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Uspekhi Mat. Nauk, 2019, Volume 74, Issue 2(446), Pages 27–80 (Mi umn9863)  

The finite-gap method and the periodic NLS Cauchy problem of anomalous waves for a finite number of unstable modes

P. G. Grinevicha, P. M. Santinibc

a Landau Institute for Theoretical Physics of the Russian Academy of Sciences
b Università di Roma "La Sapienza", Roma, Italy
c Istituto Nazionale di Fisica Nucleare (INFN), Roma, Italy

Abstract: The focusing non-linear Schrödinger (NLS) equation is the simplest universal model describing the modulation instability (MI) of quasimonochromatic waves in weakly non-linear media, and MI is considered to be the main physical mechanism for the appearance of anomalous (rogue) waves (AWs) in nature. In this paper the finite-gap method is used to study the NLS Cauchy problem for generic periodic initial perturbations of the unstable background solution of the NLS equation (here called the Cauchy problem of AWs) in the case of a finite number $N$ of unstable modes. It is shown how the finite-gap method adapts to this specific Cauchy problem through three basic simplifications enabling one to construct the solution, to leading and relevant order, in terms of elementary functions of the initial data. More precisely, it is shown that, to leading order, i) the initial data generate a partition of the time axis into a sequence of finite intervals, ii) in each interval $I$ of the partition only a subset of ${\mathscr N}(I)\leqslant N$ unstable modes are ‘visible’, and iii) for $t\in I$ the NLS solution is approximated by the ${\mathscr N}(I)$-soliton solution of Akhmediev type describing for these ‘visible’ unstable modes a non-linear interaction with parameters also expressed in terms of the initial data through elementary functions. This result explains the relevance of the $m$-soliton solutions of Akhmediev type with $m\leqslant N$ in the generic periodic Cauchy problem of AWs in the case of a finite number $N$ of unstable modes.
Bibliography: 118 titles.

Keywords: focusing non-linear Schrödinger equation, periodic Cauchy problem for anomalous waves, asymptotics in terms of elementary functions, finite-gap approximation, Riemann surfaces close to degenerate ones.

Funding Agency Grant Number
Russian Science Foundation 18-11-00316
Sapienza Università di Roma
The work of the first author was supported by the Russian Science Foundation, grant 18-11-00316. The second author was partially supported by the University “La Sapienza”, grant 2017.


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English version:
Russian Mathematical Surveys, 2019, 74:2, 211–263

Bibliographic databases:

UDC: 517.958
MSC: Primary 35Q55; Secondary 14H70, 14H81, 74J30, 78A60, 76B25, 76B15
Received: 08.11.2018

Citation: P. G. Grinevich, P. M. Santini, “The finite-gap method and the periodic NLS Cauchy problem of anomalous waves for a finite number of unstable modes”, Uspekhi Mat. Nauk, 74:2(446) (2019), 27–80; Russian Math. Surveys, 74:2 (2019), 211–263

Citation in format AMSBIB
\by P.~G.~Grinevich, P.~M.~Santini
\paper The finite-gap method and the periodic NLS Cauchy problem of anomalous waves for a finite number of unstable modes
\jour Uspekhi Mat. Nauk
\yr 2019
\vol 74
\issue 2(446)
\pages 27--80
\jour Russian Math. Surveys
\yr 2019
\vol 74
\issue 2
\pages 211--263

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