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Self-similar solutions for the 1D Schrödinger map on the hyperbolic plane. (English) Zbl 1128.35099
Summary: We study self-similar solutions for \({\mathbf X}_{t} = {\mathbf {XX}}_{ss}\), the equivalent in the Minkowski 3-space \(\mathbb R^{2,1}\) to the localized induction approximation flow, trying to adapt some results given by S. Gutiérrez, J. Rivas and L. Vega [Commun. Partial Differ. Equations 28, 927–968 (2003; Zbl 1044.35089)]. We show the existence of a one-parameter family of smooth solutions developing a corner in finite time. The main difference with respect to the Euclidean case studied by those authors is the proof of the boundedness of T, \({\mathbf e}_{1}\) and \({\mathbf e}_{2}\), the equivalents of T, b and n in \(\mathbb R^{2,1}\).

MSC:
35Q55 NLS equations (nonlinear Schrödinger equations)
76B47 Vortex flows for incompressible inviscid fluids
37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
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