The Cauchy problem for the planar spin-liquid model

Abstract

In this paper, we study the Cauchy problem of a two-dimensional model for a moving ferromagnetic continuum and prove global existence and uniqueness of solutions. In addition, equivalence to the coupled system of nonlinear Schrödinger equations interacting with a Chern-Simons gauge field is established.

Equivariante and Self-similar Standing Waves for a Hamiltonian Hyperbolic-hyperbolic Spin-field System

“…Local and global wellposedness results under various regularity, size and/or decay conditions on the data can be found in the literature (e.g. [27,22,12,13,11,16,4,15,2] and references therein). In particular, Hayashi and Saut studied the Cauchy problem of this type of systems for maps into S 2 and obtained well-posedness results for small smooth solutions with/without exponential decay assumption on initial data in [12,13], which applies to system (1.1) in the case of target manifold S 2 .…”

Schrodinger Maps and their Associated Frame Systems

“…Using this last observation a natural question to ask in the constant curvature case is: When do solutions of the GNLS represent solutions of SM? For smooth solutions this question was answered in one dimension by Terng and Uhlenbeck [TU06] and in two dimensions, for a special case, by N.H. Chang and O. Pashev [CP05].…”

On the Stability of Certain Spin Models in 2+1 Dimensions

“…4) In this paper we consider the spin-field models on R 2 × I ∂ t s = s × μ (s 11 + s 22 ) + s 1 ζ 2 − s 2 ζ 1 , ζ = 2μs · μ (s 1 × μ s 2 ), (1.5) where , μ ∈ {−1, 1} and I ⊆ R is an open interval. The functions s : R 2 × I → S μ and ζ : R 2 × I → R in (1.5) are assumed to be sufficiently smooth functions, and s 1 = ∂ x s, s 2 = ∂ y s, s 11 = ∂ 2 x s, s 22 = ∂ 2 y s, ζ 1 = ∂ x ζ , ζ 2 = ∂ y ζ , and ζ = (∂ 2 x + ∂ 2 y )ζ .…”

“…Intuitively, one could think ofḋ 1 (f, f ) andρ 1 (f, f ) as nonlinear ways to measure the "distance" between the functions f, f , at a critical level, in our geometric setting in which the usual "difference" f − f is not geometrically relevant. 4 Semidistance functions of this type have been used in recent work of Tao [17] on global regularity of wave maps. Our stability result is the following:…”

“…The proof in [4] relies on perturbative analysis of the "modified spin system" (which is derived using the generalized Hasimoto transform) and the key cancellation of the magnetic terms of the nonlinearities of the modified spin system discussed in the paragraph following (1.6) and (1.7). These are two of the main ideas we use in this paper as well.…”