New geometries associated with the nonlinear Schrödinger equation

Details New geometries associated with the nonlinear Schrödinger equation New geometries associated with the nonlinear Schrödinger equation

2001-11-23

arxiv.org/abs/nlin/0111052v1

Nonlinear Sciences

Pattern Formation and Solitons

13 pages, 3 figures

Scientific paper

10.1140/epjb/e2002-00284-8

We apply our recent formalism establishing new connections between the geometry of moving space curves and soliton equations, to the nonlinear Schr\"{o}dinger equation (NLS). We show that any given solution of the NLS gets associated with three distinct space curve evolutions. The tangent vector of the first of these curves, the binormal vector of the second and the normal vector of the third, are shown to satisfy the integrable Landau-Lifshitz (LL) equation ${\bf S}_u = {\bf S} \times {\bf S}_{ss}$, (${\bf S}^2=1$). These connections enable us to find the three surfaces swept out by the moving curves associated with the NLS. As an example, surfaces corresponding to a stationary envelope soliton solution of the NLS are obtained.

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