Maximum Modulus Estimates for the Linear Steady Stokes System

A maximum modulus estimate for the Stokes system in bounded domains of ℝⁿ (n ≥ 2) is established via methods of hydrodynamical potential theory. The method is based on the unique solvability of the boundary integral equations' system resulting from the double layer potential ansatz together with a projection onto the normal field on the boundary. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Boundary integral equations and maximum modulus estimates for the Stokes system

A maximum modulus estimate for the Stokes system in bounded domains of ℝⁿ (n ≥ 2) is established via methods of hydrodynamical potential theory. The method is based on the unique solvability of the boundary integral equations' system resulting from the double layer potential ansatz together with a projection onto the normal field on the boundary. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An integral equation of the first kind for a free boundary value problem of the stationary Stokes' equations

We investigate a free boundary value problem of the stationary Stokes' equations. In a previous paper adapted hydrodynamical potentials have been constructed and their jump relations have been discussed. Here we study a direct method to obtain an equivalent boundary integral equations' system of the first kind. Its solution properties are investigated in the framework of strongly elliptic pseudodifferential operators. For numerical purposes a suitable representation formula for the variational equation is given in terms of integro-differential operators which avoids the evaluation of hypersingular integrals.     

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.     

Transient RankingMethods for the Simplification of Nonlinear DAE Systems in Analog Circuit Design

Symbolic analysis of analog circuits using computer algebra is limited by the complexity problem: even for small circuits the symbolic equation systems get too large to be handled efficiently. In the past years simplification techniques have been developed which reduce the complexity of such equation systems. In this paper we are focusing on simpli fication techniques for equations modeling nonlinear transient analog circuits. Here it is necessary to control the dynamic behavior of the simplified system. For this, we developed two different methods which predict the influence of a simplification on the equations' transient solution. We will describe both methods and compare their efficiency and accuracy.     

Peakons,strings, and the finite Toda lattice

As is well‐known, the Toda lattice flow may be realized as an isospectral flow of a Jacobi matrix. A bijective map from a discrete string problem with positive weights to Jacobi matrices allows the pure peakon flow of the Camassa‐Holm equation to be realized as an isospectral Jacobi flow as well. This gives a unified picture of the Toda, Jacobi, and multipeakon flows, and leads to explicit solutions of the Jacobi flows via Stieltjes' determination of the continued fraction expansion of a Stieltjes transform. A simple modification produces a bijection from generalized strings, with positive and negative weights, to singular Jacobi matrices, and thus brings peakon/antipeakon flows into the same picture. © 2001 John Wiley & Sons, Inc.     

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 BRACKETS AND THE CATEGORY OF HOMOTOPY PAIRS

A new treatment is given of the cylinder-web diagram and associated diagonal sequences in homotopy pair theory. The efficiency of the diagram as a machine for computing homotopy pair groups is enhanced by a result that traces the path of a Toda bracket element through the arrows of the diagram. The diagonal factorization problem for a homotopy pair class is studied and related to the behaviour of Toda brackets. A necessary and sufficient condition for the vanishing of a Toda bracket is obtained.     

ON CUP-ONE AND TODA BRACKET

Abstract

The definition of the cup-one construction in various cases is recalled. A variation expressible in terms of the Toda bracket is given, enabling nontriviality of Toda brackets of a certain type to be established in a rather simple way from knowledge of the behaviour of cup-one.     

Series solutions of confluent Heun equations in terms of incomplete Gamma-functions

We present a simple systematic algorithm for construction of expansions of the solutions of ordinary differential equations with rational coefficients in terms of mathematical functions having indefinite integral representation. The approach employs an auxiliary equation involving only the derivatives of a solution of the equation under consideration. Using power-series expansions of the solutions of this auxiliary equation, we construct several expansions of the four confluent Heun equations'' solutions in terms of the incomplete Gamma-functions. In the cases of single- and double-confluent Heun equations the coefficients of the expansions obey four-term recurrence relations, while for the bi- and tri-confluent Heun equations the recurrence relations in general involve five terms. Other expansions for which the expansion coefficients obey recurrence relations involving more terms are also possible. The particular cases when these relations reduce to ones involving less number of terms are identified. The conditions for deriving closed-form finite-sum solutions via right-hand side termination of the constructed series are discussed.     

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.     

The Bäcklund transformation and novel solutions for the Toda lattice

The bilinear Bäcklund transformation for the Toda lattice is derived from the Darboux transformation and some novel solutions to the Toda lattice are obtained through a modified bilinear Bäcklund transformation.     

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.     

Explicit solutions of classical generalized toda models

The explicit solution of the generalized Toda models connected with the root systems of simple Lie algebras is constructed. This solution is obtained by a special projection of geodesic flows on some symmetric spaces on the phase space of the Toda models. In particular, for the usualN-body Toda model the solutions are expressed by minors of the exponent of the Jacobi matrix depending on initial data.This paper has been published early as a preprint ITEP-157 (1978)     

2D Toda lattice equation with self-consistent sources: Casoratian type solutions,bilinear Bäcklund transformation and Lax pair

The two-dimensional Toda lattice equation with self-consistent sources is proposed based on its bilinear forms. Casoratian-type solutions and Bäcklund transformation (BT) for the bilinear forms are presented. Starting from the BT, a Lax pair is derived for the 2D Toda lattice with self-consistent sources.     

Wave functions of the Toda chain with boundary interaction

We give an integral representation of the wave functions of the quantum N-particle Toda chain with boundary interaction. In the case of the Toda chain with a one-boundary interaction, we obtain the wave function by an integral transformation from the wave functions of the open Toda chain. The kernel of this transformation is given explicitly in terms of -functions. The wave function of the Toda chain with a two-boundary interaction is obtained from the previous wave functions by an integral transformation. In this case, the difference equation for the kernel of the integral transformation admits a separation of variables. The separated difference equations coincide with the Baxter equation.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 142, No. 2, pp. 346–364, February, 2005.     

NEW SYMPLECTIC MAPS： INTEGRABILITY AND LAX REPRESENTATION

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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.      

Symmetries,Conservation Laws,and Noether's Theorem for Differential‐Difference Equations

This paper mainly contributes to the extension of Noether's theorem to differential‐difference equations. For this purpose, we first investigate the prolongation formula for continuous symmetries, which makes a characteristic representation possible. The relations of symmetries, conservation laws, and the Fréchet derivative are also investigated. For nonvariational equations, because Noether's theorem is now available, the self‐adjointness method is adapted to the computation of conservation laws for differential‐difference equations. Several differential‐difference equations are investigated as illustrative examples, including the Toda lattice and semidiscretizations of the Korteweg–de Vries (KdV) equation. In particular, the Volterra equation is taken as a running example.     

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.