Unitarily-invariant integrable systems and geometric curve flows inSU(n + 1)/U(n) andSO(2n)/U(n)

Abstract

1 department of mathematics university of south florida tampa, fl 33620, usa 2 department of mathematics and statistics brock university st. catharines, on l2s3a1, canada 3 department of mathematics institute for advance studies in basic science (iasbs) 45137-66731, zanjan, iran Abstract. Bi-Hamiltonian hierarchies of soliton equations are derived from geometric nonstretching (inelastic) curve flows in the Hermitian symmetric spaces SU (n + 1)/U (n) and SO(2n)/U (n). The derivation uses Hasimoto variables defined by a moving parallel frame along the curves. As main results, new integrable multi-component versions of the Sine-Gordon (SG) equation and the modified Korteveg-de Vries (mKdV) equation, as well as a novel nonlocal multi-component version of the nonlinear Schrödinger (NLS) equation are obtained, along with their bi-Hamiltonian structures and recursion operators. These integrable systems are unitarily invariant and correspond to geometric curve flows given by a non-stretching wave map and a mKdV analog of a non-stretching Schrödinger map in the case of the SG and mKdV systems, and a generalization of the vortex filament bi-normal equation in the case of the NLS systems. N x = −κT + τB,B x = −τ N x , while the tangential coefficient a is determined by a x = κb due to the non-stretching property of the curve. When b = a = 0 and c = κ, the curve γ undergoes a bi-normal flow γ t = κB. This flow equation physically describes the motion of a vortex filament in incompressible fluids [5]. The induced flow on (κ, τ ) turns out to be equivalent to the NLS equation for the Hasimoto variable u = κ exp(i τ dx) [5]. Moreover, the Lax pair and bi-Hamiltonian operators for the NLS equation turn out to be encoded in a simple way in the structure equations of a moving frame formulation of the curve flow [3, 4], where the Hasimoto transformation from (κ, τ ) to u corresponds geometrically to a gauge transformation from a Frenet frame to a parallel frame given by rotating the vectors (N,B) in the normal plane by an angle θ(x) = − τ dx along the curve [6]. Unlike a Frenet frame, a parallel frame has a rigid gauge freedom consisting of a constant rotation φ applied to the vectors (N,B) in the normal plane. Under this rigid gauge transformation, u transforms to e iφ u by a constant phase rotation, and so u is not an invariant of the curve like (κ, τ ) but instead has the geometrical meaning of a U(1)-covariant of the curve [4].Similarly, all of the symmetries of the NLS equation themselves correspond to geometrical curve flows in Euclidean space, and in particular the first higher symmetry in the NLS hierarchy is given by γ t = κτB + κ xN + 1 2T with the Hasimoto variable u satisfying the complex modified Korteveg-de Vries (mKdV) equation u t = u xxx + 3 2 |u| 2 u x . This flow equation physically describes axial motion of a vortex filament [7]. It is also an integrable system, sharing the same integrability properties as the NLS equation.A broad generalization of parallel frames and Hasimoto variables has been obtained in work...

Hasimoto variables, generalized vortex filament equations, Heisenberg models and Schrödinger maps arising from group-invariant NLS systems

“…JQ])φ 0 , and so the Sym-Pohlmeyer curve flow associated with the mKdV system (2.33) is given by [25] γ t = −γ sss + 3 2 [γ ss , [γ ss , γ s ]] (2.44) in the Hermitian symmetric Lie algebra g. 8…”

“…[3] show that the NLS equation arises directly from a parallel frame formulation of a non-stretching geometric curve flow given by a chiral Schrödinger map equation in the Lie group SU(2), which is a variant of the Sym-Pohlmeyer curve flow in the Lie algebra su(2). This type of geometrical derivation of group-invariant integrable systems and corresponding non-stretching map equations has been pursued [30,2,5,6,7,8] for all of the basic classical (non-exceptional) Riemannian symmetric spaces, yielding many types of multi-component mKdV and sine-Gordon systems, as well as multi-component NLS systems [3], formulated in terms of Hasimoto variables.In the present paper, we adapt the general results on parallel frames and Hasimoto variables from Ref.[4] to extend the geometrical relationships among the NLS equation, the vortex filament equation, the Heisenberg spin model, and the Schrödinger map equation to the general setting of Hermitian symmetric spaces. Several main results are obtained.We give a new geometrical derivation of the Fordy-Kulish isospectral NLS flows [19], together with the associated Sym-Pohlmeyer curve flows [25], in all Lie algebras that arise from Hermitian symmetric spaces.…”

Integrable spinor/quaternion generalizations of the nonlinear Schrodinger equation

“…One novelty of this generalization is that it does not retain any U(1) symmetry. In contrast, the known multi-component generalizations of the NLS equation, such as the Fordy-Kullish NLS systems [1] associated to Hermitian symmetric Lie algebras and the unitarily-invariant NLS systems [2] associated to Riemannian symmetric Lie algebras with a unitary subalgebra, all involve an invariance group whose center is a U(1) subgroup. Their root symmetry comes from multiplication by i on their variables.…”

“…Their root symmetry comes from multiplication by i on their variables. Generalizing U(1) to SU (2), with i replaced by σ, leads to the interesting structure that the variables in the resulting SU(2) integrable system can be identified with the components of a spinor representation of Spin(3), so thus the system will describe an integrable NLS spinor equation for u = (u 1 , u 2 ).…”

“…For generalizing the NLS equation to an SU(2) system, the Euclidean Lie algebra will be replaced by a Lie algebra obtained from a similar contraction of the symmetric Lie algebra su(4)/sp (2). Another integrable SU(2) generalization will be derived by considering a contraction of the symmetric Lie algebra so(6)/u(3), which will lead to an NLS-type system involving a real scalar variable in addition to a pair of complex scalar variables.…”