arXiv subjects

P. G. Kevrekidis

Publications and source records attributed to P. G. Kevrekidis.

At least 37 records · Page 2Linked to original sources

arXiv2025

Generic transverse stability of kink structures in atomic and optical nonlinear media with competing attractive and repulsive interactions

We demonstrate the existence and stability of one-dimensional (1D) topological kink configurations immersed in higher-dimensional bosonic gases and nonlinear optical setups. Our analysis pertains, in particular, to the two- and three-dimensional extended Gross-Pitaevskii models with quantum fluctuations describing droplet-bearing environments but also to the two-dimensional cubic-quintic nonlinear Schrödinger equation containing higher-order corrections to the nonlinear refractive index. Contrary to the generic dark soliton transverse instability, the kink structures are generically robust under the interplay of low-amplitude attractive and high-amplitude repulsive interactions. A quasi-1D effective potential picture dictates the existence of these defects, while their stability is obtained numerically and analytically through linearization analysis and direct dynamics in the presence of external fluctuations showcasing their unprecedented resilience. These generic (across different models) findings should be detectable in current cold atom and optics experiments, offering insights towards controlling topological excitations.

cond-mat.quant-gas↗

arXiv2025

Stability and dynamics of massive vortices in two-component Bose-Einstein condensates

The study of structures involving vortices in one component and bright solitary waves in another has a time-honored history in two-component atomic Bose-Einstein condensates. In the present work, we revisit this topic extending considerations well-past the near-integrable regime of nearly equal scattering lengths. Instead, we focus on stationary states and spectral stability of such structures for large values of the inter-component interaction coefficient. We find that the state can manifest dynamical instabilities for suitable parameter values. We also explore a phenomenological, yet quantitatively accurate upon suitable tuning, particle model which, in line also with earlier works, offers the potential of accurately following the associated stability and dynamical features. Finally, we probe the dynamics of the unstable vortex-bright structure, observing an unprecedented, to our knowledge, instability scenario in which the oscillatory instability leads to a patch of vorticity that harbors and eventually ejects multiple vortex-bright structures.

nlin.PS↗

arXiv2025

Kinks of fractional $ϕ^4$ models: existence, uniqueness, monotonicity, stability, and sharp asymptotics

In the present work we construct kink solutions for different (parabolic and wave) variants of the fractional $ϕ^4$ model, in both the sub-Laplacian and super-Laplacian setting. We establish existence and monotonicity results (for the sub - Laplacian case), along with sharp asymptotics which are corroborated through numerical computations. Importantly, in the sub-Laplacian regime, we provide the explicit and numerically verifiable spectral condition, which guarantees uniqueness for odd kinks. We check numerically the relevant condition to confirm the uniqueness of such solutions. In addition, we show asymptotic stability for the stationary kinks in the parabolic setting and also, the spectral stability for the traveling kinks in the corresponding wave equation.

math.AP↗

arXiv2025

Exponential Asymptotics for Translational Modes in the Discrete Nonlinear Schr{ö}dinger Model

In the present work, we revisit the topic of translational eigenmodes in discrete models. We focus on the prototypical example of the discrete nonlinear Schr{ö}dinger equation, although the methodology presented is quite general. We tackle the relevant discrete system based on exponential asymptotics and start by deducing the well-known (and fairly generic) feature of the existence of two types of fixed points, namely site-centered and inter-site-centered. Then, turning to the stability problem, we not only retrieve the exponential scaling (as $ e^{-π^2/(2 \varepsilon)} $, where $ \varepsilon $ denotes the spacing between nodes) and its corresponding prefactor power-law (as $ \varepsilon^{-5/2} $), both of which had been previously obtained, but we also obtain a highly accurate leading-order prefactor and, importantly, the next-order correction, for the first time, to the best of our knowledge. This methodology paves the way for such an analysis in a wide range of lattice nonlinear dynamical equation models.

nlin.PS↗

arXiv2025

On the Fractional Dynamics of Kinks in sine-Gordon Models

In the present work we explore the dynamics of single kinks, kink-anti-kink pairs and bound states in the prototypical fractional Klein-Gordon example of the sine-Gordon equation. In particular, we modify the order $β$ of the temporal derivative to that of a Caputo fractional type and find that, for $1<β<2$, this imposes a dissipative dynamical behavior on the coherent structures. We also examine the variation of a fractional Riesz order $α$ on the spatial derivative. Here, depending on whether this order is below or above the harmonic value $α=2$, we find, respectively, monotonically attracting kinks, or non-monotonic and potentially attracting or repelling kinks, with a saddle equilibrium separating the two. Finally, we also explore the interplay of the two derivatives, when both Caputo temporal and Riesz spatial derivatives are involved.

nlin.PS↗

arXiv2024

Fractional Solitons: A Homotopic Continuation from the Biharmonic to the Harmonic $ϕ^4$ Model

In the present work we explore the path from a harmonic to a biharmonic PDE of Klein-Gordon type from a continuation/bifurcation perspective. More specifically, we make use of the Riesz fractional derivative as a tool that allows us to interpolate between these two limits. We illustrate, in particular, how the coherent kink structures existing in these models transition from the exponential tail of the harmonic operator case, via the power-law tails of intermediate fractional orders, to the oscillatory exponential tails of the biharmonic model. Importantly, we do not limit our considerations to the single kink case, but extend to the kink-antikink pair, finding an intriguing cascade of saddle-center bifurcations happening exponentially close to the biharmonic limit. Our analysis clearly explains the transition between the infinitely many stationary soliton pairs of the biharmonic case and the absence of even a single such pair in the harmonic limit. The stability of the different configurations obtained and the associated dynamics and phase portraits are also analyzed.

nlin.PS↗

arXiv2024

Stability and dynamics of nonlinear excitations in a two-dimensional droplet-bearing environment

We unravel stationary states in the form of dark soliton stripes, bubbles, and kinks embedded in a two-dimensional droplet-bearing setting emulated by an extended Gross-Pitaevskii approach. The existence of these configurations is corroborated through an effectively reduced potential picture demonstrating their concrete parametric regions of existence. The excitation spectra of such configurations are analyzed within the Bogoliubov-de-Gennes framework exposing the destabilization of dark soliton stripes and bubbles, while confirming the stability of droplets, and importantly unveiling spectral stability of the kink against transverse excitations. Additionally, a variational approach is constructed providing access to the transverse stability analysis of the dark soliton stripe for arbitrary chemical potentials and widths of the structure. This is subsequently compared with the stability analysis outcome demonstrating very good agreement at small wavenumbers. Dynamical destabilization of dark soliton stripes via the snake instability is showcased, while bubbles are found to feature both a splitting into a gray soliton pair and a transverse instability thereof. These results shed light on unexplored stability and instability properties of nonlinear excitations in environments featuring a competition of mean-field repulsion and beyond-mean-field attraction that can be probed by state-of-the-art experiments.

nlin.PS↗

arXiv2024

Dynamics of Nonlinear Lattices

In this topical review we explore the dynamics of nonlinear lattices with a particular focus to Fermi-Pasta-Ulam-Tsingou type models that arise in the study of elastic media and, more specifically, granular crystals. We first revisit the workhorse of such lattices, namely traveling waves, both from a continuum, but also from a genuinely discrete perspective, both without and with a linear force component (induced by the so-called precompression). We then extend considerations to time-periodic states, examining dark breather structures in homogeneous crystals, as well as bright breathers in diatomic lattices. The last pattern that we consider extensively is the dispersive shock wave arising in the context of suitable Riemann (step) initial data. We show how the use of continuum (KdV) and discrete (Toda) integrable approximations can be used to get a first quantitative handle of the relevant waveforms. In all cases, theoretical analysis is accompanied by numerical computations and, where possible, by a recap and illustration of prototypical experimental results. We close the chapter by offering a number of ongoing and potential future directions and associated open problems in the field.

nlin.PS↗

arXiv2024

Stability of smooth solitary waves under intensity--dependent dispersion

The cubic nonlinear Schrodinger equation (NLS) in one dimension is considered in the presence of an intensity-dependent dispersion term. We study bright solitary waves with smooth profiles which extend from the limit where the dependence of the dispersion coefficient on the wave intensity is negligible to the limit where the solitary wave becomes singular due to vanishing dispersion coefficient. We analyze and numerically explore the stability for such smooth solitary waves, showing with the help of numerical approximations that the family of solitary waves becomes unstable in the intermediate region between the two limits, while being stable in both limits. This bistability, that has also been observed in other NLS equations with the generalized nonlinearity, brings about interesting dynamical transitions from one stable branch to another stable branch, that are explored in direct numerical simulations of the NLS equation with the intensity-dependent dispersion term.

nlin.PS↗

arXiv2024

Global Bifurcations in a Damped-Driven Diatomic Granular Crystal

We revisit here the dynamics of an engineered dimer granular crystal under an external periodic drive in the presence of dissipation. Earlier findings included a saddle-node bifurcation, whose terminal point initiated the observation of chaos; the system was found to exhibit bistability and potential quasiperiodicity. We now complement these findings by the identification of unstable manifolds of saddle periodic solutions (saddle points of the stroboscopic map) within the system dynamics. We unravel how homoclinic tangles of these manifolds lead to the appearance of a chaotic attractor, upon the apparent period-doubling bifurcations that destroy invariant tori associated with quasiperiodicity. These findings complement the earlier ones, offering more concrete insights into the emergence of chaos within this high-dimensional, experimentally accessible system.

nlin.PS↗

arXiv2024

Standing and Traveling Waves in a Nonlinearly Dispersive Lattice Model

In the work of Colliander et al. (2010), a minimal lattice model was constructed describing the transfer of energy to high frequencies in the defocusing nonlinear Schrödinger equation. In the present work, we present a systematic study of the coherent structures, both standing and traveling, that arise in the context of this model. We find that the nonlinearly dispersive nature of the model is responsible for standing waves in the form of discrete compactons. On the other hand, analysis of the dynamical features of the simplest nontrivial variant of the model, namely the dimer case, yields both solutions where the intensity is trapped in a single site and solutions where the intensity moves between the two sites, which suggests the possibility of moving excitations in larger lattices. Such excitations are also suggested by the dynamical evolution associated with modulational instability. Our numerical computations confirm this expectation, and we systematically construct such traveling states as exact solutions in lattices of varying size, as well as explore their stability. A remarkable feature of these traveling lattice waves is that they are of "antidark" type, i.e., they are mounted on top of a non-vanishing background. These studies shed light on the existence, stability and dynamics of such standing and traveling states in $1+1$ dimensions, and pave the way for exploration of corresponding configurations in higher dimensions.

nlin.PS↗

arXiv2024

Practical identifiability and parameter estimation of compartmental epidemiological models

Practical parameter identifiability in ODE-based epidemiological models is a known issue, yet one that merits further study. It is essentially ubiquitous due to noise and errors in real data. In this study, to avoid uncertainty stemming from data of unknown quality, simulated data with added noise are used to investigate practical identifiability in two distinct epidemiological models. Particular emphasis is placed on the role of initial conditions, which are assumed unknown, except those that are directly measured. Instead of just focusing on one method of estimation, we use and compare results from various broadly used methods, including maximum likelihood and Markov Chain Monte Carlo (MCMC) estimation. Among other findings, our analysis revealed that the MCMC estimator is overall more robust than the point estimators considered. Its estimates and predictions are improved when the initial conditions of certain compartments are fixed so that the model becomes globally identifiable. For the point estimators, whether fixing or fitting the that are not directly measured improves parameter estimates is model-dependent. Specifically, in the standard SEIR model, fixing the initial condition for the susceptible population S(0) improved parameter estimates, while this was not true when fixing the initial condition of the asymptomatic population in a more involved model. Our study corroborates the change in quality of parameter estimates upon usage of pre-peak or post-peak time-series under consideration. Finally, our examples suggest that in the presence of significantly noisy data, the value of structural identifiability is moot.

stat.ME↗

arXiv2024

Observation of dense collisional soliton complexes in a two-component Bose-Einstein condensate

Solitons are nonlinear solitary waves which maintain their shape over time and through collisions, occurring in a variety of nonlinear media from plasmas to optics. We present an experimental and theoretical study of hydrodynamic phenomena in a two-component atomic Bose-Einstein condensate where a soliton array emerges from the imprinting of a periodic spin pattern by a microwave pulse-based winding technique. We observe the ensuing dynamics which include shape deformations, the emergence of dark-antidark solitons, apparent spatial frequency tripling, and decay and revival of contrast related to soliton collisions. For the densest arrays, we obtain soliton complexes where solitons undergo continued collisions for long evolution times providing an avenue towards the investigation of soliton gases in atomic condensates.

cond-mat.quant-gas↗

arXiv2024

Identification of moment equations via data-driven approaches in nonlinear schrodinger models

The moment quantities associated with the nonlinear Schrodinger equation offer important insights towards the evolution dynamics of such dispersive wave partial differential equation (PDE) models. The effective dynamics of the moment quantities is amenable to both analytical and numerical treatments. In this paper we present a data-driven approach associated with the Sparse Identification of Nonlinear Dynamics (SINDy) to numerically capture the evolution behaviors of such moment quantities. Our method is applied first to some well-known closed systems of ordinary differential equations (ODEs) which describe the evolution dynamics of relevant moment quantities. Our examples are, progressively, of increasing complexity and our findings explore different choices within the SINDy library. We also consider the potential discovery of coordinate transformations that lead to moment system closure. Finally, we extend considerations to settings where a closed analytical form of the moment dynamics is not available.

nlin.PS↗

arXiv2024

Hydrodynamics of a Discrete Conservation Law

The Riemann problem for the discrete conservation law $2 \dot{u}_n + u^2_{n+1} - u^2_{n-1} = 0$ is classified using Whitham modulation theory, a quasi-continuum approximation, and numerical simulations. A surprisingly elaborate set of solutions to this simple discrete regularization of the inviscid Burgers' equation is obtained. In addition to discrete analogues of well-known dispersive hydrodynamic solutions -- rarefaction waves (RWs) and dispersive shock waves (DSWs) -- additional unsteady solution families and finite time blow-up are observed. Two solution types exhibit no known conservative continuum correlates: (i) a counterpropagating DSW and RW solution separated by a symmetric, stationary shock and (ii) an unsteady shock emitting two counter-propagating periodic wavetrains with the same frequency connected to a partial DSW or a RW. Another class of solutions called traveling DSWs, (iii), consists of a partial DSW connected to a traveling wave comprised of a periodic wavetrain with a rapid transition to a constant. Portions of solutions (ii) and (iii) are interpreted as shock solutions of the Whitham modulation equations.

nlin.PS↗

arXiv2024

Compact Localized States in Electric Circuit Flatband Lattices

We generate compact localized states in an electrical diamond lattice, comprised of only capacitors and inductors, via local driving near its flatband frequency. We compare experimental results to numerical simulations and find very good agreement. We also examine the stub lattice, which features a flatband of a different class where neighboring compact localized states share lattice sites. We find that local driving, while exciting the lattice at that flatband frequency, is unable to isolate a single compact localized state due to their non-orthogonality. Finally, we introduce lattice nonlinearity and showcase the realization of nonlinear compact localized states in the diamond lattice. Our findings pave the way of applying flatband physics to complex electric circuit dynamics.

cond-mat.mes-hall↗

arXiv2024

PainleveBacklundCheck: A Sympy-powered Kivy app for the Painlevé property of nonlinear dispersive PDEs and auto-Bäcklund transformations

In the present work we revisit the Painlevé property for partial differential equations. We consider the PDE variant of the relevant algorithm on the basis of the fundamental work of Weiss, Tabor and Carnevale and explore a number of relevant examples. Subsequently, we present an implementation of the relevant algorithm in an open-source platform in Python and discuss the details of a Sympy-powered Kivy app that enables checking of the property and the derivation of associated auto-B{ä}ck{u}nd transform when the property is present. Examples of the relevant code and its implementation are also provided, as well as details of its open access for interested potential users.

nlin.SI↗

arXiv2024

On the temporal tweezing of cavity solitons

Motivated by the work of J.K.~Jang et al., Nat.~Commun.~{\bf 6}, 7370 (2015), where the authors experimentally tweeze cavity solitons in a passive loop of optical fiber, we study the amenability to tweezing of cavity solitons as the properties of a localized tweezer are varied. The system is modeled by the Lugiato-Lefever equation, a variant of the complex Ginzburg-Landau equation. We produce an effective, localized, trapping tweezer potential by assuming a Gaussian phase-modulation of the holding beam. The potential for tweezing is then assessed as the total (temporal) displacement and speed of the tweezer are varied, and corresponding phase diagrams are presented. As the relative speed of the tweezer is increased we find two possible dynamical scenarios: successful tweezing and release of the cavity soliton. We also deploy a non-conservative variational approximation (NCVA) based on a Lagrangian description which reduces the original dissipative partial differential equation to a set of coupled ordinary differential equations for the cavity soliton parameters. We illustrate the ability of the NCVA to accurately predict the separatrix between successful and failed tweezing. This showcases the versatility of the NCVA to provide a low-dimensional description of the experimental realization of the temporal tweezing.

nlin.PS↗