arXiv subjects

Paolo Maria Santini

Publications and source records attributed to Paolo Maria Santini.

12 recordsLinked to original sources

arXiv2026

x-periodic Quasi One Dimensional Anomalous (Rogue) Waves in Multidimensional Nonlinear Schrödinger Equations: Fission, Fusion, and Recurrence

In a recent work we studied the first nonlinear stage of modulation instability (NLSMI) of x-periodic anomalous (rogue, freak, extreme) waves (AWs) of physically relevant multidimensional (generalizations of the focusing) nonlinear Schrödinger (MNLS) equation, like the non integrable elliptic and hyperbolic nonlinear Schrödinger (NLS) equations in d+1 dimensions, d = 2, 3, in the quasi one dimensional (Q1D) regime in which the wavelength in the direction of propagation x is small with respect to the wavelengths in the transversal directions. We showed that, at leading order, the first NLSMI is universal, independent of the particular MNLS model, and described by suitable adiabatic deformations of the quasi-homoclinic Akhmediev breather solution of NLS, in excellent agreement with numerical simulations. In the present work we focus on the recurrence of x-periodic AWs in the Q1D regime. We show that, although the first nonlinear stage of MI is essentially universal for all MNLS equations, the recurrence dynamics exhibit significant O(1) differences among different models. Moreover, successive nonlinear stages generally display increasingly complex combinations of fission and fusion processes, leading to progressively richer dynamical choreographies. Since MNLS equations in the Q1D regime can be viewed as multidimensional perturbations of the integrable NLS equation, we use the recently developed finite gap perturbation theory of NLS AWs to give an analytic and quantitative description of the recurrence of Q1D AWs, in excellent agreement with numerical simulations. Due to the physical relevance of the MNLS equations considered in this work, and due to the universality of the processes discussed in this paper, it is plausible that they be observable in many fields of physics, like water waves, nonlinear optics, plasma physics, Bose-Einstein condensates, etc . . .

nlin.PS↗

arXiv2025

Quasi one dimensional anomalous (rogue) waves in multidimensional nonlinear Schrödinger equations 1: fission and fusion

In this paper we study the first nonlinear stage of modulation instability (NLSMI) of $x$-periodic AWs in multidimensional generalizations of the focusing nonlinear Schrödinger (NLS) equation, like the non-integrable elliptic and hyperbolic NLS equations in $2+1$ and $3+1$ dimensions. In the quasi one-dimensional (Q1D) regime, where the wavelength in the $x$ direction of propagation is significantly smaller than in the transversal directions, the behavior is universal, independent of the particular model at leading order, and described by adiabatic deformations of the Akhmediev breather solution of NLS. Varying the initial data, the first NLSMI shows various combinations of basic processes like AW growth from the unstable background, followed by fission in the slowly varying transversal directions, and the inverse process of fusion, followed by AW decay to the background. Fission and fusion are critical processes showing similarities with multidimensional wave breaking, and with phase transitions of second kind and critical exponent $1/2$. In $3+1$ dimensions with radial symmetry in the transversal slowly varying plane, fission consists in the formation of an opening smoke ring centered on the $x$ axis. In the long wave limit, the Q1D Akhmediev breather reduces to the Q1D analogue of the Peregrine instanton, rationally localized in space. Numerical experiments on the hyperbolic NLS equation show that the process of "AW growth + fission" is not restricted to the Q1D regime, extending to a finite area of the modulation instability domain. The universality of these processes suggests their observability in natural phenomena related to AWs in contexts such as water waves, nonlinear optics, and plasma physics.

math-ph↗

arXiv2023

The effect of loss/gain and hamiltonian perturbations of the Ablowitz-Ladik lattice on the recurrence of periodic anomalous waves

The Ablowitz-Ladik (AL) equations are distinguished integrable discretizations of the focusing and defocusing nonlinear Schrödinger (NLS) equations. In a previous paper (arXiv:2305.04857) we have studied the effect of the modulation instability of the homogeneous background solution of the AL equations in the periodic setting, showing in particular that both models exhibit instability properties, and studying, in terms of elementary functions, how a generic periodic perturbation of the unstable background evolves into a recurrence of anomalous waves (AWs). Using the finite gap method, in this paper we extend the recently developed perturbation theory for periodic NLS AWs to lattice equations, studying the effect of physically relevant perturbations of the $AL$ equations on the AW recurrence, like: linear loss, gain, and/or Hamiltonian corrections, in the simplest case of one unstable mode. We show that these small perturbations induce $O(1)$ effects on the periodic AW dynamics, generating three distinguished asymptotic patterns. Since dissipation and higher order Hamiltonian corrections can hardly be avoided in natural phenomena involving AWs, and since these perturbations induce $O(1)$ effects on the periodic AW dynamics, we expect that the asymptotic states described analytically in this paper will play a basic role in the theory of periodic AWs in natural phenomena described by discrete systems. The quantitative agreement between the analytic formulas of this paper and numerical experiments is excellent.

nlin.SI↗

arXiv2017

On the occurrence of gauge-dependent secularities in nonlinear gravitational waves

We study the plane (not necessarily monochromatic) gravitational waves at nonlinear quadratic order on a flat background in vacuum. We show that, in the harmonic gauge, the nonlinear waves are unstable. We argue that, at this order, this instability can not be eliminated by means of a multiscale approach, i.e. introducing suitable long variables, as it is often the case when secularities appear in a perturbative scheme. However, this is a non-physical and gauge-dependent effect that disappears in a suitable system of coordinates. In facts, we show that in a specific gauge such instability does not occur, and that it is possible to solve exactly the second order nonlinear equations of gravitational waves. Incidentally, we note that this gauge coincides with the one used by Belinski and Zakharov to find exact solitonic solutions of Einstein's equations, that is to an exactly integrable case, and this fact makes our second order nonlinear solutions less interesting. However, the important warning is that one must be aware of the existence of the instability reported in this paper, when studying nonlinear gravitational waves in the harmonic gauge.

gr-qc↗

arXiv2009

Multiscale expansions of difference equations in the small lattice spacing regime, and a vicinity and integrability test. I

We propose an algorithmic procedure i) to study the ``distance'' between an integrable PDE and any discretization of it, in the small lattice spacing epsilon regime, and, at the same time, ii) to test the (asymptotic) integrability properties of such discretization. This method should provide, in particular, useful and concrete informations on how good is any numerical scheme used to integrate a given integrable PDE. The procedure, illustrated on a fairly general 10-parameter family of discretizations of the nonlinear Schroedinger equation, consists of the following three steps: i) the construction of the continuous multiscale expansion of a generic solution of the discrete system at all orders in epsilon, following the Degasperis - Manakov - Santini procedure; ii) the application, to such expansion, of the Degasperis - Procesi (DP) integrability test, to test the asymptotic integrability properties of the discrete system and its ``distance'' from its continuous limit; iii) the use of the main output of the DP test to construct infinitely many approximate symmetries and constants of motion of the discrete system, through novel and simple formulas.

nlin.SI↗

arXiv2007

Integrable dynamics of Toda-type on the square and triangular lattices

In a recent paper we constructed an integrable generalization of the Toda law on the square lattice. In this paper we construct other examples of integrable dynamics of Toda-type on the square lattice, as well as on the triangular lattice, as nonlinear symmetries of the discrete Laplace equations on the square and triangular lattices. We also construct the $τ$ - function formulations and the Darboux-Bäcklund transformations of these novel dynamics.

nlin.SI↗

arXiv2004

The self-adjoint 5-point and 7-point difference operators, the associated Dirichlet problems, Darboux transformations and Lelieuvre formulas

We present some basic properties of two distinguished discretizations of elliptic operators: the self-adjoint 5-point and 7-point schemes on a two dimensional lattice. We first show that they allow to solve Dirichlet boundary value problems; then we present their Darboux transformations. Finally we construct their Lelieuvre formulas and we show that, at the level of the normal vector and in full analogy with their continuous counterparts, the self-adjoint 5-point scheme characterizes a two dimensional quadrilateral lattice (a lattice whose elementary quadrilaterals are planar), while the self-adjoint 7-point scheme characterizes a generic 2D lattice.

nlin.SI↗

arXiv2000

The symmetric, D-invariant and Egorov reductions of the quadrilateral lattice

We present a detailed study of the geometric and algebraic properties of the multidimensional quadrilateral lattice (a lattice whose elementary quadrilaterals are planar; the discrete analogue of a conjugate net) and of its basic reductions. To make this study, we introduce the notions of forward and backward data, which allow us to give a geometric meaning to the tau-function of the lattice, defined as the potential connecting these data. Together with the known circular lattice (a lattice whose elementary quadrilaterals can be inscribed in circles; the discrete analogue of an orthogonal conjugate net) we introduce and study two other basic reductions of the quadrilateral lattice: the symmetric lattice, for which the forward and backward data coincide, and the D-invariant lattice, characterized by the invariance of a certain natural frame along the main diagonal. We finally discuss the Egorov lattice, which is, at the same time, symmetric, circular and D-invariant. The integrability properties of all these lattices are established using geometric, algebraic and analytic means; in particular we present a D-bar formalism to construct large classes of such lattices. We also discuss quadrilateral hyperplane lattices and the interplay between quadrilateral point and hyperplane lattices in all the above reductions.

solv-int↗

arXiv1999

Integrable Discrete Geometry: the Quadrilateral Lattice, its Transformations and Reductions

We review recent results on Integrable Discrete Geometry. It turns out that most of the known (continuous and/or discrete) integrable systems are particular symmetries of the quadrilateral lattice, a multidimensional lattice characterized by the planarity of its elementary quadrilaterals. Therefore the linear property of planarity seems to be a basic geometric property underlying integrability. We present the geometric meaning of its tau-function, as the potential connecting its forward and backward data. We present the theory of transformations of the quadrilateral lattice, which is based on the discrete analogue of the theory of rectilinear congruences. In particular, we discuss the discrete analogues of the Laplace, Combescure, Levy, radial and fundamental transformations and their interrelations. We also show how the sequence of Laplace transformations of a quadrilateral surface is described by the discrete Toda system. We finally show that these classical transformations are strictly related to the basic operators associated with the quantum field theoretical formulation of the multicomponent Kadomtsev-Petviashvilii hierarchy. We review the properties of quadrilateral hyperplane lattices, which play an interesting role in the reduction theory, when the introduction of additional geometric structures allows to establish a connection between point and hyperplane lattices. We present and fully characterize some geometrically distinguished reductions of the quadrilateral lattice, like the symmetric, circular and Egorov lattices; we review also basic geometric results of the theory of quadrilateral lattices in quadrics, and the corresponding analogue of the Ribaucour reduction of the fundamental transformation.

solv-int↗

arXiv1998

Charged Free Fermions, Vertex Operators and Classical Theory of Conjugate Nets

We show that the quantum field theoretical formulation of the $τ$-function theory has a geometrical interpretation within the classical transformation theory of conjugate nets. In particular, we prove that i) the partial charge transformations preserving the neutral sector are Laplace transformations, ii) the basic vertex operators are Levy and adjoint Levy transformations and iii) the diagonal soliton vertex operators generate fundamental transformations. We also show that the bilinear identity for the multicomponent Kadomtsev-Petviashvili hierarchy becomes, through a generalized Miwa map, a bilinear identity for the multidimensional quadrilateral lattice equations.

solv-int↗

arXiv1994

The Integrable Dynamics of Discrete and Continuous Curves

We show that the following geometric properties of the motion of discrete and continuous curves select integrable dynamics: i) the motion of the curve takes place in the N dimensional sphere of radius R, ii) the curve does not stretch during the motion, iii) the equations of the dynamics do not depend explicitly on the radius of the sphere. Well known examples of integrable evolution equations, like the nonlinear Schroedinger and the sine-Gordon equations, as well as their discrete analogues, are derived in this general framework.

solv-int↗

arXiv1994

Integrable dynamics of a discrete curve and the Ablowitz-Ladik hierarchy

We show that the following elementary geometric properties of the motion of a discrete (i.e. piecewise linear) curve select the integrable dynamics of the Ablowitz-Ladik hierarchy of evolution equations: i) the set of points describing the discrete curve lie on the sphere S^3, ii) the distance between any two subsequant points does not vary in time, iii) the dynamics does not depend explicitly on the radius of the sphere. These results generalize to a discrete context our previous work on continuous curves.

solv-int↗