Schrödinger soliton from Lorentzian manifolds

Abstract

In this paper, we introduce a new notion named as Schrödinger soliton. Socalled Schrödinger solitons are defined as a class of special solutions to the Schrödinger flow equation from a Riemannian manifold or a Lorentzian manifold M into a Kähler manifold N . If the target manifold N admits a Killing potential, then the Schrödinger soliton is just a harmonic map with potential from M into N . Especially, if the domain manifold is a Lorentzian manifold, the Schrödinger soliton is a wave map with potential into N . Then we apply the geometric energy method to this wave map system, and obtain the local well-posedness of the corresponding Cauchy problem as well as global existence in 1 + 1 dimension. As an application, we obtain the existence of Schrödinger soliton of the hyperbolic Ishimori system. 2000 Mathematics Subject Classification. 58J60, 35L70, 37K25. Key words and phrases. Schrödinger soliton, Schrödinger flow, wave map with potential, Killing potential. Partially supported by 973 project of China, Grant No. 2006CB805902.(Proof. Directly computing by the definition of tension field, we getSince S t is an isomorphism, we have τ (S t ) = 0 and henceOn the other hand,The last equality holds because the single parameter group S t satisfies S t • S s = S t+s . Differentiating this at s = 0, we get dS t • V = V (S t ).2

Vortex structures for some geometric flows from pseudo-Euclidean spaces

“…A, we will show how to convert (11)- (12) and 13- (14) into (20) and (21) respectively. At the same time, we will give the explicit expressions of the parameters µ and κ in (20)-(21) from case to case.…”

“…Problem (1) can be regarded as the Schrödinger flows for maps from pseudo-Euclidean spaces into S 2 in the sense of [3] (also see [14]). While (3) can be viewed as a generalization of the σ-model with field m : R K,N → S 2 .…”

“…Before going further, we recall that in [4] W. Ding and H. Yin proposed to study the periodic solutions of the Schrödinger flow in the case where the target manifold N is a Kähler-Einstein manifold with positive scalar curvature. Such a class of special solutions can be viewed as a sort of geometric solitary wave solutions ( [14]). Recently, from the viewpoint of differential geometry C. Song, X.…”

“…In the present paper, we also use the similar method to study (1)- (4). In other words, we shall make use of the isometry groups in both the target manifold C and the pseudo-Euclidean space to transform (11)- (14) into elliptic problems and then try to construct solutions with various vortex structures of 0-dimension(single point vortices or vortex pairs) or 1-dimension(vortex helices or vortex rings).…”

“…We would like to explain the results in the hyperbolic vortex circle case with an example. Without loss of generality, let us consider the Type C solution with Hyperbolic Vortex Circles to (14). From the transformations in Section 2.1, we know that by taking the Type C function…”