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

The focusing non-linear Schrodinger (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 fnite-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 fnite number N of unstable modes. It is shown how the fnite-gap method adapts to this specifc Cauchy problem through three basic simplifcations 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 fnite intervals, ii) in each interval I of the partition only a subset of N (I) 6 N unstable modes are visible , and iii) for t ? I the NLS solution is approximated by the 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 6 N in the generic periodic Cauchy problem of AWs in the case of a fnite number N of unstable modes.