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Name:
Miloslav Znojil
Reviewer number:
9689
Email:
znojil@ujf.cas.cz
Item's zbl-Number:
DE 016 247 519
Author(s):
Tovbis, Alexander; Venakides, Stephanos
Shorttitle:
The eigenvalue problem for focusing NLS equation
Source:
Physica D 146, No. 1-4, 150-164 (2000).
Classification:
Primary Classification:
35Q55NLS-like nonlinear Schroedinger equations
Secondary Classification:
37K15Integration of completely integrable systems by inverse spectral and scattering methods
33C90Applications
35G25Initial value problems for nonlinear higher-order PDE, nonlinear evolution equations
35B30Dependence of solutions of PDE on initial and boundary data, parameters
35P30Nonlinear eigenvalue problems, nonlinear spectral theory for PDO
47A68Factorization theory including Wiener-Hopf and spectral factorizations
34L40Particular operators Dirac, one-dimensional Schroedinger, etc.
Keywords:
Zakharov-Shabat eigenvalue problem; semi-classical limit; hypergoemetric functions; focusing nonlinear Schroedinger equation;
Review:

It is well known that the nonlinear Schroedinger (NLS) equation
describes waves in nonlinear media and that its initial-value
problem is ill posed in the so called focusing regime (i.e., for
the plus sign at the nonlinearity). At the same time the NLS
equation is tractable by the inverse scattering method and its
analysis near the ``semi-classical" limit becomes reducible to the
non-self-adjoint Zakharov-Shabat (ZS) eigenvalue problem. In 1974,
Satsume and Yajima revealed that for a set of ``modulated" initial
waves the ZS equations degenerate to the Gauss hypergeometric
differential equation with the well known special-function
solutions. The present authors extend the latter result showing
that the formal reduction to the Gauss equation also emerges for
the whole one-parametric family of the (suitably non-linearized)
initial phases S(x). They contemplate the two cases characterized
by the respective asymptotically vanishing and constant initial
amplitudes A(x) and obtain their main result: The pure point
spectrum becomes empty in the second case and beyond certain
critical asymptotic decrease of S(x) in the first case.
Remarks to the editors:

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