arXiv subjects

Ziad H. Musslimani

Publications and source records attributed to Ziad H. Musslimani.

At least 19 recordsLinked to original sources

arXiv2026

Pure-Quartic Optical Shock Waves

In this paper, we study the emergence and dynamics of pure-quartic optical dispersive shock waves governed by the nonlinear Schrödinger (NLS) equation with self-defocusing Kerr nonlinearity and fourth-order dispersion. The corresponding dispersionless hydrodynamic system reveals a distinct mechanism for wave breaking, driven by the nonlinear self-steepening of the hydrodynamic velocity. We illustrate this mechanism through optical dam-break configurations described by Riemann problems and show that, in contrast to the classical quadratic NLS equation, wave breaking occurs prior to wave splitting even in the small amplitude regime. Numerical simulations of the pure-quartic NLS (PQNLS) equation and its hydrodynamic approximation confirm these qualitatively distinct dynamical regimes.

nlin.PS↗

arXiv2025

On the integrable six-wave interaction system and its space-time shifted reduction

The multi-dimensional six-wave interaction system is derived in the context of nonlinear optics. Starting from Maxwell's equations, a reduced system of equations governing the dynamics of the electric and polarization fields are obtained. Using a space-time multi-scale asymptotic expansion, a hierarchy of coupled equations describing the spatio-temporal evolution of the perturbed electric and polarization fields are derived. The leading order equation admits a six-wave ansatz satisfying a triad resonance condition. By removing secular terms at next order, a first order in space and time quadratically nonlinear coupled six-wave interaction system is obtained. This resulting system is tied to its integrable counterpart which was postulated by Ablowitz and Haberman in the 1970s. A reduction to a space-time shifted nonlocal three-wave system is presented. The resulting system is solved using the inverse scattering transform, which employs nonlocal symmetries between the associated eigenfunctions and scattering data; soliton solutions are then obtained. Finally, an infinite set of conservation laws for the six-wave system is derived; one is shown to be connected to its Hamiltonian structure.

nlin.SI↗

arXiv2025

Tensorial symmetries and optical lattice thermalization

This paper presents a comprehensive and systematic study of the possible connection between thermalization of cubic nonlinear lattices with nearest-neighbor coupling and the structure of the mixing tensor that arises due to the presence of nonlinearities. The approach is based on rewriting the underlying lattice system as a nonlinear evolution equation governing the dynamics of the modal amplitudes (or projection coefficients). In this formulation, the linear coupling become diagonalizable, whereas all cubic nonlinear terms transform into a combinatorial sum over a product of three modal amplitudes weighted by a fourth-order mixing tensor. We have identified the exact structure of several mixing tensors (corresponding to different types of cubic nonlinearities) and tied their symmetry properties (quasi-Hermiticity and permutation symmetries associated with two lattice conservation laws) with thermalization or lack thereof. Furthermore, we have observed through direct numerical simulations that the modal occupancies of lattices preserving these tensorial symmetries approach a Rayleigh-Jeans distribution at thermal equilibrium. In addition, we provided few examples that indicate that cubic lattices with broken tensorial symmetries end up not to equilibrate to a Rayleigh-Jeans distribution. Finally, an inverse approach to the study of thermalization of cubic nonlinear lattices is developed. It establishes a duality property between lattices in local and modal bases. The idea is to establish a trade-off between the type of nonlinearities in local base and their respective interactions in supermode base. With this at hand, we were able to identify a large class of nonlinear lattices that are embedded in the modal space and admit a simple form that can be used to shed more light on the role that localization (or delocalization) of the supermodes play in thermalization processes.

nlin.PS↗

arXiv2024

Inverse scattering transform for continuous and discrete space-time shifted integrable equations

Nonlocal integrable partial differential equations possessing a spatial or temporal reflection have constituted an active research area for the past decade. Recently, more general classes of these nonlocal equations have been proposed, wherein the nonlocality appears as a combination of a shift (by a real or a complex parameter) and a reflection. This new shifting parameter manifests itself in the inverse scattering transform (IST) as an additional phase factor in an analogous way to the classical Fourier transform. In this paper, the IST is analyzed in detail for several examples of such systems. Particularly, time, space, and space-time shifted nonlinear Schrödinger (NLS) and space-time shifted modified Korteweg-de Vries (mKdV) equations are studied. Additionally, the semi-discrete IST is developed for the time, space and space-time shifted variants of the Ablowitz-Ladik integrable discretization of the NLS. One soliton solutions are constructed for all continuous and discrete cases.

nlin.SI↗

arXiv2024

Collapse dynamics for two-dimensional space-time nonlocal nonlinear Schrodinger equations

The question of collapse (blow-up) in finite time is investigated for the two-dimensional (non-integrable) space-time nonlocal nonlinear Schrodinger equations. Starting from the two-dimensional extension of the well known AKNS q,r system, three different cases are considered: (i) partial and full parity-time (PT) symmetric, (ii) reverse-time (RT) symmetric, and (iii) general q,r system. Through extensive numerical experiments, it is shown that collapse of Gaussian initial conditions depends on the value of its quasi-power. The collapse dynamics (or lack thereof) strongly depends on whether the nonlocality is in space or time. A so-called quasi-variance identity is derived and its relationship to blow-up is discussed. Numerical simulations reveal that this quantity reaching zero in finite time does not (in general) guarantee collapse. An alternative approach to the study of wave collapse is presented via the study of transverse instability of line soliton solutions. In particular, the linear stability problem for perturbed solitons is formulated for the nonlocal RT and PT symmetric nonlinear Schrodinger (NLS) equations. Through a combination of numerical and analytical approaches, the stability spectrum for some stationary one soliton solutions is found. Direct numerical simulations agree with the linear stability analysis which predicts filamentation and subsequent blow-up.

nlin.PS↗

arXiv2022

Real space Mott-Anderson electron localization with long-range interactions: exact and approximate descriptions

Real materials always contain, to some extent, randomness in the form of defects or irregularities. It is known since the seminal work of Anderson that randomness can drive a metallic phase to an insulating one, and the mechanism responsible for this transition is intrinsically different from the one of the interaction-induced transitions discovered by Mott. Lattice Hamiltonians, with their conceptual and computational advantages, permitted to investigate broadly the interplay of both mechanisms. However, a clear understanding of the differences (or not) with their real-space counterparts is lacking, especially in the presence of long-range Coulomb interactions. This work aims at shedding light on this challenging question by investigating a real-space one-dimensional model of interacting electrons in the presence of a disordered potential. The transition between delocalized and localized phases are characterized using two different indicators, namely the single-particle occupation entropy and the position-space information entropy. In addition, the performance of density functional approximations to reproduce the exact ground-state densities of this many-body localization model are gauged.

physics.chem-ph↗

arXiv2020

Large coupling-strength expansion of the Møller-Plesset adiabatic connection: From paradigmatic cases to variational expressions for the leading terms

We study in detail the first three leading terms of the large coupling-strength limit of the adiabatic connection that has as weak-interaction expansion the Møller-Plesset perturbation theory. We first focus on the H atom, both in the spin-polarized and the spin-unpolarized case, reporting numerical and analytical results. In particular, we derive an asymptotic equation that turns out to have simple analytical solutions for certain channels. The asymptotic H atom solution for the spin-unpolarized case is then shown to be variationally optimal for the many-electron spin-restricted closed-shell case, providing expressions for the large coupling-strength density functionals up to the third leading order. We also analyze the H2 molecule and the uniform electron gas.

physics.chem-ph↗

arXiv2020

Kohn-Sham equations with functionals from the strictly-correlated regime: Investigation with a spectral renormalization method

We re-adapt a spectral renormalization method, introduced in nonlinear optics, to solve the Kohn-Sham (KS) equations of density functional theory (DFT), with a focus on functionals based on the strictly-correlated electrons (SCE) regime, which are particularly challenging to converge. Important aspects of the method are: (i) the eigenvalues and the density are computed simultaneously; (ii) it converges using randomized initial guesses; (iii) easy to implement. Using this method we could converge for the first time the Kohn-Sham equations with functionals that include the next leading term in the strong-interaction limit of density functional theory, the so-called zero-point energy (ZPE) functional as well as with an interaction-strength-interpolation (ISI) functional that includes both the exact SCE and ZPE terms. This work is the first building block for future studies on quantum systems confined in low dimensions with different statistics and long-range repulsions, such as localization properties of fermions and bosons with strong long-range repulsive interactions in the presence of a random external potential.

cond-mat.str-el↗

arXiv2019

Integrable nonlocal asymptotic reductions of physically significant nonlinear equations

Quasi-monochromatic complex reductions of a number of physically important equations are obtained. Starting from the cubic nonlinear Klein-Gordon (NLKG), the Korteweg-deVries (KdV) and water wave equations, it is shown that the leading order asymptotic approximation can be transformed to the well-known integrable AKNS system [6] associated with second order (in space) nonlinear wave equations. This in turn establishes, for the first time, an important physical connection between the recently discovered nonlocal integrable reductions of the AKNS system and physically interesting equations. Reductions include the parity-time, reverse space-time and reverse time nonlocal nonlinear Schrodinger equations.

nlin.SI↗

arXiv2018

Computing eigenfunctions and eigenvalues of boundary value problems with the orthogonal spectral renormalization method

The spectral renormalization method was introduced in 2005 as an effective way to compute ground states of nonlinear Schrödinger and Gross-Pitaevskii type equations. In this paper, we introduce an orthogonal spectral renormalization (OSR) method to compute ground and excited states (and their respective eigenvalues) of linear and nonlinear eigenvalue problems. The implementation of the algorithm follows four simple steps: (i) reformulate the underlying eigenvalue problem as a fixed point equation, (ii) introduce a renormalization factor that controls the convergence properties of the iteration, (iii) perform a Gram-Schmidt orthogonalization process in order to prevent the iteration from converging to an unwanted mode; and (iv) compute the solution sought using a fixed-point iteration. The advantages of the OSR scheme over other known methods (such as Newton's and self-consistency) are: (i) it allows the flexibility to choose large varieties of initial guesses without diverging, (ii) easy to implement especially at higher dimensions and (iii) it can easily handle problems with complex and random potentials. The OSR method is implemented on benchmark Hermitian linear and nonlinear eigenvalue problems as well as linear and nonlinear non-Hermitian $\mathcal{PT}$-symmetric models.

physics.comp-ph↗

arXiv2017

General soliton solution to a nonlocal nonlinear Schrödinger equation with zero and nonzero boundary conditions

General soliton solutions to a nonlocal nonlinear Schrödinger (NLS) equation with PT-symmetry for both zero and nonzero boundary conditions {are considered} via the combination of Hirota's bilinear method and the Kadomtsev-Petviashvili (KP) hierarchy reduction method. First, general $N$-soliton solutions with zero boundary conditions are constructed. Starting from the tau functions of the two-component KP hierarchy, it is shown that they can be expressed in terms of either Gramian or double Wronskian determinants. On the contrary, from the tau functions of single component KP hierarchy, general soliton solutions to the nonlocal NLS equation with nonzero boundary conditions are obtained. All possible soliton solutions to nonlocal NLS with Parity (PT)-symmetry for both zero and nonzero boundary conditions are found in the present paper.

nlin.SI↗

arXiv2017

Time-dependent spectral renormalization method

The spectral renormalization method was introduced by Ablowitz and Musslimani in 2005, [Opt. Lett. 30, pp. 2140-2142] as an effective way to numerically compute (time-independent) bound states for certain nonlinear boundary value problems. % of the nonlinear Schrödinger (NLS), Gross-Pitaevskii and water wave type equations to mention a few. In this paper, we extend those ideas to the time domain and introduce a time-dependent spectral renormalization method as a numerical means to simulate linear and nonlinear evolution equations. The essence of the method is to convert the underlying evolution equation from its partial or ordinary differential form (using Duhamel's principle) into an integral equation. The solution sought is then viewed as a fixed point in both space and time. The resulting integral equation is then numerically solved using a simple renormalized fixed-point iteration method. Convergence is achieved by introducing a time-dependent renormalization factor which is numerically computed from the physical properties of the governing evolution equation. The proposed method has the ability to incorporate physics into the simulations in the form of conservation laws or dissipation rates. This novel scheme is implemented on benchmark evolution equations: the classical nonlinear Schrödinger (NLS), integrable $PT$ symmetric nonlocal NLS and the viscous Burgers' equations, each of which being a prototypical example of a conservative and dissipative dynamical system. Numerical implementation and algorithm performance are also discussed.

nlin.PS↗

arXiv2017

Vortices in Bose-Einstein condensates with $\mathcal{PT}$-symmetric gain and loss

We investigate vortex excitations in dilute Bose-Einstein condensates in the presence of complex $\mathcal{PT}$-symmetric potentials. These complex potentials are used to describe a balanced gain and loss of particles and allow for an easier calculation of stationary states in open systems than in a full dynamical calculation including the whole environment. We examine the conditions under which stationary vortex states can exist and consider transitions from vortex to non-vortex states. In addition, we study the influences of $\mathcal{PT}$ symmetry on the dynamics of non-stationary vortex states placed at off-center positions.

quant-ph↗

arXiv2017

Inverse scattering transform for the nonlocal reverse space-time Sine-Gordon, Sinh-Gordon and nonlinear Schrödinger equations with nonzero boundary conditions

The reverse space-time (RST) Sine-Gordon, Sinh-Gordon and nonlinear Schrödinger equations were recently introduced and shown to be integrable infinite-dimensional dynamical systems. The inverse scattering transform (IST) for rapidly decaying data was also constructed. In this paper, IST for these equations with nonzero boundary conditions (NZBCs) at infinity is presented. The NZBC problem is more complicated due to the associated branching structure of the associated linear eigenfunctions. With constant amplitude at infinity, four cases are analyzed; they correspond to two different signs of nonlinearity and two different values of the phase at infinity. Special soliton solutions are discussed and explicit 1-soliton and 2-soliton solutions are found. In terms of IST, the difference between the RST Sine-Gordon/Sinh-Gordon equations and the RST NLS equation is the time dependence of the scattering data. Spatially dependent boundary conditions are also briefly considered.

math-ph↗

arXiv2016

Wave propagation through disordered media without backscattering and intensity variations

A fundamental manifestation of wave scattering in a disordered medium is the highly complex intensity pattern the waves acquire due to multi-path interference. Here we show that these intensity variations can be entirely suppressed by adding disorder-specific gain and loss components to the medium. The resulting constant-intensity (CI) waves in such non-Hermitian scattering landscapes are free of any backscattering and feature perfect transmission through the disorder. An experimental demonstration of these unique wave states is envisioned based on spatially modulated pump beams that can flexibly control the gain and loss components in an active medium.

physics.optics↗

arXiv2016

Inverse scattering transform for the nonlocal nonlinear Schrödinger equation with nonzero boundary conditions

In 2013 a new nonlocal symmetry reduction of the well-known AKNS scattering problem was found; it was shown to give rise to a new nonlocal $PT$ symmetric and integrable Hamiltonian nonlinear Schrödinger (NLS) equation. Subsequently, the inverse scattering transform was constructed for the case of rapidly decaying initial data and a family of spatially localized, time periodic one soliton solution were found. In this paper, the inverse scattering transform for the nonlocal NLS equation with nonzero boundary conditions at infinity is presented in the four cases when the data at infinity have constant amplitudes. The direct and inverse scattering problems are analyzed. Specifically, the direct problem is formulated, the analytic properties of the eigenfunctions and scattering data and their symmetries are obtained. The inverse scattering problem is developed via a left-right Riemann-Hilbert problem in terms of a suitable uniformization variable and the time dependence of the scattering data is obtained. This leads to a method to linearize/solve the Cauchy problem. Pure soliton solutions are discussed and explicit 1-soliton solution and two 2-soliton solutions are provided for three of the four different cases corresponding to two different signs of nonlinearity and two different values of the phase difference between plus and minus infinity. In the one other case there are no solitons.

nlin.SI↗

arXiv2016

Integrable nonlocal nonlinear equations

A nonlocal nonlinear Schrödinger (NLS) equation was recently found by the authors and shown to be an integrable infinite dimensional Hamiltonian equation. Unlike the classical (local) case, here the nonlinearly induced "potential" is $PT$ symmetric thus the nonlocal NLS equation is also $PT$ symmetric. In this paper, new {\it reverse space-time} and {\it reverse time} nonlocal nonlinear integrable equations are introduced. They arise from remarkably simple symmetry reductions of general AKNS scattering problems where the nonlocality appears in both space and time or time alone. They are integrable infinite dimensional Hamiltonian dynamical systems. These include the reverse space-time, and in some cases reverse time, nonlocal nonlinear Schrödinger, modified Korteweg-deVries (mKdV), sine-Gordon, $(1+1)$ and $(2+1)$ dimensional three-wave interaction, derivative NLS, "loop soliton", Davey-Stewartson (DS), partially $PT$ symmetric DS and partially reverse space-time DS equations. Linear Lax pairs, an infinite number of conservation laws, inverse scattering transforms are discussed and one soliton solutions are found. Integrable reverse space-time and reverse time nonlocal discrete nonlinear Schrödinger type equations are also introduced along with few conserved quantities. Finally, nonlocal Painlevé type equations are derived from the reverse space-time and reverse time nonlocal NLS equations.

nlin.SI↗

arXiv2016

Generalized Uncertainty Principle and Analogue of Quantum Gravity in Optics

The design of optical systems capable of processing and manipulating ultra-short pulses and ultra-focused beams is highly challenging with far reaching fundamental technological applications. One key obstacle routinely encountered while implementing sub-wavelength optical schemes is how to overcome the limitations set by standard Fourier optics. A strategy to overcome these difficulties is to utilize the concept of generalized uncertainty principle (G-UP) that has been originally developed to study quantum gravity. In this paper we propose to use the concept of G-UP within the framework of optics to show that the generalized Schrodinger equation describing short pulses and ultra-focused beams predicts the existence of a minimal spatial or temporal scale which in turn implies the existence of maximally localized states. Using a Gaussian wavepacket with complex phase, we derive the corresponding generalized uncertainty relation and its maximally localized states. We numerically show that the presence of nonlinearity helps the system to reach its maximal localization. Our results may trigger further theoretical and experimental tests for practical applications and analogues of fundamental physical theories.

nlin.SI↗