described by a W∞ matrix model in D = 1 that can be solved exactly via the collective field theory method. We describe why a quantization of 4D gravity could be attained via a 2D quantum W∞ gauge theory coupled to an infinite-component scalar-multiplet. A proof that noncritical W∞ (super)strings are devoid of BRST anomalies in dimensions D = 27(D = 11), respectively, follows and which coincide with the critical (super)membrane dimensions D = 27(D = 11). We establish the correspondence between the states associated with the quasifinite highest weights irreducible representations of W∞, [Formula: see text] algebras and the quantum states of the continuous Toda molecule. Schrödinger-like quantum mechanics wave functional equations are derived and solutions are found in the zeroth-order approximation. Since higher-conformal spin W∞ symmetries are very relevant in the study of 2DW∞ gravity, the quantum Hall effect, large N QCD, strings, membranes, … it is warranted to explore further the interplay among all these theories.

The box–ball system and the N-soliton solution of the ultradiscrete KdV equation

Any state of the box–ball system (BBS) together with its time evolution is described by the N-soliton solution (with appropriate choice of N) of the ultradiscrete KdV equation. It is shown that simultaneous elimination of all ‘10’-walls in a state of the BBS corresponds exactly to reducing the parameters that determine ‘the size of a soliton’ by one. This observation leads to an expression for the solution to the initial-value problem (IVP) for the BBS. Expressions for the solution to the IVP for the ultradiscrete Toda molecule equation and the periodic BBS are also presented.

ULTRADISCRETIZATION OF THE TZITZEICA EQUATION

The trilinear form of the discrete Tzitzeica equation by Schief is found to be a discrete Toda molecule equation with a special boundary condition. Based on this fact, a higher order discrete Tzitzeica equation and an ultradiscrete Tzitzeica equation are obtained.

SOME ASPECTS OF THE TODA MOLECULE

A $q$ -discrete analog of the Toda molecule equation and its $N$ -soliton solution are constructed by using the bilinear method. The solution is expressed in the Casorati determinant form whose elements are given in terms of the $q$ -orthogonal polynomials.

Polynomials associated with equilibria of affine Toda–Sutherland systems

An affine Toda-Sutherland system is a quasi-exactly solvable multi-particle dynamics based on an affine simple root system. It is a `cross' between two well-known integrable multi-particle dynamics, an affine Toda molecule and a Sutherland system. Polynomials describing the equilibrium positions of affine Toda-Sutherland systems are determined for all affine simple root systems.

Affine Toda–Sutherland systems

A cross between two well-known integrable multi-particle dynamics, an affine Toda molecule and a Sutherland system, is introduced for any affine root system. Though it is not completely integrable but partially integrable, or quasi exactly solvable, it inherits many remarkable properties from the parents. The equilibrium position is algebraic, i.e. proportional to the Weyl vector. The frequencies of small oscillations near equilibrium are proportional to the affine Toda masses, which are essential ingredients of the exact factorisable S-matrices of affine Toda field theories. Some lower lying frequencies are integer times a coupling constant for which the corresponding exact quantum eigenvalues and eigenfunctions are obtained. An affine Toda-Calogero system, with a corresponding rational potential, is also discussed.

Conserved quantities of box and ball system

HTML view is not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button. Conserved quantities of box and ball system(BBS) are presented from the hungry Toda molecule equation, an inverse ultra-discrete limit of the BBS.

The Discrete Relativistic Toda Molecule Equation and a Padé Approximation Algorithm

The relativistic Toda molecule equation (RTM) describes a one-parameter deformation of coefficients of the recurrence relation of a class of biorthogonal polynomials including the Szego polynomials. In this paper, we present (i) explicit solutions of the discrete relativistic Toda molecule equation (d-RTM), (ii) a new Pade approximation algorithm for a given power series.

Proof of solitonical nature of box and ball systems by means of inverse ultra-discretization

A soliton cellular automaton, which represents movement of a finite number of balls in an array of boxes, is investigated. Its dynamics is described by an ultra-discrete equation obtained from an extended Toda molecule equation. The rules for soliton interactions and factorization property of the scattering matrices (Yang-Baxter relation) are proved by means of inverse ultra-discretization. The conserved quantities are also presented and used for another proof of the solitonical nature.

Soliton cellular automaton, Toda molecule equation and sorting algorithm

A direct connection between a soliton cellular automaton (SCA) and an ultra-discrete analogue of the Toda molecule equation (uTM equation) is clarified. A solution to the SCA is presented by means of that to the uTM equation. A sorting algorithm based on this connection is also constructed.

Dynamics of the finite Toda molecule over finite fields and a decoding algorithm

The finite Toda molecule over finite fields is introduced whose dynamics are completely classified by conserved quantities. The trajectories blow up in a finite time or are periodic. A BCH-Goppa decoding algorithm is designed. The number of Toda particles is congruent with the maximal number of errors to be decoded.