# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [1] Arms R.J. and Hama F.R. (1965). Localized-induction concept on a curved vortex and motion of an elliptic vortex ring. Phys. Fluids 8: 553 · doi:10.1063/1.1761268 [2] Bănică, V., Vega, L.: On the Dirac delta as initial condition for non-linear Schrödinger equations (submitted) [3] Betchov R. (1965). On the curvature and torsion of an isolated vortex filament. J. Fluid Mech. 22: 471 · Zbl 0133.43803 · doi:10.1017/S0022112065000915 [4] Da Rios L.S. (1906). On the motion of an unbounded fluid with a vortex filament of any shape. Rend. Circ. Mat. Palermo 22: 117 · JFM 37.0764.01 · doi:10.1007/BF03018608 [5] Ding, Q.: A note on the NLS and the Schrödinger flow of maps. Physics Letters A, Volume 248 (Issue 1), 49–56 (1998) · Zbl 1115.35368 [6] Ding Q. (1999). The gauge equivalence of the NLS and the Schrödinger flow of maps in 2 + 1 dimensions. J. Phys. A: Math. Gen. 32: p. 5087–5096 · Zbl 0941.35104 [7] Gutiérrez S., Rivas J. and Vega L. (2003). Formation of singularities and self-similar vortex motion under the localized induction approximation. Comm. PDE 28: 927–968 · Zbl 1044.35089 · doi:10.1081/PDE-120021181 [8] Hasimoto H. (1972). A soliton on a vortex filament. J. Fluid Mech. 51: 477–485 · Zbl 0237.76010 · doi:10.1017/S0022112072002307 [9] Thaller B. (1992). The Dirac Equation. Springer, Heidelberg · Zbl 0765.47023
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.