Abstract. A bi-Hamiltonian hierarchy of quaternion soliton equations is derived from geometric non-stretching flows of curves in the quaternionic projective space HP n . The derivation adapts the method and results in recent work by one of us on the Hamiltonian structure of non-stretching curve flows in Riemannian symmetric spacesas a symmetric space in terms of compact real symplectic groups and quaternion unitary groups. As main results, scalar-vector (multi-component) versions of the sine-Gordon (SG) equation and the modified Korteweg-de Vries (mKdV) equation are obtained along with their bi-Hamiltonian integrability structure consisting of a shared hierarchy of quaternionic symmetries and conservation laws generated by a hereditary recursion operator. The corresponding geometric curve flows in HP n are shown to be described by a non-stretching wave map and a mKdV analog of a non-stretching Schrödinger map.

department of mathematics, brock university, st. catharines, on canada department of mathematics, institute for advanced studies in basic sciences, gava zang, zanjan, iranAbstract. Bi-Hamiltonian hierarchies of symplectically-invariant soliton equations are derived from geometric non-stretching flows of curves in the Riemannian symmetric spaces Sp(n+1)/Sp(1)×Sp(n) and SU (2n)/Sp(n). The derivation uses Hasimoto variables defined by a moving parallel frame along the curves. As main results, two new multi-component versions of the sine-Gordon (SG) equation and the modified Korteweg-de Vries (mKdV) equation exhibiting Sp(1) × Sp(n − 1) invariance are obtained along with their bi-Hamiltonian integrability structure consisting of a hierarchy of symmetries and conservation laws generated by a hereditary recursion operator. The corresponding geometric curve flows in both Sp(n + 1)/Sp(1) × Sp(n) and SU (2n)/Sp(n) are shown to be described by a non-stretching wave map and a mKdV analog of a non-stretching Schrödinger map.8

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...

1 department of mathematics and statistics brock university st. catharines, on l2s3a1, canada 2 department of mathematics institute for advance studies in basic science (iasbs) 45137-66731, zanjan, iran Abstract. The deep geometrical relationships holding among the NLS equation, the vortex filament equation, the Heisenberg spin model, and the Schrödinger map equation are extended to the general setting of Hermitian symmetric spaces. New results are obtained by utilizing a generalized Hasimoto variable which arises from applying the general theory of parallel moving frames. The example of complex projective space CP N = SU (N + 1)/U (N ) is used to illustrate the method and results.of the curve, while u = κe i τ dx has the geometrical meaning of a U(1)-covariant of the curve, with (Re u, Im u) being the components of the Cartan matrix of a parallel frame [18] given by the tangent vector T and the pair of normal vectors Re(e i τ dx (N + iB)), Im(e i τ dx (N + iB)). The phase-rotation symmetry on u corresponds to a U(1) ≃ SO(2) gauge group of transformations in SO(3) preserving the structure of a parallel frame by rigid rotations in the normal plane along the curve. As shown by Hasimoto [22], the bi-normal equation also describes the physical motion of a vortex filament in fluid mechanics. Furthermore, the bi-normal equation can be interpreted as a Heisenberg spin model in R 3 , or equivalently as a Schrödinger map equation on the sphere S 2 . The geometrical interrelationships among these physical and mathematical formulations of the NLS equation have been explored in Ref. [12]. Abstract formulations involving differential invariants, Hamiltonian structures, and loop groups can be found in Refs. [33,15,16].A generalization of linear isospectral flows yielding group invariant multi-component integrable NLS systems is known [19] for all Lie algebras associated with Hermitian symmetric spaces (namely, Riemannian symmetric spaces with a compatible complex structure). These isospectral flows, called Fordy-Kulish systems, each have an associated Sym-Pohlmeyer curve flow which can be viewed as a Lie-algebra version of a bi-normal equation [25].A generalization of parallel frames and Hasimoto variables yielding group-invariant integrable systems is known [4] for all Riemannian symmetric spaces. This includes [3] semisimple Lie groups, viewed in a natural way as diagonal Riemannian symmetric spaces. In particular, the results in Ref. [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 syst...

Abstract. Quaternionic vector mKDV equations are derived from the Cartan structure equation in the symmetric space ‫ސވ‬ n = Sp(n + 1)/Sp(1) × Sp(n). The derivation of the soliton hierarchy utilizes a moving parallel frame and a Cartan connection 1-form ω related to the Cartan geometry on ‫ސވ‬ n modelled on (sp n+1 , sp 1 × sp n ). The integrability structure is shown to be geometrically encoded by a PoissonNijenhuis structure and a symplectic operator.