arXiv subjects

Panayotis Kevrekidis

Publications and source records attributed to Panayotis Kevrekidis.

At least 19 recordsLinked to original sources

arXiv2026

Wedge problems and dispersive shock waves in the two-dimensional Toda lattice

We study the formation and interaction of dispersive shock waves (DSWs) in the two-dimensional Toda lattice subject to wedge-type initial conditions, and show that their interaction gives rise to a discrete analog of Mach reflection for dispersive shock waves in discrete systems. The initial jump across each leg of the wedge acts locally as a Riemann problem for the one-dimensional Toda lattice, producing two oblique DSWs whose leading-edge soliton amplitude is determined explicitly by the one-dimensional Whitham modulation theory. The two-dimensional nature of the problem manifests when these oblique DSWs meet along the symmetry axis. We show that, for compressive wedges (i.e., when the initial conditions are such that two oblique DSWs that are generated propagate toward each other), a critical slope $q_{\mathrm{cr}}$ separates two regimes: in the subcritical regime ($q q_{\mathrm{cr}}$) the interaction is ordinary and produces a localized peak whose amplitude is determined analytically. We also show qualitatively that a similar dichotomy between two regimes exists for expansive wedges (i.e., when the initial conditions are such that the two oblique DSWs propagate away from each other). We confirm all analytical predictions by comparing them with the results of direct numerical simulations. Finally, we show that the continuum limit of the result is consistent with the analogous theory for the Kadomtsev-Petviashvili equation, providing an independent validation of the analytical framework.

nlin.PS↗

arXiv2026

Thermalization in a nonlinear variant of the discrete nonlinear Schrödinger Equation

We study the thermalization properties of a fully nonlinear lattice model originally derived from the two-dimensional cubic defocusing nonlinear Schr{ö}dinger equation (NLS) using analytical and numerical methods. The model conserves both energy and norm, whose densities define a microcanonical energy-norm parameter space, while the nonlinear nearest-neighbor coupling is controlled by a parameter $D$. Within this space, our analysis identifies broad parameter regimes in which the dynamics is ergodic not only within but also outside the standard Gibbs region, indicating the need for a modified statistical description. At higher energies, the system instead exhibits long-lived compacton-mediated localization and signatures of weak nonergodicity, as evidenced by finite-time variances, excursion-time statistics, and probability distributions of local amplitudes. We show that stronger coupling $D$ enhances fluctuations and accelerates the crossover of the finite-time variance of the local norm density from an initial $\sim T^{-1/2}$ decay toward the faster $\sim T^{-1}$ decay characteristic of ergodic thermalization, where $T$ denotes the averaging time. In the high-energy regime, weak coupling $D\leq 1$ favors persistent single-site compacton localization, whereas stronger coupling $D>1$ yields long-lived two-site localization. Our results provide insights into the interplay between thermalization, localization, and non-Gibbs statistical behavior in genuinely nonlinear systems.

cond-mat.stat-mech↗

arXiv2026

Hamiltonian Graph Inference Networks: Joint structure discovery and dynamics prediction for lattice Hamiltonian systems from trajectory data

Lattice Hamiltonian systems underpin models across condensed matter, nonlinear optics, and biophysics, yet learning their dynamics from data is obstructed by two unknowns: the interaction topology and whether node dynamics are homogeneous. Existing graph-based approaches either assume the graph is given or, as in $α$-separable graph Hamiltonian network, infer it only for separable Hamiltonians with homogeneous node dynamics. We introduce the Hamiltonian Graph Inference Network (HGIN), which jointly recovers the interaction graph and predicts long-time trajectories from state data alone, for both separable and non-separable Hamiltonians and under heterogeneous node dynamics. HGIN couples a structure-learning module -- a learnable weighted adjacency matrix trained under a Hamilton's-equations loss -- with a trajectory-prediction module that partitions edges into physically distinct subgraphs via $k$-means clustering, assigning each subgraph its own encoder and thereby breaking the parameter-sharing bottleneck of conventional GNNs. On three benchmarks -- a Klein--Gordon lattice with long-range interactions and two discrete nonlinear Schrödinger lattices (homogeneous and heterogeneous) -- HGIN reduces long-time energy prediction error and trajectory prediction error by six to thirteen orders of magnitude relative to baselines. A symmetry argument on the Hamiltonian loss further shows that the learned weights encode the parity of the underlying pair potential, yielding an interpretable readout of the system's interaction structure.

cs.LG↗

arXiv2026

Direct observation of long-range many-body coherence in quasi-one-dimensional attractive Bose gases

Macroscopic coherence is an important feature of quantum many-body systems exhibiting collective behaviors, with examples ranging from atomic Bose-Einstein condensates, and quantum liquids to superconductors. Probing many-body coherence in a dynamically unstable regime, however, presents an intriguing and outstanding challenge in out-of-equilibrium quantum many-body physics. Here, we experimentally study the first- and second-order coherence of degenerate quasi-one-dimensional (1D) Bose gases quenched from repulsive to modulationally unstable attractive interaction regimes. The resulting dynamics, monitored by in-situ density and matter-wave interference imaging, reveals phase-coherent density wave evolutions, clearly distinguished from iconic soliton trains previously observed in attractive gases. This arises from the interplay between noise-amplified density modulations and dispersive shock waves of broad interest in nonlinear physics, plasmas, granular systems, and beyond. At longer times, the gases become phase-scrambled, exhibiting a finite correlation length. Interestingly, following an interaction quench back to the repulsive regime, we observe that quasi-long-range coherence can be spontaneously re-established. This captivating rephasing dynamics can be attributed to the nucleation and annihilation of density defects in the quasi-1D geometry. These results shed light on out-of-equilibrium phase coherence in quantum many-body systems in a regime where beyond mean-field effects may arise and theoretical approaches have not been well-established.

cond-mat.quant-gas↗

arXiv2025

Collapse of a quantum vortex in an attractive two-dimensional Bose gas

We experimentally and numerically study the collapse dynamics of a quantum vortex in a two-dimensional atomic superfluid following a fast interaction ramp from repulsion to attraction. We find the conditions and time scales for a superfluid vortex to radially converge into a quasi-stationary density profile, demonstrating the spontaneous formation of a vortex soliton-like structure in an atomic Bose gas. We record an emergent self-similar dynamics caused by an azimuthal modulational instability, which amplifies initial density perturbations and leads to the eventual splitting of a solitonic ring profile or direct fragmentation of a superfluid into disordered, but roughly circular arrays of Townes soliton-like wavepackets. These dynamics are qualitatively reproduced by simulations based on the Gross-Pitaevskii equation. However, a discrepancy in the magnitude of amplified density fluctuations predicted by our mean-field analysis suggests the presence of effects beyond the mean-field approximation. Our study sets the stage for exploring out-of-equilibrium dynamics of vortex quantum matter quenched to attractive interactions and their universal characteristics.

cond-mat.quant-gas↗

arXiv2025

Hydromagnetic shock waves in a cold weakly collisional plasma

In this work we revisit the topic of existence of hydrodynamic shock waves in a cold weakly collisional plasma. For this purpose we consider the well established Adlam-Allen model with the addition of a dashpot term associated with the dissipation of the motion of the electrons relative to the ions. We establish the connection between this model and the Korteweg-de-Vries Burgers equation via an asymptotic multiscale analysis. This fact suggests the possibility that this system may support shock wave solutions. Accordingly, by considering a corresponding dynamical system arising through a co-traveling frame reduction, we identify such orbits via a phase-plane analysis. We then leverage such initial conditions within systematic simulations of the original modified Adlam-Allen model, revealing a variety of supported robust wavefronts depending on the magnitude of the dissipation considered.

nlin.PS↗

arXiv2025

Computer Assisted Discovery of Integrability via SILO: Sparse Identification of Lax Operators

We formulate the discovery of Lax integrability of Hamiltonian dynamical systems as a symbolic regression problem, which, loosely speaking, seeks to maximize the compatibility between a pair of Lax operators and the known Hamiltonian of the dynamical system. Our approach is first tested on the simple harmonic oscillator. We then move on to the Henon-Heiles system, i.e. a two-degree-of-freedom system of nonlinear oscillators. The integrability of the Henon-Heiles system is critically dependent on a set of three parameters within its Hamiltonian, a fact that we leverage to assess the robustness of our approach in detecting the integrability of this system with respect to the parameter dependence of the Hamiltonian. We then adapt our method to canonical examples of Hamiltonian partial differential equations, including the Korteweg-de Vries and cubic nonlinear Schrödinger equations, again testing robustness against nonintegrable perturbations of their respective Hamiltonians. In all examples, our approach reliably confirms or denies the integrability of the equations of interest. Moreover, by appropriately adjusting the loss function and applying thresholded $l^0$ regularization to enforce sparsity in the operator weights, we successfully recover accurate forms of the Lax pairs despite wide initial hypotheses on the operators. Some of the relevant Lax pairs, notably for the Henon-Heiles system and the Korteweg-deVries equation, are distinct from the ones that are typically reported in the literature. The Lax pairs that our methodology discovers warrant further mathematical and computational investigation, and we discuss extensively the opportunities for further improvement of SILO as a viable tool for interpretable exploration of integrable Hamiltonian dynamical systems.

nlin.SI↗

arXiv2025

Experimentally Tractable Generation of High-Order Rogue Waves in Bose-Einstein Condensates

In this work, we study a prototypical, experimentally accessible scenario that enables the systematic generation of so-called high-order rogue waves in atomic Bose-Einstein condensates. These waveforms lead to significantly and controllably more extreme focusing events than the famous Peregrine soliton. In one spatial dimension, we showcase conclusive numerical evidence that our scheme generates the focusing behavior associated with the first four rogue waves from the relevant hierarchy. We then extend considerations to anisotropic two-dimensional and even three-dimensional settings, establishing that the scheme can generate second order rogue waves despite the well-known limitation of finite-time blow up of focusing nonlinear Schrödinger equations.

nlin.PS↗

arXiv2024

$α$-SGHN: A Robust Model for Learning Particle Interactions in Lattice Systems

We propose an $α$-separable graph Hamiltonian network ($α$-SGHN) that reveals complex interaction patterns between particles in lattice systems. Utilizing trajectory data, $α$-SGHN infers potential interactions without prior knowledge about particle coupling, overcoming the limitations of traditional graph neural networks that require predefined links. Furthermore, $α$-SGHN preserves all conservation laws during trajectory prediction. Experimental results demonstrate that our model, incorporating structural information, outperforms baseline models based on conventional neural networks in predicting lattice systems. We anticipate that the results presented will be applicable beyond the specific onsite and inter-site interaction lattices studied, including the Frenkel-Kontorova model, the rotator lattice, and the Toda lattice.

math.DS↗

arXiv2024

An epidemical model with nonlocal spatial infections

The SIR model is one of the most prototypical compartmental models in epidemiology. Generalizing this ordinary differential equation (ODE) framework into a spatially distributed partial differential equation (PDE) model is a considerable challenge. In the present work, we extend a recently proposed model based on nearest-neighbor spatial interactions by one of the authors in~\cite{vaziry2022modelling} towards a nonlocal, nonlinear PDE variant of the SIR prototype. We then seek to develop a set of tools that provide insights for this PDE framework. Stationary states and their stability analysis offer a perspective on the early spatial growth of the infection. Evolutionary computational dynamics enable visualization of the spatio-temporal progression of infection and recovery, allowing for an appreciation of the effect of varying parameters of the nonlocal kernel, such as, e.g., its width parameter. These features are explored in both one- and two-dimensional settings. At a model-reduction level, we develop a sequence of interpretable moment-based diagnostics to observe how these reflect the total number of infections, the epidemic's epicenter, and its spread. Finally, we propose a data-driven methodology based on the sparse identification of nonlinear dynamics (SINDy) to identify approximate closed-form dynamical equations for such quantities. These approaches may pave the way for further spatio-temporal studies, enabling the quantification of epidemics.

q-bio.QM↗

arXiv2023

On the Whitham modulation equations for the Toda lattice and the quantitative characterization of its dispersive shocks

The aim of this work is multifold. Firstly, it intends to present a complete, quantitative and self-contained description of the periodic traveling wave solutions and Whitham modulation equations for the Toda lattice, combining results from different previous works in the literature. Specifically, we connect the Whitham modulation equations and a detailed expression for the periodic traveling wave solutions of the Toda lattice. Along the way, some key details are filled in, such as the explicit expression of the characteristic speeds of the genus-one Toda-Whitham system. Secondly, we use these tools to obtain a detailed quantitative characterization of the dispersive shocks of the Toda system. Lastly, we validate the relevant analysis by performing a detailed comparison with direct numerical simulations.

nlin.SI↗

arXiv2020

On the generation and propagation of solitary waves in integrable and non-integrable nonlinear lattices

We investigate the generation and propagation of solitary waves in the context of the Hertz chain and Toda lattice, with the aim to highlight the similarities, as well as differences between these systems. We begin by discussing the kinetic and potential energy of a solitary wave in these systems, and show that under certain circumstances the kinetic and potential energy profiles in these systems (i.e., their spatial distribution) look reasonably close to each other. While this and other features, such as the connection between the amplitude and the total energy of the wave bear similarities between the two models, there are also notable differences, such as the width of the wave. We then study the dynamical behavior of these systems in response to an initial velocity impulse. For the Toda lattice, we do so by employing the inverse scattering transform, and we obtain analytically the ratio between the energy of the resulting solitary wave and the energy of the impulse, as a function of the impulse velocity; we then compare the dynamics of the Toda system to that of the Hertz system, for which the corresponding quantities are obtained through numerical simulations. In the latter system, we obtain a universality in the fraction of the energy stored in the resulting solitary traveling wave irrespectively of the size of the impulse. This fraction turns out to only depend on the nonlinear exponent. Finally, we investigate the relation between the velocity of the resulting solitary wave and the velocity of the impulse. In particular, we provide an alternative proof for the numerical scaling rule of Hertz type systems.

nlin.PS↗

arXiv2019

Toroidal Vortex Filament Knots & Links: Existence, Stability and Dynamics

Using the Klein-Majda-Damodaran model of nearly-parallel vortex filaments, we construct vortex knots and links on a torus involving periodic boundary conditions and analyze their stability. For a special class of vortex knots -- toroidal knots -- we give a full characterization of both their energetic and dynamical stability. In addition to providing explicit expressions for the relevant waveforms, we derive explicit formulas for their stability boundaries. These include simple links and different realizations of a trefoil knot. It is shown that a ring of more than 7 filaments can potentially be stablized by giving it a slight twist and connecting neighbouring filaments on a torus. In addition to rings, (helical) filament lattice configurations are also considered and are found to be dynamically stable for all rotation frequencies and also energetically stable for sufficiently fast rotations. Numerical simulations are used to compare the Klein-Majda-Damodaran model with the full three-dimensional (3D) Gross-Pitaevskii equations as well as to confirm the analytical theory. Potential differences between the quasi-one-dimensional and the fully 3D description are also discussed.

nlin.PS↗

arXiv2016

Breathers in a locally resonant granular chain with precompression

We study a locally resonant granular material in the form of a precompressed Hertzian chain with linear internal resonators. Using an asymptotic reduction, we derive an effective nonlinear Schrödinger (NLS) modulation equation. This, in turn, leads us to provide analytical evidence, subsequently corroborated numerically, for the existence of two distinct types of discrete breathers related to acoustic or optical modes: (a) traveling bright breathers with a strain profile exponentially vanishing at infinity and (b) stationary and traveling dark breathers, exponentially localized, time-periodic states mounted on top of a non-vanishing background. The stability and bifurcation structure of numerically computed exact stationary dark breathers is also examined. Stationary bright breathers cannot be identified using the NLS equation, which is defocusing at the upper edges of the phonon bands and becomes linear at the lower edge of the optical band.

nlin.PS↗

arXiv2016

Wave mixing in coupled phononic crystals via a variable stiffness mechanism

We investigate wave mixing effects in a phononic crystal that couples the wave dynamics of two channels -- primary and control ones -- via a variable stiffness mechanism. We demonstrate analytically and numerically that the wave transmission in the primary channel can be manipulated by the control channel's signal. We show that the application of control waves allows the selection of a specific mode through the primary channel. We also demonstrate that the mixing of two wave modes is possible whereby a modulation effect is observed. A detailed study of the design parameters is also carried out to optimize the switching capabilities of the proposed system. Finally, we verify that the system can fulfill both switching and amplification functionalities, potentially enabling the realization of an acoustic transistor.

nlin.PS↗

arXiv2015

Nonlinear waves in a strongly nonlinear resonant granular chain

We explore a recently proposed locally resonant granular system bearing harmonic internal resonators in a chain of beads interacting via Hertzian elastic contacts. In this system, we propose the existence of two types of configurations: (a) small-amplitude periodic traveling waves and (b) dark-breather solutions, i.e., exponentially localized, time periodic states mounted on top of a non-vanishing background. We also identify conditions under which the system admits long-lived bright breather solutions. Our results are obtained by means of an asymptotic reduction to a suitably modified version of the so-called discrete p-Schr{ö}dinger (DpS) equation, which is established as controllably approximating the solutions of the original system for large but finite times (under suitable assumptions on the solution amplitude and the resonator mass). The findings are also corroborated by detailed numerical computations. A remarkable feature distinguishing our results from other settings where dark breathers are observed is the complete absence of precompression in the system, i.e., the absence of a linear spectral band.

nlin.PS↗

arXiv2015

Non-holonomic constraints and their impact on discretizations of Klein-Gordon lattice dynamical models

We explore a new type of discretizations of lattice dynamical models of the Klein-Gordon type relevant to the existence and long-term mobility of nonlinear waves. The discretization is based on non-holonomic constraints and is shown to retrieve the "proper" continuum limit of the model. Such discretizations are useful in exactly preserving a discrete analogue of the momentum. It is also shown that for generic initial data, the momentum and energy conservation laws cannot be achieved concurrently. Finally, direct numerical simulations illustrate that our models yield considerably higher mobility of strongly nonlinear solutions than the well-known "standard" discretizations, even in the case of highly discrete systems when the coupling between the adjacent nodes is weak. Thus, our approach is better suited for cases where an accurate description of mobility for nonlinear traveling waves is important.

nlin.PS↗

arXiv2014

Chaotic behavior of three interacting vortices in a confined Bose-Einstein condensate

Motivated by recent experimental works, we investigate a system of vortex dynamics in an atomic Bose-Einstein condensate (BEC), consisting of three vortices, two of which have the same charge. These vortices are modeled as a system of point particles which possesses a Hamiltonian structure. This tripole system constitutes a prototypical model of vortices in BECs exhibiting chaos. By using the angular momentum integral of motion we reduce the study of the system to the investigation of a two degree of freedom Hamiltonian model and acquire quantitative results about its chaotic behavior. Our investigation tool is the construction of scan maps by using the Smaller ALignment Index (SALI) as a chaos indicator. Applying this approach to a large number of initial conditions we manage to accurately and efficiently measure the extent of chaos in the model and its dependence on physically important parameters like the energy and the angular momentum of the system.

nlin.CD↗