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

Abstract

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:

On nonlocal reductions of a generalized Heisenberg ferromagnet equation

“…Moreover, we have derived recursion operators that allowed us to describe an integrable hierarchy related to (8), see (34). We would like to stress here on an important difference between the integrable hierarchy for (2) as obtained in [13] and that one we have derived in previous section. Unlike the integrable hierarchy for the coupled system, (34) is not the most general one.…”

“…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). Thus, it is a natural generalization of the coupled system (2) already studied.…”

“…The form of the matrix coefficients in (3), (4), (5), (13), (17) and (18) implies that the Lax operators are subject to the following symmetry conditions:…”

“…In order to treat both cases simultaneously we shall refer to condition (21) as a pseudo-Hermitian reduction when Q m+n = 1 1 m+n and a Hermitian one when Q m+n = 1 1 m+n . Let us consider the linear problem L(λ)ψ(x, t, λ) = 0 (22) and denote the set of its fundamental solutions by F. Then (20) and (21) could be considered consequences of a Mikhailov-type reduction imposed on the linear spectral problem, see [9,13]. Indeed, consider the following maps:…”

“…In a series of papers [4,5,12,13,15], the properties of the pair of nonlinear evolution equations (NLEEs)…”