General solutions of the two-dimensional system of Volterra equations which realize the B�cklund transformation for the Toda lattice

Leznov, A. N.; Savel'ev, M. V.; Смирнов, В. Г.

doi:10.1007/bf01086394

Cited by 10 publications

(10 citation statements)

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“…Similar representations for solutions of integrable equations are widely known, see e.g. papers [43,44] devoted to the Toda lattices and the monograph [45] containing a lot of examples. The proof of formulae ( 22) and ( 23) given below is based on the Jacobi identity for Wronskians…”

Section: Determinant Formulaementioning

confidence: 90%

Bogoyavlensky lattices and generalized Catalan numbers

Adler¹

2022

Preprint

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We study the problem of the decay of initial data in the form of a unit step for the Bogoyavlensky lattices. In contrast to the Gurevich-Pitaevskii problem of the decay of initial discontinuity for the KdV equation, it turns out to be exactly solvable, since the dynamics is linearizable due to termination on the half-line. The answer is written in terms of generalized hypergeometric functions, which serve as exponential generating functions for generalized Catalan numbers. This can be proved by the fact that the generalized Hankel determinants for these numbers are equal to 1, which is a well-known result in combinatorics. Another method is based on a non-autonomous symmetry reduction consistent with the dynamics. It reduces the lattice equation to a finite-dimensional system and makes it possible to solve the problem for a more general finite-parameter family of initial data.

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Section: Determinant Formulaementioning

confidence: 90%

Bogoyavlensky lattices and generalized Catalan numbers

Adler¹

2022

Preprint

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“…As was mentioned in section 2, the authors of [5,6] derived general solutions of the 2DVE in the case of finite chain. Here we would like to present several other classes of solutions for this system, namely ones that can be obtained from the results presented in this paper using (3.19a) and (3.19b).…”

Section: 33)mentioning

confidence: 98%

“…System (2.5) is known since the works of Leznov, Savel'ev and Smirnov [5,6] where the authors demonstrated that this system, which was named the 'two-dimensional Volterra equation', represents the Bäcklund transformations for the 2DTL and constructed its general solutions in the finite case using the results of [14,15]. Another model that we would like to discuss is the RTC [8,9,10,11] which can be presented as a Hamiltonian system…”

Section: Wgc 2dve 2dtl and Rtcmentioning

confidence: 99%

“…where ∂ and ∂ stand for ∂/∂z and ∂/∂ z, is that it can be viewed as a connecting link between almost all Toda-like chains. In the following section, we present its relationships with the Wu-Geng chain (WGC) [4], the 'two-dimensional Volterra equation' (2DVE) [5,6], the two-dimensional Toda lattice (2DTL) [7] and the relativistic Toda chain (RTC) [8,9,10,11] The main goal of this work is to obtain some explicit solutions for (1.4). We will not elaborate the inverse scattering transform or the algebro-geometric approach from scratch.…”

Section: Introductionmentioning

confidence: 99%

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Explicit solutions for a (2+1)-dimensional Toda-like chain

Vekslerchik

2013

Preprint

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We consider a (2+1)-dimensional Toda-like chain which can be viewed as a two-dimensional generalization of the Wu-Geng model and which is closely related to the two-dimensional Volterra, two-dimensional Toda and relativistic Toda lattices. In the framework of the Hirota direct approach, we present equations describing this model as a system of bilinear equations that belongs to the Ablowitz-Ladik hierarchy. Using the Jacobi-like determinantal identities and the Fay identity for the thetafunctions, we derive its Toeplitz, dark-soliton and quasiperiodic solutions as well as the similar set of solutions for the two-dimensional Volterra chain.

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“…Proceeding from the known solutions of the original system, solutions of (1.20) can easily be constructed in explicit form as continued fractions (see [18]). …”

Section: Kdv)mentioning

confidence: 99%

Exactly and completely integrable nonlinear dynamical systems

Leznov

Saveliev

1989

Acta Appl Math

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This survey is devoted to a consistent exposition of the group-algebraic methods for the integration of systems of nonlinear partial differential equations possessing a nontrivial internal symmetry algebra. Samples of exactly and completely integrable wave and evolution equations are considered in detail, including the generalized (periodic and finite nonperiodic) Toda lattice, nonlinear Schr6dinger, Korteweg~le Vries, Lotka-Volterra equations, etc.). For exactly integrable systems, the general solutions of the Cauchy and Goursat problems are given in an explicit form, while for completely integrable systems an effective method for the construction of their soliton solutions is developed.Application of the developed methods to a differential geometry problem of the classification of integrable manifolds embeddings is discussed.Supersymmetric extensions are constructed for exactly integrable systems. Using an example of the generalized Toda lattice, a quantization scheme is developed. It includes an explicit derivation of the corresponding Pleisenberg operators and their description in terms of Hopf-type quantum algebras.Among multidimensional systems, the four-dimensional self-dual Yang-Mills equations are investigated with the aim of constructing their general solutions.

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General solutions of the two-dimensional system of Volterra equations which realize the B�cklund transformation for the Toda lattice

Cited by 10 publications

References 10 publications

Bogoyavlensky lattices and generalized Catalan numbers

Bogoyavlensky lattices and generalized Catalan numbers

Explicit solutions for a (2+1)-dimensional Toda-like chain

Exactly and completely integrable nonlinear dynamical systems

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