Lie Algebras of Slow Growth and Klein-Gordon PDE

Миллионщиков, Дмитрий Владимирович

doi:10.1007/s10468-018-9794-4

Cited by 21 publications

(21 citation statements)

References 24 publications

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“…The first idea of the proof [33], [34] is to represent the characteristic Lie algebra (sinh( )) as the Lie algebra ℒ ( 0 , 1 , 2 ) generated not by two, but by three elements 0 , 1 , 2 . The advantage of the representation (sinh( )) = ℒ ( 0 , 1 , 2 ) is the following two commutation relations:…”

Section: New Operators Can Be Written As Followsmentioning

confidence: 99%

“…In paper [21], the characteristic algebra of the sine-Gordon equation, having a growth rate of 3 2 was studied, and in [25], there was studied the characteristic algebra of the Tzitzeica equation growing at an average rate of 4 3 . In papers [33], [34] infinite linear bases for these Lie algebras were constructed and it was shown that they are isomorphic to the nonnegative part of the loop algebra s (2, C) ⊗ C[ , −1 ] and to the twisted loop algebra of the simple Lie algebra s (3, C), respectively.…”

Section: Introductionmentioning

confidence: 99%

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Characteristic algebras and integrable exponential systems

Миллионщиков

Смирнов

2021

Ufimsk. Mat. Zh.

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In the present paper we study characteristic algebras for exponential systems corresponding to degenerate Cartan matrices. These systems generalize hyperbolic sine-Gordon and Tzitzeica equations well-known in the theory of integrable systems. For such systems, corresponding to Cartan matrices of rank 2, we describe explicitly characteristic algebras in terms of generators and relations and we prove that they have linear growth. We study the relations between the higher symmetries of these systems and the structure of their characteristic algebras. We describe completely the higher symmetries of exponential systems corresponding to the Cartan matrix of affine Lie algebra(1) 2 . We also obtain partial results on symmetries of such systems corresponding to other degenerate Cartan matrices of rank 2. We propose a conjecture on the structure of higher symmetries of arbitrary exponential system corresponding to a degenerate Cartan matrix. We study an interesting combinatorics related to an operator generating a characteristic algebra in the simplest case for a Darboux integrable Liouville equation. The found combinatorial properties can be very useful for proving the aforementioned conjecture on the structure of higher symmetries. Moreover, in the present paper we give a rigorous meaning to the concept of a characteristic algebra of a hyperbolic system used for a long time in the literature. We do this by means of the notion of Lie-Rinehart algebra and at the examples we demonstrate that such formalization is indeed needed.

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Section: New Operators Can Be Written As Followsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Characteristic algebras and integrable exponential systems

Миллионщиков

Смирнов

2021

Ufimsk. Mat. Zh.

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“…These equalities means that for any Z ∈ L y and any a ∈ A, the element aZ ∈ L y . In this case the algebra L y is called the Lie-Rinehart algebra [20,22].…”

Section: Introductionmentioning

confidence: 99%

“…20) Now we will work with equations (2.10)-(2.12) to clarify functions F 2 , F 4 . Let us express F 4,uz , F 2,uz from (2.10) and substitute them into (2.12).…”

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confidence: 99%

Lax Pair for a Novel Two-Dimensional Lattice

Кузнецова¹

2021

SIGMA

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In paper by I.T. Habibullin and our joint paper the algorithm for classification of integrable equations with three independent variables was proposed. This method is based on the requirement of the existence of an infinite set of Darboux integrable reductions and on the notion of the characteristic Lie-Rinehart algebras. The method was applied for the classification of integrable cases of different subclasses of equations u n,xy = f (u n+1 , u n , u n−1 , u n,x , u n,y ) of special forms. Under this approach the novel integrable chain was obtained. In present paper we construct Lax pair for the novel chain. To construct the Lax pair, we use the scheme suggested in papers by E.V. Ferapontov. We also study the periodic reduction of the chain.

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“…An effective criterion of the Darboux integrability of the system is connected with properties of an associated algebraic structures. More precisely, the characteristic Lie-Rinehart algebras [20,21] assigned to both characteristic directions have to be of a finite dimension. Since the obtained hyperbolic system is of a very specific form, this allows us to study effectively the characteristic algebras.…”

Section: Introductionmentioning

confidence: 99%

Classification of a subclass of quasilinear two-dimensional lattices by means of characteristic algebras

Nikolaevna

2019

Ufimsk. Mat. Zh.

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We consider a classification problem of integrable cases of the Toda type twodimensional lattices , = ( +1 , , −1 , , , , ). The function = ( 1 , 2 , · · · 5 ) is assumed to be analytic in a domain ⊂ C 5 . The sought function = ( , ) depends on real , and integer . Equations with three independent variables are complicated objects for study and especially for classification. It is commonly accepted that for a given equation, the existence of a large class of integrable reductions indicates integrability. Our classification algorithm is based on this observation. We say that a constraint 0 = ( , ) defines a degenerate cutting off condition for the lattice if it divides this lattice into two independent semi-infinite lattices defined on the intervals −∞ < < 0 and 0 < < +∞, respectively. We call a lattice integrable if there exist cutting off boundary conditions allowing us to reduce the lattice to an infinite number of hyperbolic type systems integrable in the sense of Darboux. Namely, we require that lattice is reduced to a finite system of such kind by imposing degenerate cutting off conditions at two different points = 1 , = 2 for arbitrary pair of integers 1 , 2 . Recall that a system of hyperbolic equations is called Darboux integrable if it admits a complete set of integrals in both characteristic directions. An effective criterion of the Darboux integrability of the system is connected with properties of an associated algebraic structures. More precisely, the characteristic Lie-Rinehart algebras assigned to both characteristic directions have to be of a finite dimension. Since the obtained hyperbolic system is of a very specific form, the characteristic algebras are effectively studied. Here we focus on a subclass of quasilinear lattices of the form , = ( −1 , , +1 ) , + ( −1 , , +1 ) , + ( −1 , , +1 ).

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Lie Algebras of Slow Growth and Klein-Gordon PDE

Cited by 21 publications

References 24 publications

Characteristic algebras and integrable exponential systems

Characteristic algebras and integrable exponential systems

Lax Pair for a Novel Two-Dimensional Lattice

Classification of a subclass of quasilinear two-dimensional lattices by means of characteristic algebras

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