|1- Anco, S C., Asadi, E., Dogonchi, A., "Integrable systems with unitary invariance from non-stretching geometric curve flows in the Hermitian symmetric space Sp(n)/U(n)
", Int. J. Mod. Phys. Conf. Ser., 38, 1560071-1-1560071-15, (2015).|
A moving parallel frame method is applied to geometric non-stretching curve flows in the Hermitian symmetric space Sp(n)/U(n) to derive new integrable systems with unitary invariance. These systems consist of a bi-Hamiltonian modified Korteweg-de Vries equation and a Hamiltonian sine-Gordon (SG) equation, involving a scalar variable coupled to a complex vector variable. The Hermitian structure of the symmetric space Sp(n)/U(n) is used in a natural way from the beginning in formulating a complex matrix representation of the tangent space 𝔰𝔭(n)/𝔲(n) and its bracket relations within the symmetric Lie algebra (𝔲(n), 𝔰𝔭(n)).
|2- C. Anco, S., Asadi, E., "Symplectically invariant soliton equations from
non-stretching geometric curve flows", J. Phys. A: Math. Theor, 45, 475207-1 -475207-37, (2012).|
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 amoving parallel frame along the curves. As main results, two new multi-component versions of the sine–Gordon 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 shared hierarchy of symmetries and conservation laws generated by a hereditary recursion operator. The corresponding geometric curve flows in 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 analogue of a nonstretching Schr¨odinger map.