arXiv subjects

P. M. Santini

Publications and source records attributed to P. M. Santini.

At least 19 recordsLinked to original sources

arXiv2023

The periodic N breather anomalous wave solution of the Davey-Stewartson equations; first appearance, recurrence, and blow up properties

The integrable focusing Davey-Stewarson (DS) equations, multidimensional generalizations of the focusing cubic nonlinear Schrödinger (NLS) equation, provide ideal mathematical models for describing analytically the dynamics of 2+1 dimensional anomalous (rogue) waves (AWs). In this paper i) we construct the $N$-breather AW solution of Akhmediev type of the DS1 and DS2 equations, describing the nonlinear interaction of $N$ unstable modes over the constant background solution. ii) For the simplest multidimensional solution of DS2 we construct its limiting subcases, and we identify the constraint on its arbitrary parameters giving rise to blow up at finite time. iii) We use matched asymptotic expansions to describe the relevance of the constructed AW solutions in the doubly periodic Cauchy problem for small initial perturbations of the background, in the case of one and two unstable modes. We also show, in the case of two unstable modes, that no blow up takes place generically, although the AW amplitude can be arbitrarily large. All the results are expressed in terms of elementary functions.

nlin.SI↗

arXiv2023

Modulation instability, periodic anomalous wave recurrence, and blow up in the Ablowitz-Ladik lattices

The Ablowitz-Ladik equations, hereafter called $AL_+$ and $AL_-$, are distinguished integrable discretizations of respectively the focusing and defocusing nonlinear Schrödinger (NLS) equations. In this paper we first study the modulation instability of the homogeneous background solutions of $AL_{\pm}$ in the periodic setting, showing in particular that the background solution of $AL_{-}$ is unstable under a monochromatic perturbation of any wave number if the amplitude of the background is greater than $1$, unlike its continuous limit, the defocusing NLS. Then we use Darboux transformations to construct the exact periodic solutions of $AL_{\pm}$ describing such instabilities, in the case of one and two unstable modes, and we show that the solutions of $AL_-$ are always singular on curves of spacetime. At last, using matched asymptotic expansion techniques, we describe in terms of elementary functions how a generic periodic perturbation of the background solution i) evolves according to $AL_{+}$ into a recurrence of the above exact solutions, in the case of one and two unstable modes, and ii) evolves according to $AL_{-}$ into a singularity in finite time if the amplitude of the background is greater than $1$. The quantitative agreement between the analytic formulas of this paper and numerical experiments is perfect.

nlin.SI↗

arXiv2022

The finite gap method and the periodic Cauchy problem of $2+1$ dimensional anomalous waves for the focusing Davey-Stewartson 2 equation

The focusing Nonlinear Schrödinger (NLS) equation is the simplest universal model describing the modulation instability (MI) of $1+1$ dimensional quasi monochromatic waves in weakly nonlinear media, and MI is considered the main physical mechanism for the appearence of anomalous (rogue) waves (AWs) in nature. In analogy with the recently developed analytic theory of periodic AWs of the focusing NLS equation, in this paper we extend these results to a $2+1$ dimensional context, concentrating on the focusing Davey-Stewartson 2 (DS2) equation, an integrable $2+1$ dimensional generalization of the focusing NLS equation. More precisely, we use the finite gap theory to solve, to leading order, the doubly periodic Cauchy problem of the focusing DS2 equation for small initial perturbations of the unstable background solution, what we call the periodic Cauchy problem of the AWs. As in the NLS case, we show that, to leading order, the solution of this Cauchy problem is expressed in terms of elementary functions of the initial data.

math-ph↗

arXiv2020

The linear and nonlinear instability of the Akhmediev breather

The Akhmediev breather (AB) and its M-soliton generalization $AB_M$ are exact solutions of the focusing NLS equation periodic in space and exponentially localized in time over the constant unstable background; they describe the appearance of $M$ unstable nonlinear modes and their interaction, and they are expected to play a relevant role in the theory of periodic anomalous (rogue) waves (AWs) in nature. It is rather well established that they are unstable with respect to small perturbations of the NLS equation. Concerning perturbations of these solutions within the NLS dynamics, there is the following common believe in the literature. Let the NLS background be unstable with respect to the first $N$ modes; then i) if the $M$ unstable modes of the $AB_M$ solution are strictly contained in this set ($M 1$.

nlin.PS↗

arXiv2019

The effect of a small loss or gain in the periodic NLS anomalous wave dynamics. I

The focusing Nonlinear Schrödinger (NLS) equation is the simplest universal model describing the modulation instability (MI) of quasi monochromatic waves in weakly nonlinear media, and MI is considered the main physical mechanism for the appearence of anomalous (rogue) waves (AWs) in nature. Using the finite gap method, two of us (PGG and PMS) have recently solved, to leading order and in terms of elementary functions of the initial data, the NLS Cauchy problem for generic periodic initial perturbations of the unstable background solution of NLS (what we call the Cauchy problem of the AWs), in the case of a finite number of unstable modes. In this paper, concentrating on the simplest case of a single unstable mode, we study the periodic Cauchy problem of the AWs for the NLS equation perturbed by a linear loss or gain term. Using the finite gap method and the theory of perturbations of soliton PDEs, we construct the proper analytic model describing quantitatively how the solution evolves, after a suitable transient, into slowly varying lower dimensional patterns (attractors) in the $(x,t)$ plane, characterized by $ΔX=L/2$ in the case of loss, and by $ΔX=0$ in the case of gain, where $ΔX$ is the $x$-shift of the position of the AW during the recurrence, and $L$ is the period. This process is described, to leading order, in terms of elementary functions of the initial data. Since dissipation can hardly be avoided in all natural phenomena involving AWs, and since a small dissipation induces $O(1)$ effects on the periodic AW dynamics, generating the slowly varying loss/gain attractors analytically described in this paper, we expect that these attractors, together with their generalizations corresponding to more unstable modes, will play a basic role in the theory of periodic AWs in nature.

nlin.SI↗

arXiv2018

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

The focusing Nonlinear Schrödinger (NLS) equation is the simplest universal model describing the modulation instability (MI) of quasi monochromatic waves in weakly nonlinear media, and MI is considered the main physical mechanism for the appearence of anomalous (rogue) waves (AWs) in nature. In this paper we study, using the finite gap method, the NLS Cauchy problem for generic periodic initial perturbations of the unstable background solution of NLS (what we call the Cauchy problem of the AWs), in the case of a finite number $N$ of unstable modes. We show how the finite gap method adapts to this specific Cauchy problem through three basic simplifications, allowing one to construct the solution, at the leading and relevant order, in terms of elementary functions of the initial data. More precisely, we show that, at the 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 ${\cal N}(I)\le N$ unstable modes are "visible", and iii) the NLS solution is approximated, for $t\in I$, by the ${\cal N}(I)$-soliton solution of Akhmediev type, describing the nonlinear interaction of these visible unstable modes, whose parameters are expressed in terms of the initial data through elementary functions. This result explains the relevance of the $m$-soliton solutions of Akhmediev type, with $m\le N$, in the generic periodic Cauchy problem of the AWs, in the case of a finite number $N$ of unstable modes.

math-ph↗

arXiv2017

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

The focusing Nonlinear Schrödinger (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.

nlin.SI↗

arXiv2017

Numerical instability of the Akhmediev breather and a finite-gap model of it

In this paper we study the numerical instabilities of the NLS Akhmediev breather, the simplest space periodic, one-mode perturbation of the unstable background, limiting our considerations to the simplest case of one unstable mode. In agreement with recent theoretical findings of the authors, in the situation in which the round-off errors are negligible with respect to the perturbations due to the discrete scheme used in the numerical experiments, the split-step Fourier method (SSFM), the numerical output is well-described by a suitable genus 2 finite-gap solution of NLS. This solution can be written in terms of different elementary functions in different time regions and, ultimately, it shows an exact recurrence of rogue waves described, at each appearance, by the Akhmediev breather. We discover a remarkable empirical formula connecting the recurrence time with the number of time steps used in the SSFM and, via our recent theoretical findings, we establish that the SSFM opens up a vertical unstable gap whose length can be computed with high accuracy, and is proportional to the inverse of the square of the number of time steps used in the SSFM. This neat picture essentially changes when the round-off error is sufficiently large. Indeed experiments in standard double precision show serious instabilities in both the periods and phases of the recurrence. In contrast with it, as predicted by the theory, replacing the exact Akhmediev Cauchy datum by its first harmonic approximation, we only slightly modify the numerical output. Let us also remark, that the first rogue wave appearance is completely stable in all experiments and is in perfect agreement with the Akhmediev formula and with the theoretical prediction in terms of the Cauchy data.

nlin.PS↗

arXiv2017

The finite gap method and the analytic description of the exact rogue wave recurrence in the periodic NLS Cauchy problem. 1

The focusing 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 appearance 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 appearance, 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 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 1-breather solution, and how perturbations due to numerics and/or real experiments can affect this result.

nlin.SI↗

arXiv2015

On the dispersionless Kadomtsev-Petviashvili equation with arbitrary nonlinearity and dimensionality: exact solutions, longtime asymptotics of the Cauchy problem, wave breaking and discontinuous shocks

We study the generalization of the dispersionless Kadomtsev - Petviashvili (dKP) equation in n+1 dimensions and with nonlinearity of degree m+1, a model equation describing the propagation of weakly nonlinear, quasi one dimensional waves in the absence of dispersion and dissipation, and arising in several physical contexts, like acoustics, plasma physics, hydrodynamics and nonlinear optics. In 2+1 dimensions and with quadratic nonlinearity, this equation is integrable through a novel IST, and it has been recently shown to be a prototype model equation in the description of the two dimensional wave breaking of localized initial data. In higher dimensions and with higher nonlinearity, the generalized dKP equations are not integrable, but their invariance under motions on the paraboloid allows one to construct in this paper a family of exact solutions describing waves constant on their paraboloidal wave front and breaking simultaneously in all points of it, developing after breaking either multivaluedness or single valued discontinuous shocks. Then such exact solutions are used to build the longtime behavior of the solutions of the Cauchy problem, for small and localized initial data, showing that wave breaking of small initial data takes place in the longtime regime if and only if $m(n-1)\le 2$. At last, the analytic aspects of such a wave breaking are investigated in detail in terms of the small initial data, in both cases in which the solution becomes multivalued after breaking or it develops a discontinuous shock. These results, contained in the 2012 master thesis of one of the authors (FS), generalize those obtained by one of the authors (PMS) and S.V.Manakov for the dKP equation in n+1 dimensions with quadratic nonlinearity, and are obtained following the same strategy.

nlin.SI↗

arXiv2015

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).

nlin.SI↗

arXiv2015

Nonlocality and the inverse scattering transform for the Pavlov equation

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 $v_{xt}+v_{yy}+v_xv_{xy}-v_yv_{xx}=0$, in this paper we establish the following. 1. The non-local term $\partial_x^{-1}$ arising from its evolutionary form $v_{t}= v_{x}v_{y}-\partial^{-1}_{x}\,\partial_{y}\,[v_{y}+v^2_{x}]$ corresponds to the asymmetric integral $-\int_x^{\infty}dx'$. 2. Smooth and well-localized initial data $v(x,y,0)$ evolve in time developing, for $t>0$, the constraint $\partial_y {\cal M}(y,t)\equiv 0$, where ${\cal M}(y,t)=\int_{-\infty}^{+\infty} \left[v_{y}(x,y,t) +(v_{x}(x,y,t))^2\right]\,dx$. 3. Since no smooth and well-localized initial data can satisfy such constraint at $t=0$, the initial ($t=0+$) dynamics of the Pavlov equation can not be smooth, although, as it was already established, small norm solutions remain regular for all positive times. We expect that the techniques developed in this paper to prove the above results, should be successfully used in the study of the non-locality of other basic examples of integrable dispersionless PDEs in multidimensions.

nlin.SI↗

arXiv2015

Geometry of Winter Model

By constructing the Riemann surface controlling the resonance structure of Winter model, we determine the limitations of perturbation theory. We then derive explicit non-perturbative results for various observables in the weak-coupling regime, in which the model has an infinite tower of long-lived resonant states. The problem of constructing proper initial wavefunctions coupled to single excitations of the model is also treated within perturbative and non-perturbative methods.

hep-th↗

arXiv2015

The Inverse Spectral Transform for the Dunajski hierarchy and some of its reductions, I: Cauchy problem and longtime behavior of solutions

In this paper we apply the formal Inverse Spectral Transform for integrable dispersionless PDEs arising from the commutation condition of pairs of one-parameter families of vector fields, recently developed by S. V. Manakov and one of the authors, to one distinguished class of equations, the so-called Dunajski hierarchy. We concentrate, for concreteness, i) on the system of PDEs characterizing a general anti-self-dual conformal structure in neutral signature, ii) on its first commuting flow, and iii) on some of their basic and novel reductions. We formally solve their Cauchy problem and we use it to construct the longtime behavior of solutions, showing, in particular, that unlike the case of soliton PDEs, different dispersionless PDEs belonging to the same hierarchy of commuting flows evolve in time in very different ways, exhibiting either a smooth dynamics or a gradient catastrophe at finite time.

nlin.SI↗

arXiv2014

The Cauchy problem for the Pavlov equation

Commutation of multidimensional vector fields leads to integrable nonlinear dispersionless PDEs arising in various problems of mathematical physics and intensively studied in the recent literature. This report is aiming to solve the scattering and inverse scattering problem for integrable dispersionless PDEs, recently introduced just at a formal level, concentrating on the prototypical example of the Pavlov equation, and to justify an existence theorem for global bounded solutions of the associated Cauchy problem with small data.

nlin.SI↗

arXiv2014

Renormalization in Winter Model

We show that metastable states in Winter model can be related to the eigenstates of a particle in a box by means of renormalization and mixing.

quant-ph↗

arXiv2013

Integrable dispersionless PDEs arising as commutation condition of pairs of vector fields

We review some results about the theory of integrable dispersionless PDEs arising as commutation condition of pairs of one-parameter families of vector fields, developed by the authors during the last years. We review, in particular, the formal aspects of a novel Inverse Spectral Transform including, as inverse problem, a nonlinear Riemann - Hilbert (NRH) problem, allowing one i) to solve the Cauchy problem for the target PDE; ii) to construct classes of RH spectral data for which the NRH problem is exactly solvable; iii) to construct the longtime behavior of the solutions of such PDE; iv) to establish if a localized initial datum breaks at finite time. We also comment on the existence of recursion operators and Backlünd - Darboux transformations for integrable dispersionless PDEs.

nlin.SI↗

arXiv2012

Holomorphic eigenfunctions of the vector field associated with the dispersionless Kadomtsev-Petviashvili equation

Vector fields naturally arise in many branches of mathematics and physics. Recently it was discovered that Lax pairs for many important multidimensional integrable partial differential equations (PDEs) of hydrodynamic type (also known as dispersionless PDEs) consist of vector field equations. These vector fields have complex coefficients and their analytic, in the spectral parameter, eigenfunctions play an important role in the formulations of the direct and inverse spectral transforms. In this paper we prove existence of eigenfunctions of the basic vector field associated with the celebrated dispersionless Kadomtsev-Petviashvili equation, which are holomorphic in the spectral parameter $λ$ in the strips $|\Imλ|> C_0$.

math-ph↗