Generalized Hasimoto Transform of One-Dimensional Dispersive Flows into Compact Riemann Surfaces

Abstract

Abstract. We study the structure of differential equations of one-dimensional dispersive flows into compact Riemann surfaces. These equations geometrically generalize two-sphere valued systems modeling the motion of vortex filament. We define a generalized Hasimoto transform by constructing a good moving frame, and reduce the equation with values in the induced bundle to a complex valued equation which is easy to handle. We also discuss the relationship between our reduction and the theory of linear dispersive partial differential equations.

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

“…Furthermore, it follows from [19] that the S 2 -valued model (1.3) is reformulated as (1.4) where N is the canonical two-sphere S 2 with S = 1 and b…”

“…It is to be commented that another geometric generalization of (1.3) has been proposed in [19]. The equation can be formulated by…”

“…for u = u(t, x) : R × T → N, where (N, J, h) is a compact Kähler manifold, and b 1 = 0, b 2 , b 3 , b 4 ∈ R are real constants. As in [19], the equation (1.3) is actually generalized also as (1.4) where N = S 2 , b 1 = a 1 , b 2 = 1, b 3 = a 2 − a 1 , and b 4 = −5a 1 + 2a 2 . In fact, (1.1) and (1.4) are essentially the same, if (N, J, h) is a Riemann surface with constant Gaussian curvature.…”

A third order dispersive flow for closed curves into almost Hermitian manifolds

“…They constructed a good moving frame along the map and reduced (1) to a simple complex-valued semilinear Schrödinger equation under the assumption that u(t, x) has a fixed base point as x → +∞. Similarly, Onodera reduced (1) with a = 0 and a one-dimensional fourth order dispersive flow to complex-valued equations in [23]. Generally speaking, these reductions require some restrictions on the range of the mappings, and one cannot make use of them to solve the initial value problem for the original equations without restrictions on the range of the initial data.…”

Schrödinger flow into almost Hermitian manifolds

“…They constructed a good moving frame along the map and reduced (1) to a simple complex-valued equation when u(t, x) has a fixed base point as x → +∞. Similarly, Onodera reduced a one-dimensional third or fourth order dispersive flow to a complex-valued equation in [25]. In [20] and [21], Nahmod, Stefanov and Uhlenbeck obtained a system of semilinear Schrödinger equations from the equation of the Schrödinger map of the Euclidean space to the two-sphere when the Schrödinger map never takes values in some open set of the two-sphere.…”