arXiv subjects

Francesco Coppini

Publications and source records attributed to Francesco Coppini.

6 recordsLinked to original sources

arXiv2026

Novel periodic solutions and rogue waves of the defocusing scalar and coupled Ablowitz-Ladik systems on a nonzero background

In this paper we apply Hirota's bilinear method to the scalar and coupled Ablowitz-Ladik systems in the defocusing dispersion regime under the assumption of a background amplitude $0<\rho<1$. We first establish, in the scalar case, the correspondence between the Hirota's parameters and the spectral parameters of the inverse scattering transform. Then we show that when the Hirota parameter associated to the discrete eigenvalue is chosen outside the range corresponding to a discrete dark soliton, novel solutions of the Ablowitz-Ladik system emerge. In general, these solutions are singular, but there exists a class of time-periodic solutions for which it is possible to choose the soliton parameters so that the breathers remain regular on the lattice for all times. We also discuss the interactions between a dark soliton and a regular breather, and between two regular breathers. For the coupled Ablowitz-Ladik system, by including in the background discrete, counter-propagating plane waves, we use Hirota's method to derive novel Akhmediev-type (i.e., space-periodic) discrete breathers which are regular for all times. Finally, taking the limit of the discrete Akhmediev breathers as the period approaches infinity (i.e., as the wavenumber approaches zero) we obtain novel rogue wave solutions of the coupled Ablowitz-Ladik system.

nlin.SI↗

arXiv2026

x-periodic Quasi One Dimensional Anomalous (Rogue) Waves in Multidimensional Nonlinear Schr\"odinger 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 Schr\"odinger (MNLS) equation, like the non integrable elliptic and hyperbolic nonlinear Schr\"odinger (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↗

arXiv2026

Pseudovorticity of 2+1D optical solitons

In the hydrodynamic representation of a quantum fluid or optical field, vorticity vanishes wherever the phase is well defined, and is instead localized at phase singularities, or quantum vortices. Pseudovorticity, by contrast, characterizes local rotational structures, even in regions without singularities or net orbital angular momentum. We study both experimentally and numerically pseudovorticity in photorefractive solitons and show that a detailed phase and amplitude analysis unveils a complex rotational flow dynamic: bright 2+1D solitons are found to carry a pseudovorticity dipole, while quadrupoles emerge in soliton fusion. The phenomenon, also explained using geometrical considerations, suggests a general picture according to which stable high-dimensional solitons naturally carry a hierarchy of pseudovorticity multipoles, encoded in the local perturbed phase and amplitude.

physics.optics↗

arXiv2025

Quasi one dimensional anomalous (rogue) waves in multidimensional nonlinear Schr\"odinger 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 Schr\"odinger (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↗

arXiv2024

Evidence of 1+1D photorefractive stripe solitons deep in the Kerr limit

The Kerr nonlinearity allows for exact analytic soliton solutions in 1+1D. While nothing excludes that these solitons form in naturally-occurring real-world 3D settings as solitary walls or stripes, their observation has previously been considered unfeasible because of the strong transverse instability intrinsic to the extended nonlinear perturbation. We report the observation of solitons that are fully compatible with the 1+1D Kerr paradigm limit hosted in a 2+1D system. The waves are stripe spatial solitons in bulk copper doped potassium-lithium-tantalate-niobate (KLTN) supported by the unsaturated photorefractive screening nonlinearity. The parameters of the stripe solitons fit well, in the whole existence domain, with the 1+1D existence curve that we derive for the first time in closed form starting from the saturable model of propagation. Transverse instability, that accompanies the solitons embedded in the 3D system, is found to have a gain length much longer than the crystal. Findings establish our system as a versatile platform for investigating exact soliton solutions in bulk settings and in exploring the role of dimensionality at the transition from integrable to non-integrable regimes of propagation.

nlin.PS↗

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 Schr\"odinger (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↗