Integration of equation of  Toda periodic chain kind

Babajanov, Bazar; Khasanov, A. B.

doi:10.13108/2017-9-2-17

Cited by 10 publications

(2 citation statements)

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“…The method of invariant manifolds is one of the most common methods for constructing solutions to nonlinear equations (see, for example, [8]). As alternative approaches to the problem of constructing solutions to nonlinear equations, we should mention the method of conditional symmetries [6,7,9], the Puiseux series method [10], the method of asymptotic expansions [11] and the method of finite-gap integration [12][13][14][15][16][17][18][19][20][21].…”

Section: Introductionmentioning

confidence: 99%

Construction of exact solutions to the Ruijsenaars–Toda lattice via generalized invariant manifolds

2022

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The article discusses a new method for constructing particular solutions of nonlinear integrable lattices, based on the concept of a generalized invariant manifold (GIM). In contrast to the finite-gap integration method, instead of the eigenfunctions of the Lax operators, we use a joint solution of the linearized equation and GIM. This makes it possible to derive Dubrovin type equations not only in the time variable t, but also in the spatial discrete variable n. We illustrate the efficiency of the method using the Ruijsenaars–Toda lattice as an example, for which we find some new solutions: a real and bounded particular solution in the form of a kink, a periodic solution expressed in terms of Jacobi elliptic functions and a solution expressed through Weierstrass ℘ -function.

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Section: Introductionmentioning

confidence: 99%

Construction of exact solutions to the Ruijsenaars–Toda lattice via generalized invariant manifolds

2022

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“…The finite-gape integration method is effectively applied to nonlinear integrable chains as well. Finite-gap solutions of nonlinear Volterra and Toda lattices, as well as the relativistic Toda lattice, were investigated in a number of works [8,[10][11][12][15][16][17][18][19]. In them one can find explicit formulas for the solutions of these chains in terms of the Riemann theta functions, as well as analogues of Dubrovin's equations (1.3), which determine the dynamics in time t. As far as the authors know, analogs of the Dubrovin equations describing the dynamics in the spatial variable n ∈ Z have not been presented before.…”

Section: Introductionmentioning

confidence: 99%

Construction of exact solutions to the Ruijsenaars-Toda lattice via generalized invariant manifolds

Habibullin¹,

Khakimova²,

Smirnov³

2021

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The article discusses a new method for constructing algebro-geometric solutions of nonlinear integrable lattices, based on the concept of a generalized invariant manifold (GIM). In contrast to the finite-gap integration method, instead of the eigenfunctions of the Lax operators, we use a joint solution of the linearized equation and GIM. This makes it possible to derive Dubrovin type equations not only in the time variable t, but also in the spatial discrete variable n. We illustrate the efficiency of the method using the Ruijsenaars-Toda lattice as an example, for which we have derived a real and bounded particular solution in the form of a kink, a periodic solution expressed in terms of Jacobi elliptic functions and a solution expressed through Weierstrass ℘-function.

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