We establish a new connection between the theory of totally positive Grassmannians and the theory of M-curves using the finite-gap theory for solitons of the KP equation.Here and in the following KP equation denotes the Kadomtsev-Petviashvili 2 equation (see (1)), which is the first flow from the KP hierarchy. We also assume that all KP times are real. We associate to any point of the real totally positive Grassmannian Gr TP (N, M ) a reducible curve which is a rational degeneration of an M-curve of minimal genus g = N (M − N ), and we reconstruct the real algebraic-geometric dataá la Krichever for the underlying real bounded multiline KP soliton solutions. From this construction it follows that these multiline solitons can be explicitly obtained by degenerating regular real finite-gap solutions corresponding to smooth M-curves. In our approach we rule the addition of each new rational component to the spectral curve via an elementary Darboux transformation which corresponds to a section of a specific projection Gr TP (r + 1, M − N + r + 1) → Gr TP (r, M − N + r). 34 7. Γ(ξ) and the vacuum divisor for soliton data in Gr TP (2, 4) 36 7.1. Γ(ξ) and its desingularization for generic soliton data in Gr TP (2, 4) 36 7.2. The leading order coefficients and vectors of the vacuum wavefunction 37 7.3. The vacuum divisor for soliton data in Gr TP (2, 4) 38 Appendix A. Points in Gr TP (N, M ) and totally positive matrices in classical sense 39 Appendix B. Lemmas for the proof of Theorem 6 42 of the wavefunction is in the remaining (infinite) oval. Finally, for a fixed spectral curve, such solutions are parametrized by degree g non-special divisors of the Baker-Akhiezer function with exactly one pole in each finite oval. In the solitonic limit a certain number of cycles shrinks and the non-singular curve degenerates to a reducible curve of rational type.Let us fix the M phases, κ 1 < · · · < κ M . Then generic regular bounded multi-line KP solitons are parametrized by points in Gr TP (N, M ) ⊂ Gr TNN (N, M ), i.e. points in the Grassmannian with all Plücker coordinates strictly positive [48]. These solutions then depend on N (M − N ) parameters (the dimension of the Grassmannian). We show that for any compact subset in Gr TP (N, M ) we may fix a spectral curve, which is a rational degeneration of regular M-curves of genus N (M − N ), and the solutions are parametrized by N (M − N ) point divisors on it.The starting point of the construction is the following observation: for any soliton data in Gr TP (N, M ), the Sato dressed wave function is defined on CP 1 , which we denote Γ 0 , with M + 1 marked points (the phases κ 1 , . . . , κ M and the essential singularity P 0 ). On Γ 0 , the normalized Sato dressed wave function is a Baker-Akhiezer function for the given soliton data with a real N -point divisor. To obtain a degree N (M −N ) divisor, in our construction we attach additional components to Γ 0 in such a way that the resulting reducible curve possesses the N (M − N ) + 1 real components (ovals), and we extend the Baker...

The focusing Nonlinear Schrödinger (NLS) equation is the simplest universal model describing the modulation instability (MI) of quasi monochromatic waves in weakly nonlinear media, considered the main physical mechanism for the appearence of rogue (anomalous) waves (RWs) in Nature. In this paper we study, using the finite gap method, the NLS Cauchy problem for periodic initial perturbations of the unstable background solution of NLS exciting just one of the unstable modes. We distinguish two cases. In the case in which only the corresponding unstable gap is theoretically open, the solution describes an exact deterministic alternate recurrence of linear and nonlinear stages of MI, and the nonlinear RW stages are described by the 1-breather Akhmediev solution, whose parameters, different at each RW appearence, are always given in terms of the initial data through elementary functions. If the number of unstable modes is > 1, this uniform in t dynamics is sensibly affected by perturbations due to numerics and/or real experiments, provoking O(1) corrections to the result. In the second case in which more than one unstable gap is open, a detailed investigation of all these gaps is necessary to get a 1 arXiv:1707.05659v2 [nlin.SI] 22 Jul 2017 uniform in t dynamics, and this study is postponed to a subsequent paper. It is however possible to obtain the elementary description of the first nonlinear stage of MI, given again by the Akhmediev 1breather solution, and how perturbations due to numerics and/or real experiments can affect this result. Since the solution of the Cauchy problem is given in terms of different elementary functions in different time intervals, obviously matching in the corresponding overlapping regions, an alternative approach, based on matched asymptotic expansions, is suggested and presented in a separate paper in which the RW recurrence, in the case of a finite number of unstable modes and of a generic initial perturbation exciting all of them, is again described in term of elementary functions.