Dynamics of a coupled modified (1 + 1)‐dimensional Toda equation of BKP type

In this paper, we present a new coupled modified (1 + 1)‐dimensional Toda equation of BKP type (Kadomtsev‐Petviashvilli equation of B‐type), which is a reduction of the (2 + 1)‐dimensional Toda equation. Two‐soliton and three‐soliton solutions to the coupled system are derived. Furthermore, the N‐soliton solution is presented in the form of Pfaffian. The asymptotic analysis of two‐soliton solutions is studied to explain their collision properties. It is shown that the coupled system exhibit richer interaction phenomena including soliton fission, fusion, and mixed collision. Copyright © 2015 John Wiley & Sons, Ltd.     

Reduction and realization in toda and volterra

We construct a new symplectic, bi-Hamiltonian realization of the KM-system by reducing the corresponding one for the Toda lattice. The bi-Hamiltonian pair is constructed using a reduction theorem of Fernandes and Vanhaecke. In this paper we also review the important work of Moser on the Toda and KM-systems.      

New Solutions to the Ultradiscrete Soliton Equations

New solutions to the ultradiscrete soliton equations, such as the Box–Ball system, the Toda equation, etc. are obtained. One of the new solutions which we call a "negative-soliton" satisfies the ultradiscrete KdV equation (Box–Ball system) but there is not a corresponding traveling wave solution for the discrete KdV equation. The other one which we call a "static-soliton" satisfies the ultradiscrete Toda equation but there is not a corresponding traveling wave solution for the discrete Toda equation. A collision of a soliton with a negative-soliton generates many balls in a box over the capacity of the box in the Box–Ball system, while a collision of a soliton with the static-soliton describes, in the ultradiscrete limit, transmission of a soliton through junctions of a "nonuniform Toda equation." We have obtained exact solutions describing these phenomena.     

Toda equation and special polynomials associated with the Garnier system

We prove that a certain sequence of τ-functions of the Garnier system satisfies Toda equation. We construct algebraic solutions of the system by the use of Toda equation; then show that the associated τ-functions are expressed in terms of the universal character which is a generalization of Schur polynomial attached to a pair of partitions.     

Webs, Lenard schemes, and the local geometry of bi-Hamiltonian Toda and Lax structures

We introduce a criterion that a given bi-Hamiltonian structure admits a local coordinate system where both brackets have constant coefficients. This criterion is applied to the bi-Hamiltonian open Toda lattice in a generic point, which is shown to be locally isomorphic to a Kronecker odd-dimensional pair of brackets with constant coefficients. This shows that the open Toda lattice cannot be locally represented as a product of two bi-Hamiltonian structures. Near, a generic point, the bi-Hamiltonian periodic Toda lattice is shown to be isomorphic to a product of two open Toda lattices (one of which is a (trivial) structure of dimension 1). While the above results might be obtained by more traditional methods, we use an approach based on general results on geometry of webs. This demonstrates the possibility of applying a geometric language to problems on bi-Hamiltonian integrable systems; such a possibility may be no less important than the particular results proved in this paper. Based on these geometric approaches, we conjecture that decompositions similar to the decomposition of the periodic Toda lattice exist in local geometry of the Volterra system, the complete Toda lattice, the multidimensional Euler top, and a regular bi-Hamiltonian Lie coalgebra. We also state general conjectures about the geometry of more general "homogeneous" finite-dimensional bi-Hamiltonian structures. The class of homogeneous structures is shown to coincide with the class of systems integrable by Lenard scheme. The bi-Hamiltonian structures which admit a non-degenerate Lax structure are shown to be locally isomorphic to the open Toda lattice.     

Quantum generalized Toda system

We construct a ??spectral curve?? for the generalized Toda system, which allows efficiently finding its quantization. In turn, the quantization is realized using the technique of the quantum characteristic polynomial for the Gaudin system and an appropriate Alder-Kostant-Symes reduction. We also discuss some relations of this result to the recent consideration of the Drinfeld Zastava space, the monopole space, and corresponding symmetries of the Borel Yangian.     

q-Discretization of the two-dimensional Toda equations

q-Discrete versions of the two-dimensional Toda molecule equation and the two-dimensional Toda lattice equation are proposed through the direct method. The Bäcklund transformation and the Lax pair of the former are obtained. Moreover, the reduction to theq-discrete cylindrical Toda equations is also discussed.Department of Electronical Engeneering, Doshisha University, Kyoto 610-03, Japan. Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606, Japan. (On leave from Department of Applied Mathematics, Faculty of Engineering, Hiroshima University.) Department of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153, Japan. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 99, No. 3, pp. 390–398, June, 1994.     

Solutions for Toda system on Riemann surface with boundary

In this paper, we study the solutions for Toda system on Riemann surface with boundary. We prove a sufficient condition for the existence of solution of Toda system in the critical case.     

Sharp estimates for fully bubbling solutions of a SU(3) Toda system

In this paper, we obtain sharp estimates of fully bubbling solutions of SU(3) Toda system in a compact Riemann surface. In geometry, the SU(n?+?1) Toda system is related to holomorphic curves, harmonic maps or harmonic sequences of the Riemann surface to ${\mathbb{CP}^n}$ . In order to compute the Leray?CSchcuder degree for the Toda system, we have to obtain accurate approximations of the bubbling solutions. Our main goals in this paper are (i) to obtain a sharp convergence rate, (ii) to completely determine the locations, and (iii) to derive the ${\partial _z^2}$ condition, a unexpected and important geometric constraint.     

Construction of trigonometric Todar-matrices via a hamiltonian reduction of the cotangent bundle over loop groups

It is shown how the classical trigonometric τ-matrices of the Toda model can be obtained by a Hamiltonian reduction of the cotangent bundle over loop groups. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 113, No. 1, pp. 3–12, October, 1997.     

Involutive solutions for toda and langmuir lattices through nonlinearization of lax pairs

Under the constraint determined by a relation (a _n ,b _n )^T={f(?)} _n between the reflectionless potentials and the eigenfunctions of the general discrete Schrödinger eigenvalue problem, the Lax pair of the Toda lattice is nonlinearized to be a finite-dimensional difference system and a nonlinear evolution equation, while the solution varietyN of the former is an invariant set of S-flows determined by the latter, and the constants of the motion for the algebraic system are presented.f maps the solution of the algebraic system into the solution of a certain stationary Toda equation. Similar results concerning the Langmuir lattice are given, and a relation between the two difference systems, which are the spatial parts of the nonlinearized Lax pairs of the Toda lattice and Langmuir lattice, is discussed.     

Cluster characters and the combinatorics of Toda systems

We survey some connections between Toda systems and cluster algebras. One of these connections is based on representation theory: it is known that Laurent expansions of cluster variables are generating functions of Euler characteristics of quiver Grassmannians, and the same turns out to be true of the Hamiltonians of the open relativistic Toda chain. Another connection is geometric: the closed nonrelativistic Toda chain can be regarded as a meromorphic Hitchin system and studied from the standpoint of spectral networks. From this standpoint, the combinatorial formulas for the Hamiltonians of the open relativistic system are sums of trajectories of differential equations defined by the closed nonrelativistic spectral curves.     

Phase portraits of the full symmetric Toda systems on rank-2 groups

We continue investigations begun in our previous works where we proved that the phase diagram of the Toda system on special linear groups can be identified with the Bruhat order on the symmetric group if all eigenvalues of the Lax matrix are distinct or with the Bruhat order on permutations of a multiset if there are multiple eigenvalues. We show that the phase portrait of the Toda system and the Hasse diagram of the Bruhat order coincide in the case of an arbitrary simple Lie group of rank 2. For this, we verify this property for the two remaining rank-2 groups, Sp(4,?) and the real form of G₂.     

A convexity theorem for three tangled Hamiltonian torus actions,and super-integrable systems

A completely integrable system on a symplectic manifold is called super-integrable when the number of independent integrals of motion is more than half the dimension of the manifold. Several important completely integrable systems are super-integrable: the harmonic oscillators, the Kepler system, the non-periodic Toda lattice, etc. Motivated by an additional property of the super-integrable system of the Toda lattice (Agrotis et al., 2006) [2], we will give a generalization of the Atiyah and Guillemin–Sternberg?s convexity theorem.     

Integrability of the semi-discrete Toda equation with self-consistent sources

A new procedure to construct and solve soliton equations with self-consistent sources (SESCSs) is applied to the semi-discrete Toda equation, based on its bilinear from. Bilinear Bäcklund transformation (BT) for the semi-discrete Toda ESCS is presented. Starting from the BT, a Lax pair is derived for the semi-discrete Toda ESCS.     

A systematic study of the Toda lattice in the context of the Hamilton-Jacobi theory

Integrability of the Toda lattice is studied in the framework of the Hamilton-Jacobi theory of separation of variables. It is shown based on the Benenti theory that only the two dimensional, non-periodic Toda lattice is separable in the sense of the existence of a point transformation to a separable system of coordinates. The separation of variables is exhibited in this case explicitly.     

Integrals of open two-dimensional lattices

We present an explicit formula for integrals of the open two-dimensional Toda lattice of type A_n. This formula is applicable for various reductions of this lattice. As an illustration, we find integrals of the G₂ Toda lattice. We also reveal a connection between the open An Toda and Shabat-Yamilov lattices.     

Symmetry groups of differential-difference equations and their compatibility

It is shown that the intrinsic determining equations of a given differential-difference equation (DDE) can be derived by the compatibility between the original equation and the intrinsic invariant surface condition. The (2+1)-dimensional Toda lattice, the special Toda lattice and the DD-KP equation serving as examples are used to illustrate this approach. Then, Bäcklund transformations of the (2+1)-dimensional DDEs including the special Toda lattice, the modified Toda lattice and the DD-KZ equation are presented by using the non-intrinsic direct method. In addition, the Clarkson-Kruskal direct method is developed to find similarity reductions of the DDEs.     

A hierarchy of differential-difference equations,conservation laws and new integrable coupling system

Starting from a discrete spectral problem with two arbitrary parameters, a hierarchy of nonlinear differential-difference equations is derived. The new hierarchy not only includes the original hierarchy, but also the well-known Toda equation and relativistic Toda equation. Moreover, infinitely many conservation laws for a representative discrete equation are given. Further, a new integrable coupling system of the resulting hierarchy is constructed.     

Hypergeometric Solutions of Soliton Equations

We consider multivariable hypergeometric functions related to Schur functions and show that these hypergeometric functions are tau functions of the KP hierarchy and are simultaneously the ratios of Toda lattice tau functions evaluated at certain values of higher Toda lattice times. The variables of the hypergeometric functions are related to the higher times of those hierarchies via a Miwa change of variables. The discrete Toda lattice variable shifts the parameters of the hypergeometric functions. We construct the determinant representation and the integral representation of a special type for the KP tau functions. We write a system of linear differential and difference equations on these tau functions, which play the role of string equations.