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Hyperelliptic quasi-periodic and soliton solutions of the nonlinear Schrödinger equation. (English) Zbl 0578.35086
In order to examine the nonlinear Schrödinger equations \[ (NLS)_{1,2}\quad u_{xx}-iu_ t=\pm 2 | u|^ 2 u. \] The author considers the complexified version of the equations \[ (NLS)\quad u_{xx}-iu_ t=-2u^ 2v,\quad v_{xx}+iv_ t=-2v^ 2u, \] where u and v are appropriate holomorphic functions. She constructs quasi- periodic solutions of (NLS) by hyperelliptic theta functions (using methods from algebraic geometry) and discusses solitonary behaviour of certain solutions of (NLS). She also gives some applications of her results, for example to inverse spectral problems for finite-genus periodic potentials.
Reviewer: N.Jacob

MSC:
35Q99 Partial differential equations of mathematical physics and other areas of application
35P25 Scattering theory for PDEs
14K25 Theta functions and abelian varieties
35G20 Nonlinear higher-order PDEs
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