arXiv subjects

E. G. Charalampidis

Publications and source records attributed to E. G. Charalampidis.

At least 19 recordsLinked to original sources

arXiv2026

Blowup of Multi-Peaked Waveforms in the Two-Dimensional Nonlinear Schroedinger Model

In the present work, we explore the self-focusing and resulting collapse of two-dimensional waveforms involving multiple pulses in a nonlinear Schroedinger equation with a general power-law nonlinearity. We find that a wide range of multi-peaked states bifurcate from the critical threshold of the cubic nonlinearity, thus representing ``bifurcations from infinity'', i.e., the relevant pulses start at infinite distance in the critical limit and draw nearer, as the nonlinearity exponent increases past that threshold. We identify the resulting ``interacting particle system'' as amounting to a force balance between the exponentially interacting tails (modulated by a suitable power law) and a linear phase-induced force. The equilibria emerging from this force balance are found to be in excellent agreement with the identified steady states of the partial differential equation. The spectral stability of multi-peaked configurations is analyzed, leading to the conclusion that all the relevant states are less stable than the single-peak collapsing solution whose stability was analyzed earlier. Indeed, we reveal both symmetry-breaking, as well as motion-inducing destabilizing dynamics, with the former ones among them being dominant and ultimately leading to a single dominant collapse spot. Moreover, we characterize systematically both the real and imaginary eigenvalues of multi-peaked configurations, partitioning them in groups of different sizes, described by powers of the solution's blowup rate G.

nlin.PS↗

arXiv2026

The continuous spectrum of bound states in expulsive potentials

On the contrary to the common intuition, which suggests that a steep expulsive potential makes quantum states widely delocalized, we demonstrate that one- and two-dimensional (1D and 2D) Schroedinger equations, which include expulsive potentials that are steeper than the quadratic ones, give rise to normalizable eigenstates, which may be considered as a manifestation of effective self-trapping in the linear system. These states constitute full continuous spectra in both the 1D and 2D cases. In 1D, they are spatially even and odd eigenstates. The 2D states may carry any value of the vorticity (alias magnetic quantum number). Asymptotic expressions for wave functions of the 1D and 2D eigenstates, valid far from the center, are derived analytically, demonstrating excellent agreement with full numerical solutions. Special exact solutions for vortex states are obtained in the 2D case. These findings suggest an extension of the concept of bound states in the continuum, in quantum mechanics and paraxial photonics. Gross-Pitaevskii equations are briefly considered as the nonlinear extension of the 1D and 2D settings. In 1D, the cubic nonlinearity slightly deforms the eigenstates, maintaining their stability

quant-ph↗

arXiv2026

Stability and mixed phases of three-component droplets in one dimension

We explore the ground state properties and excitation spectra of one-dimensional three-component bosonic mixtures accommodating a droplet in two of the species and a third minority component. Relying on the suitable Lee-Huang-Yang framework, we reveal a plethora of distinct self-bound droplet phases and their phase transitions through variations of either the particle number of the majority components or the intercomponent coupling. The ensuing phases demonstrate that the minority component is being un-trapped, partially trapped, or fully trapped by the majority droplet species. These states are characterized by their binding energies captured by the chemical potentials and their low-amplitude excitation spectrum, including mode crossings at the particle-emission threshold. We further derive effective reduced models which are valid in the high-imbalance limit, and accurately reproduce the numerically computed ground states, while providing analytical insights into the role of quantum fluctuations. Our results map out the stability and structure of mixed droplet phases offering guidance into forthcoming experimental and theoretical studies of multicomponent quantum droplets.

cond-mat.quant-gas↗

arXiv2023

Self-similar blow-up solutions in the generalized Korteweg-de Vries equation: Spectral analysis, normal form and asymptotics

In the present work we revisit the problem of the generalized Korteweg-de Vries equation parametrically, as a function of the relevant nonlinearity exponent, to examine the emergence of blow-up solutions, as traveling waveforms lose their stability past a critical point of the relevant parameter $p$, here at $p=5$. We provide a {\it normal form} of the associated collapse dynamics and illustrate how this captures the collapsing branch bifurcating from the unstable traveling branch. We also systematically characterize the linearization spectrum of not only the traveling states, but importantly of the emergent collapsing waveforms in the so-called co-exploding frame where these waveforms are identified as stationary states. This spectrum, in addition to two positive real eigenvalues which are shown to be associated with the symmetries of translation and scaling invariance of the original (non-exploding) frame features complex patterns of negative eigenvalues that we also fully characterize. We show that the phenomenology of the latter is significantly affected by the boundary conditions and is far more complicated than in the corresponding symmetric Laplacian case of the nonlinear Schr{ö}dinger problem that has recently been explored. In addition, we explore the dynamics of the unstable solitary waves for $p>5$ in the co-exploding frame.

nlin.PS↗

arXiv2023

Existence, stability and spatio-temporal dynamics of time-quasiperiodic solutions on a finite background in discrete nonlinear Schrödinger models

In the present work we explore the potential of models of the discrete nonlinear Schrödinger (DNLS) type to support spatially localized and temporally quasiperiodic solutions on top of a finite background. Such solutions are rigorously shown to exist in the vicinity of the anti-continuum, vanishing coupling limit of the model. We then use numerical continuation to illustrate their persistence for finite coupling, as well as to explore their spectral stability. We obtain an intricate bifurcation diagram showing a progression of such solutions from simpler ones bearing single- and two-site excitations to more complex, multi-site ones with a direct connection of the branches of the self-focusing and self-defocusing nonlinear regime. We further probe the variation of the solutions obtained towards the limit of vanishing frequency for both signs of the nonlinearity. Our analysis is complemented by exploring the dynamics of the solutions via direct numerical simulations.

nlin.PS↗

arXiv2022

A Spectral Analysis of the Nonlinear Schroedinger Equation in the Co-Exploding Frame

The nonlinear Schroedinger model is a prototypical dispersive wave equation that features finite time blowup, either for supercritical exponents (for fixed dimension) or for supercritical dimensions (for fixed nonlinearity exponent). Upon identifying the self-similar solutions in the so-called "co-exploding frame", a dynamical systems analysis of their stability is natural, yet is complicated by the mixed Hamiltonian-dissipative character of the relevant frame. In the present work, we study the spectral picture of the relevant linearized problem. We examine the point spectrum of 3 eigenvalue pairs associated with translation, $U(1)$ and conformal invariances, as well as the continuous spectrum. We find that two eigenvalues become positive, yet are attributed to symmetries and are thus not associated with instabilities. In addition to a vanishing eigenvalue, 3 more are found to be negative and real, while the continuous spectrum is nearly vertical and on the left-half (spectral) plane. Finally, the subtle effects of the boundaries are also assessed and their role in the observed weak eigenvalue oscillations is clarified.

nlin.PS↗

arXiv2021

Existence, Stability and Dynamics of Monopole and Alice Ring Solutions in Anti-Ferromagnetic Spinor Condensates

In this work we study the existence, stability, and dynamics of select topological point and line defects in anti-ferromagnetic, polar phase, $F=1$ $^{23}$Na spinor condensates. Specifically, we leverage fixed-point and numerical continuation techniques in three spatial dimensions to identify solution families of monopole and Alice rings as the chemical potential (number of atoms) and trapping strengths are varied within intervals of realizable experimental parameters. We are able to follow the monopole from the linear limit of small atom number all the way to the Thomas-Fermi regime of large atom number. Additionally, and importantly, our studies reveal the existence of {\em two} Alice ring solution branches, corresponding to, relatively, smaller and larger ring radii, that bifurcate from each other in a saddle-center bifurcation as the chemical potential is varied. We find that the monopole solution is always dynamically unstable in the regimes considered. In contrast, we find that the larger Alice ring is indeed stable close to the bifurcation point until it destabilizes from an oscillatory instability bubble for a larger value of the chemical potential. We also report on the possibility of dramatically reducing, yet not completely eliminating, the instability rates for the smaller Alice ring by varying the trapping strengths. The dynamical evolution of the different unstable waveforms is also probed via direct numerical simulations.

cond-mat.quant-gas↗

arXiv2020

Deflation-based Identification of Nonlinear Excitations of the 3D Gross--Pitaevskii equation

We present previously unknown solutions to the 3D Gross--Pitaevskii equation describing atomic Bose-Einstein condensates. This model supports elaborate patterns, including excited states bearing vorticity. The discovered coherent structures exhibit striking topological features, involving combinations of vortex rings and multiple, possibly bent vortex lines. Although unstable, many of them persist for long times in dynamical simulations. These solutions were identified by a state-of-the-art numerical technique called deflation, which is expected to be applicable to many problems from other areas of physics.

nlin.PS↗

arXiv2020

Nonlinear Localized Modes in Two-Dimensional Hexagonally-Packed Magnetic Lattices

We conduct an extensive study of nonlinear localized modes (NLMs), which are temporally periodic and spatially localized structures, in a two-dimensional array of repelling magnets. In our experiments, we arrange a lattice in a hexagonal configuration with a light-mass defect, and we harmonically drive the center of the chain with a tunable excitation frequency, amplitude, and angle. We use a damped, driven variant of a vector Fermi- Pasta-Ulam-Tsingou lattice to model our experimental setup. Despite the idealized nature of the model, we obtain good qualitative agreement between theory and experiments for a variety of dynamical behaviors. We find that the spatial decay is direction-dependent and that drive amplitudes along fundamental displacement axes lead to nonlinear resonant peaks in frequency continuations that are similar to those that occur in one-dimensional damped, driven lattices. However, driving along other directions leads to the creation of asymmetric NLMs that bifurcate from the main solution branch, which consists of symmetric NLMs. When we vary the drive amplitude, we observe such behavior both in our experiments and in our simulations. We also demonstrate that solutions that appear to be time-quasi-periodic bifurcate from the branch of symmetric time-periodic NLMs.

nlin.PS↗

arXiv2020

Kuznetsov-Ma breather-like solutions in the Salerno model

The Salerno model is a discrete variant of the celebrated nonlinear Schrödinger (NLS) equation interpolating between the discrete NLS (DNLS) equation and completely integrable Ablowitz-Ladik (AL) model by appropriately tuning the relevant homotopy parameter. Although the AL model possesses an explicit time-periodic solution known as the Kuznetsov-Ma (KM) breather, the existence of time-periodic solutions away from the integrable limit has not been studied as of yet. It is thus the purpose of this work to shed light on the existence and stability of time-periodic solutions of the Salerno model. In particular, we vary the homotopy parameter of the model by employing a pseudo-arclength continuation algorithm where time-periodic solutions are identified via fixed-point iterations. We show that the solutions transform into time-periodic patterns featuring small, yet non-decaying far-field oscillations. Remarkably, our numerical results support the existence of previously unknown time-periodic solutions {\it even} at the integrable case whose stability is explored by using Floquet theory. A continuation of these patterns towards the DNLS limit is also discussed.

nlin.PS↗

arXiv2018

Lattices With Internal Resonator Defects

We consider a variety of settings involving chains with one or more defects stemming from the introduction of nodes bearing internal resonators. Motivated by experimental results in woodpile elastic lattices with one or two defects, we consider a variety of different theoretical scenarios. These include multi-defect chains and their ability to transmit, reflect, and especially trap energy; they also include settings with linear vs. nonlinear defects of variable interaction exponent. Moreover, they involve defects which are spatially separated and either statically, or more effectively dynamically, enable the confinement of energy between the separated defects. Wherever possible, comparisons of the experiments with numerical simulations, as well as with theoretical intuition are also offered, to provide a justification for the observed findings.

nlin.PS↗

arXiv2018

Phononic Rogue Waves

We present a theoretical study of extreme events occurring in phononic lattices. In particular, we focus on the formation of rogue or freak waves, which are characterized by their localization in both spatial and temporal domains. We consider two examples. The first one is the prototypical nonlinear mass-spring system in the form of a homogeneous Fermi-Pasta-Ulam-Tsingou (FPUT) lattice with a polynomial potential. By deriving an approximation based on the nonlinear Schroedinger (NLS) equation, we are able to initialize the FPUT model using a suitably transformed Peregrine soliton solution of the NLS, obtaining dynamics that resembles a rogue wave on the FPUT lattice. We also show that Gaussian initial data can lead to dynamics featuring rogue wave for sufficiently wide Gaussians. The second example is a diatomic granular crystal exhibiting rogue wave like dynamics, which we also obtain through an NLS reduction and numerical simulations. The granular crystal (a chain of particles that interact elastically) is a widely studied system that lends itself to experimental studies. This study serves to illustrate the potential of such dynamical lattices towards the experimental observation of acoustic rogue waves.

nlin.PS↗

arXiv2017

Extreme events in near integrable lattices

In the present work, we examine the potential robustness of extreme wave events associated with large amplitude fluctuations of the Peregrine soliton type, upon departure from the integrable analogue of the discrete nonlinear Schrödinger (DNLS) equation, namely the Ablowitz-Ladik (AL) model. Our model of choice will be the so-called Salerno model, which interpolates between the AL and the DNLS models. We find that rogue wave events essentially are drastically distorted even for very slight perturbations of the homotopic parameter connecting the two models off of the integrable limit. Our results suggest that the Peregrine soliton structure is a rather sensitive feature of the integrable limit, which may not persist under "generic" perturbations of the limiting integrable case.

nlin.PS↗

arXiv2017

Discrete BPS Skyrmions

A discrete analogue of the extended Bogomolny-Prasad-Sommerfeld (BPS) Skyrme model that admits time-dependent solutions is presented. Using the spacing h of adjacent lattice nodes as a parameter, we identify the spatial profile of the solution and the continuation of the relevant branch of solutions over the lattice spacing for different values of the potential (free) parameter α. In particular, we explore the dynamics and stability of the obtained solutions, finding that, while they generally seem to be prone to instabilities, for suitable values of the lattice spacing and for sufficiently large value of α, they may be long-lived in direct numerical simulations.

hep-th↗

arXiv2017

Revisiting Diffusion: Self-similar Solutions and the $t^{-1/2}$ Decay in Initial and Initial-Boundary Value Problems

The diffusion equation is a universal and standard textbook model for partial differential equations (PDEs). In this work, we revisit its solutions, seeking, in particular, self-similar profiles. This problem connects to the classical theory of special functions and, more specifically, to the Hermite as well as the Kummer hypergeometric functions. Reconstructing the solution of the original diffusion model from novel self-similar solutions of the associated self-similar PDE, we infer that the $t^{-1/2}$ decay law of the diffusion amplitude is {\it not necessary}. In particular, it is possible to engineer setups of {\it both} the Cauchy problem and the initial-boundary value problem in which the solution decays at a {\it different rate}. Nevertheless, we observe that the $t^{-1/2}$ rate corresponds to the dominant decay mode among integrable initial data, i.e., ones corresponding to finite mass. Hence, unless the projection to such a mode is eliminated, generically this decay will be the slowest one observed. In initial-boundary value problems, an additional issue that arises is whether the boundary data are \textit{consonant} with the initial data; namely, whether the boundary data agree at all times with the solution of the Cauchy problem associated with the same initial data, when this solution is evaluated at the boundary of the domain. In that case, the power law dictated by the solution of the Cauchy problem will be selected. On the other hand, in the non-consonant cases a decomposition of the problem into a self-similar and a non-self-similar one is seen to be beneficial in obtaining a systematic understanding of the resulting solution.

math.AP↗

arXiv2017

Computing stationary solutions of the two-dimensional Gross-Pitaevskii equation with Deflated continuation

In this work we employ a recently proposed bifurcation analysis technique, the deflated continuation algorithm, to compute steady-state solitary waveforms in a one-component, two dimensional nonlinear Schrödinger equation with a parabolic trap and repulsive interactions. Despite the fact that this system has been studied extensively, we discover a wide variety of previously unknown branches of solutions. We analyze the stability of the newly discovered branches and discuss the bifurcations that relate them to known solutions both in the near linear (Cartesian, as well as polar) and in the highly nonlinear regimes. While deflated continuation is not guaranteed to compute the full bifurcation diagram, this analysis is a potent demonstration that the algorithm can discover new nonlinear states and provide insights into the energy landscape of complex high-dimensional Hamiltonian dynamical systems.

nlin.PS↗

arXiv2016

Rogue Waves in Ultracold Bosonic Seas

In this work, we numerically consider the initial value problem for nonlinear Schrödinger (NLS) type models arising in the physics of ultracold boson gases, with generic Gaussian wavepacket initial data. The corresponding Gaussian's width and, wherever relevant also its amplitude, serve as control parameters. First we explore the one-dimensional, standard NLS equation with general power law nonlinearity, in which large amplitude excitations reminiscent of Peregrine solitons or regular solitons appear to form, as the width of the relevant Gaussian is varied. Furthermore, the variation of the nonlinearity exponent aims at a first glimpse of the interplay between rogue or soliton formation and collapse features. The robustness of the main features to noise in the initial data is also confirmed. To better connect our study with the physics of atomic condensates, and explore the role of dimensionality effects, we also consider the nonpolynomial Schrödinger equation (NPSE), as well as the full three-dimensional NLS equation, and examine the degree to which relevant considerations generalize.

nlin.PS↗

arXiv2016

Vortex-soliton complexes in coupled nonlinear Schrödinger equations with unequal dispersion coefficients

We consider a two-component, two-dimensional nonlinear Schrödinger system with unequal dispersion coefficients and self-defocusing nonlinearities, chiefly with equal strengths of the self- and cross-interactions. In this setting, a natural waveform with a nonvanishing background in one component is a vortex, which induces an effective potential well in the second component, via the nonlinear coupling of the two components. We show that the potential well may support not only the fundamental bound state, but also multi-ring excited radial state complexes for suitable ranges of values of the dispersion coefficient of the second component. We systematically explore the existence, stability, and nonlinear dynamics of these states. The complexes involving the excited radial states are weakly unstable, with a growth rate depending on the dispersion of the second component. Their evolution leads to transformation of the multi-ring complexes into stable VB solitons ones with the fundamental state in the second component. The excited states may be stabilized by a harmonic-oscillator trapping potential, as well as by unequal strengths of the self- and cross-repulsive nonlinearities.

nlin.PS↗