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

Habibullin, I. T.; Poptsova, Mariya Nikolaevna

doi:10.1088/1751-8113/48/11/115203

Cited by 5 publications

(6 citation statements)

References 33 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%

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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%

“…To construct the conservation laws, we apply the method of formal diagonalization suggested in [32,33] and developed in [34].…”

Section: Construction Of the Recursion Operators And Conservation Law...mentioning

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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“…The proof of Theorem 1 was given in work [7]. It should be said that in the proof of the theorem, there was constructed a formal series = satisfying the equation It follows from Theorem 1 that the discrete operator = −1 is reduced to the quasidiagonal form 0 = −1 ℎ by the transformation…”

Section: Asymptotic Diagonalization Of the Discrete Operator In The Vmentioning

confidence: 99%

“…An algorithm for solving the problem on asymptotic diagonalization of a discrete operator in the vicinity of the singular point and its applications in the theory of integrable nonlinear discrete equations was discussed in details in works [5,6,7]. Interesting results on non-autonomous discrete dynamical systems were obtained by using the formal diagonalization method in works [8,9].…”

Section: Introductionmentioning

confidence: 99%

Symmetries and conservation laws for a two-component discrete potentiated Korteweg - de Vries equation

Poptsova¹,

Habibullin²

2016

Ufimskii Matematicheskii Zhurnal

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In the work we discuss briefly a method for constructing a formal asymptotic solution to a system of linear difference equations in the vicinity of a special value of the parameter. In the case when the system is the Lax pair for some nonlinear equation on a square graph, the found formal asymptotic solution allows us to describe the conservation laws and higher symmetries for this nonlinear equation. In the work we give a complete description of a series of conservation laws and the higher symmetries hierarchy for a discrete potential two-component Korteweg-de Vries equation.

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“…Unfortunately, our Lax pairs for A N are not given in terms of the Cartan-Weyl basis of the algebra as it was in the case of (1.1) (see [10]), so there are problems with generalization to other algebras. Note that alternative examples of discretizations of the Toda lattices related with the algebra A (1) N −1 are investigated in [21]- [24].…”

Section: Introductionmentioning

confidence: 99%

Discrete exponential type systems on a quad graph, corresponding to the affine Lie algebras $A^{(1)}_{N-1}$

Habibullin¹,

Khakimova²

2019

Preprint

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The article deals with the problem of the integrable discretization of the well-known Drinfeld-Sokolov hierarchies related to the Kac-Moody algebras. A class of discrete exponential systems connected with the Cartan matrices has been suggested earlier in [1] which coincide with the corresponding Drinfeld-Sokolov systems in the continuum limit. It was conjectured that the systems in this class are all integrable and the conjecture has been approved by numerous examples. In the present article we study those systems from this class which are related to the algebras A(1) N −1 . We found the Lax pair for arbitrary N , briefly discussed the possibility of using the method of formal diagonalization of Lax operators for describing a series of local conservation laws and illustrated the technique using the example of N = 3. Higher symmetries of the system A (1) N −1 are presented in both characteristic directions. Found recursion operator for N = 3. It is interesting to note that this operator is not weakly nonlocal.

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

Cited by 5 publications

References 33 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

Symmetries and conservation laws for a two-component discrete potentiated Korteweg - de Vries equation

Discrete exponential type systems on a quad graph, corresponding to the affine Lie algebras $A^{(1)}_{N-1}$

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