Local existence of a fourth-order dispersive curve flow on locally Hermitian symmetric spaces and its application

Uniqueness of 1D Generalized Bi-Schrödinger Flow

“…Let us go back to our problem (1.4)- (1.5). Recently, the present author in [29] showed only the local existence of the solution to (1.4)-(1.5) for time-dependent maps into a compact locally Hermitian symmetric space. Indeed, by the mix of the parabolic regularization and the geometric energy method combined with a gauge transformation acting on Γ(u −1 T N ), he showed the following: Theorem 1.2 ([29],Theorem 1.3.).…”

“…Another unified formulation of (1.1) was derived by the present author in [29] for timedependent maps from the real line R or the one-dimensional flat torus T := R/2πZ into a locally Hermitian symmetric space. Let (N, J, h) be a locally Hermitian symmetric space, which is a complex manifold characterized by the condition ∇ N J = ∇ N R N = 0.…”

“…Z for any X, Y, Z ∈ Γ(T N ). Then, the result of [29] shows (1.1) for time-dependent maps u from R or T into (N, J, h) can be written by…”

“…Indeed, the S 2 -valued model (1.3) in this context is derived by a geometric reformulation of a continuum limit of the Heisenberg spin chain systems with nearest neighbor bilinear and bi-quadratic exchange interactions( [18]) . One can consult with [29,Section 2.2] for the reformulation of the physical model as (1.3). Moreover, the S 2 -valued model (1.3) also occurs in relation with the so-called Fukumoto-Moffatt model equation( [11,12]) describing the motion of a vortex filament in an incompressible prefect fluid in R 3 .…”

On the vortex filament in 3-spaces and its generalizations

“…Hence, surprisingly, the Fukumoto-Moffatt's model is nonintegrable in general. We should point out that, except the physical background (see [34]) and the geometric characterization (see [22]), the analytical properties of the Fukumoto-Moffatt's model, such as the existence and uniqueness of the solutions are investigated in [59]. The third-order correction model in the Minkowski 3-space R 2,1 was proposed in [20], which consists of the solutions to (2.3) adjusted by assuming (λ, ν and µ are also real parameters)…”