The finitegap method and the periodic NLS Cauchy problem of anomalous waves for a finite number of unstable modes
Abstract
The focusing nonlinear Schrödinger (NLS) equation is the simplest universal model describing the modulation instability (MI) of quasimonochromatic waves in weakly nonlinear 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 finitegap 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 finitegap 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 nonlinear 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.
 Publication:

Russian Mathematical Surveys
 Pub Date:
 April 2019
 DOI:
 10.1070/RM9863
 arXiv:
 arXiv:1810.09247
 Bibcode:
 2019RuMaS..74..211G
 Keywords:

 35Q55;
 Mathematical Physics;
 Nonlinear Sciences  Exactly Solvable and Integrable Systems
 EPrint:
 LaTeX, 67 pages