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Name:  
Miloslav Znojil  
Reviewer number:  
9689  
Email:  
znojil@ujf.cas.cz  
Item's zblNumber:  
DE 016 247 519  
Author(s):  
Tovbis, Alexander; Venakides, Stephanos  
Shorttitle:  
The eigenvalue problem for focusing NLS equation  
Source:  
Physica D 146, No. 14, 150164 (2000).  
Classification:  
Primary Classification:  
 
Secondary Classification:  
 
Keywords:  
ZakharovShabat eigenvalue problem; semiclassical 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 initialvalue 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 ``semiclassical" limit becomes reducible to the nonselfadjoint ZakharovShabat (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 specialfunction solutions. The present authors extend the latter result showing that the formal reduction to the Gauss equation also emerges for the whole oneparametric family of the (suitably nonlinearized) 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:  