The exact rogue wave recurrence in the NLS periodic setting via matched asymptotic expansions, for 1 and 2 unstable modes

The focusing Nonlinear Schr\"odinger (NLS) equation is the simplest universal model describing the modulation instability (MI) of quasi monochromatic waves in weakly nonlinear media, the main physical mechanism for the generation of rogue (anomalous) waves (RWs) in Nature. In this paper we investigate the $x$-periodic Cauchy problem for NLS for a generic periodic initial perturbation of the unstable constant background solution, in the case of $N=1,2$ unstable modes. We use matched asymptotic expansion techniques to show that the solution of this problem describes an exact deterministic alternate recurrence of linear and nonlinear stages of MI, and that the nonlinear RW stages are described by the N-breather solution of Akhmediev type, whose parameters, different at each RW appearence, are always given in terms of the initial data through elementary functions. This paper is motivated by a preceeding work of the authors in which a different approach, the finite gap method, was used to investigate periodic Cauchy problems giving rise to RW recurrence.Comment: 20 pages. arXiv admin note: text overlap with arXiv:1708.00762 and substantial text overlap with arXiv:1707.0565

Discrete SL2 Connections and Self-Adjoint Difference Operators on the Triangulated 2-manifold

Discretization Program of the famous Completely Integrable Systems and associated Linear Operators was developed in 1990s. In particular, specific properties of the second order difference operators on the triangulated manifolds and equilateral triangle lattices were studied in the works of S.Novikov and I.Dynnikov since 1996. They involve factorization of operators, the so-called Laplace Transformations, new discretization of Complex Analysis and new discretization of $GL_n$ connections on the triangulated $n$-manifolds. The general theory of the new type discrete $GL_n$ connections was developed. However, the special case of $SL_n$-connections (and unimodular $SL_n^{\pm}$ connections such that $\det A=\pm 1$) was not selected properly. As we prove in this work, it plays fundamental role (similar to magnetic field in the continuous case) in the theory of self-adjoint discrete Schrodinger operators for the equilateral triangle lattice in \RR^2. In Appendix~1 we present a complete characterization of rank 1 unimodular $SL_n^{\pm}$ connections. Therefore we correct a mistake made in the previous versions of our paper (we wrongly claimed that for $n>2$ every unimodular $SL_n^{\pm}$ Connection is equivalent to the standard Canonical Connection). Using communications of Korepanov we completely clarify connection of classical theory of electric chains and star-triangle with discrete Laplace transformation on the triangle latticesComment: LaTeX, 23 pages, We correct a mistake made in the previous versions of our paper (we wrongly claimed that for $n>2$ every unimodular $SL_n^{\pm}$ Connection is equivalent to the standard Canonical Connection

An integral geometry lemma and its applications: the nonlocality of the Pavlov equation and a tomographic problem with opaque parabolic objects

As in the case of soliton PDEs in 2+1 dimensions, the evolutionary form of integrable dispersionless multidimensional PDEs is non-local, and the proper choice of integration constants should be the one dictated by the associated Inverse Scattering Transform (IST). Using the recently made rigorous IST for vector fields associated with the so-called Pavlov equation $v_{xt}+v_{yy}+v_xv_{xy}-v_yv_{xx}=0$, we have recently esatablished that, in the nonlocal part of its evolutionary form $v_{t}= v_{x}v_{y}-\partial^{-1}_{x}\,\partial_{y}\,[v_{y}+v^2_{x}]$, the formal integral $\partial^{-1}_{x}$ corresponding to the solutions of the Cauchy problem constructed by such an IST is the asymmetric integral $-\int_x^{\infty}dx'$. In this paper we show that this results could be guessed in a simple way using a, to the best of our knowledge, novel integral geometry lemma. Such a lemma establishes that it is possible to express the integral of a fairly general and smooth function $f(X,Y)$ over a parabola of the $(X,Y)$ plane in terms of the integrals of $f(X,Y)$ over all straight lines non intersecting the parabola. A similar result, in which the parabola is replaced by the circle, is already known in the literature and finds applications in tomography. Indeed, in a two-dimensional linear tomographic problem with a convex opaque obstacle, only the integrals along the straight lines non-intersecting the obstacle are known, and in the class of potentials $f(X,Y)$ with polynomial decay we do not have unique solvability of the inverse problem anymore. Therefore, for the problem with an obstacle, it is natural not to try to reconstruct the complete potential, but only some integral characteristics like the integral over the boundary of the obstacle. Due to the above two lemmas, this can be done, at the moment, for opaque bodies having as boundary a parabola and a circle (an ellipse).Comment: LaTeX, 13 pages, 3 figures. arXiv admin note: substantial text overlap with arXiv:1507.0820