Formal diagonalization of a discrete lax operator and conservation laws and symmetries of dynamical systems

Habibullin, I. T.; Yangubaeva, Marina

doi:10.1007/s11232-013-0125-y

Cited by 18 publications

(57 citation statements)

References 25 publications

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“…To construct the conservation laws, we apply the method of the formal diagonalization suggested in [32,33] and developed in [34]. The first equation in (6.10) has singular point λ = ∞ (f has a pole at λ = ∞).…”

Section: Conservation Laws Via the Lax Pair For A Volterra Type Chainmentioning

confidence: 99%

“…p n do not vanish if the variable p n satisfy the inequality p n = 0 for all n. According to the Proposition 1 in [33] the first system in (6.10) can be diagonalized, i.e. there exist formal series 6.19) such that the formal change of the variables ψ = T ϕ converts the system (6.16) to the system of the diagonal form…”

Section: Conservation Laws Via the Lax Pair For A Volterra Type Chainmentioning

confidence: 99%

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On a method for constructing the Lax pairs for nonlinear integrable equations

Habibullin

Khakimova

Poptsova³

2015

J. Phys. A: Math. Theor.

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Abstract.We suggest a direct algorithm for searching the Lax pairs for nonlinear integrable equations. It is effective for both continuous and discrete models. The first operator of the Lax pair corresponding to a given nonlinear equation is found immediately, coinciding with the linearization of the considered nonlinear equation. The second one is obtained as an invariant manifold to the linearized equation. A surprisingly simple relation between the second operator of the Lax pair and the recursion operator is discussed: the recursion operator can immediately be found from the Lax pair. Examples considered in the article are convincing evidence that the found Lax pairs differ from the classical ones. The examples also show that the suggested objects are true Lax pairs which allow the construction of infinite series of conservation laws and hierarchies of higher symmetries. In the case of the hyperbolic type partial differential equation our algorithm is slightly modified; in order to construct the Lax pairs from the invariant manifolds we use the cutting off conditions for the corresponding infinite Laplace sequence. The efficiency of the method is illustrated by application to some equations given in the Svinolupov-Sokolov classification list for which the Lax pairs and the recursion operators have not been found earlier.

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Section: Conservation Laws Via the Lax Pair For A Volterra Type Chainmentioning

confidence: 99%

Section: Conservation Laws Via the Lax Pair For A Volterra Type Chainmentioning

confidence: 99%

On a method for constructing the Lax pairs for nonlinear integrable equations

Habibullin

Khakimova

Poptsova³

2015

J. Phys. A: Math. Theor.

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“…Even for comparatively well studied systems of linear differential equations the similar problem does not have a complete solution. Theorem 1 shows that the key step in solving the problem in the discrete case is to reduce equation (1.1) to the special form (2.8) at the vicinity of the singular point λ = λ 0 (see [1,2]). Below we discuss a method for searching such a special form for a given equation (1.1).…”

Section: Appendix Algorithm Of Finding Special Form Of the Linear DImentioning

confidence: 99%

“…In [1,2] a method of asymptotic diagonalization of the Lax pairs associated with nonlinear discrete and semidiscrete models was suggested allowing to describe conservation laws and higher symmetries for the corresponding nonlinear models. Efficiency of the method was approved by application to numerous examples in [2,3,4].…”

Section: Introductionmentioning

confidence: 99%

Asymptotic diagonalization of the discrete Lax pair around singularities and conservation laws for dynamical systems

Habibullin¹,

Poptsova²

2015

J. Phys. A: Math. Theor.

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Abstract. A method of the formal diagonalization of the discrete linear operator with a parameter is studied. In the case when the operator provides a Lax operator for a nonlinear quad system the formal diagonalization method allows one to describe effectively conservation laws and generalized symmetries for this system. Asymptotic representation of the Lax operators eigenfunctions are constructed and infinite series of conservation laws are described for the quad system connected with A (1) 3 affine Lie algebra, for the modified discrete Boussinesq system and for the discrete Tzitzeica equation. For a newly found multiquadratic discrete model conservation laws and several generalized symmetries are presented.

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“…Here D τ ′ and D t ′ k are the operators of total derivatives in virtue of equations (15) and (16), respectively, with a definition shown in (17).…”

Section: Generalized Symmetries In the M-directionmentioning

confidence: 99%

An Unusual Series of Autonomous Discrete Integrable Equations on a Square Lattice

Garifullin

Yamilov

2019

Theor Math Phys

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We present an infinite series of autonomous discrete equations on the square lattice possessing hierarchies of autonomous generalized symmetries and conservation laws in both directions. Their orders in both directions are equal to κN , where κ is an arbitrary natural number and N is equation number in the series. Such a structure of hierarchies is new for discrete equations in the case N > 2.Symmetries and conservation laws are constructed by means of the master symmetries. Those master symmetries are found in a direct way together with generalized symmetries. Such construction scheme seems to be new in the case of conservation laws. One more new point is that, in one of directions, we introduce the master symmetry time into coefficients of discrete equations.In most interesting case N = 2 we show that a second order generalized symmetry is closely related to a relativistic Toda type integrable equation. As far as we know, this property is very rare in the case of autonomous discrete equations.

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Formal diagonalization of a discrete lax operator and conservation laws and symmetries of dynamical systems

Abstract: An alternative method of constructing the formal diagonalization for the discrete Lax operators is proposed which can be used to calculate conservation laws and in some cases generalized symmetries for discrete dynamical systems. Discrete

Cited by 18 publications

References 25 publications

On a method for constructing the Lax pairs for nonlinear integrable equations

On a method for constructing the Lax pairs for nonlinear integrable equations

Asymptotic diagonalization of the discrete Lax pair around singularities and conservation laws for dynamical systems

An Unusual Series of Autonomous Discrete Integrable Equations on a Square Lattice

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