Geometry of the Recursion Operators for the GMV System

Yanovski, A.B.; Vilasi, G.

doi:10.1142/s1402925112500234

Cited by 10 publications

(20 citation statements)

References 28 publications

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“…In the present report, we have considered new multicomponent NLEE of HF type, see (8), (15) and (19). That NLEE is S-integrable with a Lax pair associated with the symmetric space SU(m + n)/S(U(m) × U(n)), see (3)- (5), (13), (14), (17) and (18).…”

Section: Resultsmentioning

confidence: 99%

On nonlocal reductions of a generalized Heisenberg ferromagnet equation

Valchev

Myrzakulov

Nugmanova

et al. 2019

Renewable Energy Sources and Technologies

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Section: Resultsmentioning

confidence: 99%

On nonlocal reductions of a generalized Heisenberg ferromagnet equation

Valchev

Myrzakulov

Nugmanova

et al. 2019

Renewable Energy Sources and Technologies

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“…Though the covariance of the linear problems is a strong condition, it does not determine the dressing factor G uniquely. Indeed, after comparing (13) and (14) with (20), we see that G must solve the system of linear partial differential equations:…”

Section: Dressing Methods and Linear Bundles In Pole Gaugementioning

confidence: 99%

Pseudo-Hermitian Reduction of a Generalized Heisenberg Ferromagnet Equation. II. Special Solutions

Valchev

Yanovski

2021

JNMP

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This paper is a continuation of our previous work in which we studied a sl (3) Zakharov-Shabat type auxiliary linear problem with reductions of Mikhailov type and the integrable hierarchy of nonlinear evolution equations associated with it. Now, we shall demonstrate how one can construct special solutions over constant background through Zakharov-Shabat's dressing technique. That approach will be illustrated on the example of a generalized Heisenberg ferromagnet equation related to the linear problem for sl (3). In doing this, we shall discuss the difference between the Hermitian and pseudo-Hermitian cases.

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“…We have seen that the consideration of the NLEEs hierarchy (59) shows that the operator 'moving' the equations along the hierarchy isΛ 2 ± and it does not depend on the real form. The geometric interpretation of the hierarchies and their conservation laws which is developed for the case of GMV + [23] also suggests that the appropriate operator we must consider isΛ 2 ± . It remains to consider the third important aspect of the recursion operators -their relations to the so-called expansions over the adjoint solutions.…”

Section: Completeness Relations For the Gmv ± Systemmentioning

confidence: 99%

Pseudo-Hermitian Reduction of a Generalized Heisenberg Ferromagnet Equation. I. Auxiliary System and Fundamental Properties

Yanovski¹,

Valchev²

2021

JNMP

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We consider an auxiliary spectral problem originally introduced by Gerdjikov, Mikhailov and Valchev (GMV system) and its modification called pseudo-Hermitian reduction which is extensively studied here for the first time. We describe the integrable hierarchies of both systems in a parallel way and construct recursion operators. Using the concept of gauge equivalence, we construct expansions over the eigenfunctions of recursion operators. This permits us to obtain the expansions for both GMV systems with arbitrary constant asymptotic values of the potential functions in the auxiliary linear problems.In case ǫ = +1 the matrix g belongs to the group SU (3) (g † = g −1 ) and when ǫ = −1 to the group. Further on we shall use the general notation SU (ǫ) referring to both cases, i.e. it implies SU (ǫ) ≡ SU (3) when ǫ = 1 and SU (ǫ) ≡ SU (2, 1) when ǫ = −1.Since g(x) ∈ SU (ǫ), the values of S(x) will be in the orbit O J0 (SU (ǫ)) of J 0 with respect to SU (ǫ) (it is a submanifold of isu(ǫ)). Thus S(x) ∈ O J0 (SU (ǫ)) ∩ g 1 where g 1 is the space of the matrices X in sl (3, C) such that HXH = −X, see (15) for the reason for this notation. Let us also note that conversely, if we assume that S(x) ∈ O J0 (SU (ǫ)) ∩ g 1 then on the first place S has the form as in (1). Next, as easily checked, the eigenvalues of the matrix Sare µ 1 = 0, µ 2 = −µ 3 = ǫ|u| 2 + |v| 2 .But since they coincide with 0, ±1 we must have ǫ|u| 2 + |v| 2 = 1. Our approach to the GMV ± system will be based on the fact that it is gauge equivalent to a generalized Zakharov-Shabat auxiliary systems (GZS systems) on the algebra sl (3, C), see section 3. Generalized Zakharov-Shabat systems, called Caudrey-Beals-Coifman (CBC) systems [2] when J is complex, are probably the best known auxiliary linear problems. In their most general form they are written on an arbitrary fixed simple Lie algebra g in some finite dimensional irreducible representation and have the form:Here, q(x) and J belong to g in some fixed representation, ψ belongs to the corresponding group. The element J must be such that the kernel of ad J (ad J (X) ≡ [J, X], X ∈ g) is a Cartan subalgebra h J of g. The potential q(x) belongs to the orthogonal complement h J ⊥ of h with respect to the Killing form:

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Geometry of the Recursion Operators for the GMV System

Abstract: We consider the Recursion Operator approach to the soliton equations related to a auxiliary linear system introduced recently by Gerdjikov, Mikhailov and Valchev (GMV system) and their interpretation as dual of Nijenhuis tensors on the manifold of potentials

Cited by 10 publications

References 28 publications

On nonlocal reductions of a generalized Heisenberg ferromagnet equation

On nonlocal reductions of a generalized Heisenberg ferromagnet equation

Pseudo-Hermitian Reduction of a Generalized Heisenberg Ferromagnet Equation. II. Special Solutions

Pseudo-Hermitian Reduction of a Generalized Heisenberg Ferromagnet Equation. I. Auxiliary System and Fundamental Properties

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