arXiv subjects

Vassilis Koukouloyannis

Publications and source records attributed to Vassilis Koukouloyannis.

12 recordsLinked to original sources

arXiv2026

Machine Learning of Nonlinear Waves: Data-Driven Methods for Computer-Assisted Discovery of Equations, Symmetries, Conservation Laws, and Integrability

The purpose of this article is to provide a perspective---admittedly, a rather subjective one---of recent developments at the interface of machine learning (ML)/data-driven methods and nonlinear wave studies. We review some recent pillars of the rapidly evolving landscape of scientific ML, including deep learning, data-driven equation discovery, Koopman-based methods, and operator learning, among others. We then showcase these methods in applications ranging from learning lattice dynamical models and reduced order modeling of effective dynamics to discovery of conservation laws and potential identification of integrability of ordinary differential equations (DEs) and partial DE models. Our intention is to make clear that these ML methods are complementary to the preexisting powerful tools of the nonlinear waves community, and should be integrated into this toolkit to augment and enable mathematical discoveries and computational capabilities in the age of data.

nlin.PS↗

arXiv2026

On the Riemann problem for the Adlam-Allen model

In the present work, we revisit the Adlam-Allen (AA) model in order to investigate its numerically observed rarefaction and dispersive shock waves that arise in numerical simulations of the Riemann problem associated with the model. On the one hand, we perform a direct analysis of the rarefaction and dispersive shock waves of the AA model via examining its corresponding dispersionless system and leveraging the DSW-fitting method to obtain theoretical predictions on various edge features of the dispersive shock waves. On the other hand, we review the KdV reduction of the AA model and utilize the KdV dispersive shock wave to approximate that of the AA model. Relevant numerical comparisons demonstrate the good performance of not only the direct analysis on the AA dispersive shock wave, but also of the approximation via the KdV DSW. These methodologies provide a systematic toolbox for analyzing the outcome of Riemann problems in not only this fundamental setting of cold plasmas but also potentially in related plasma-physics problems.

nlin.PS↗

arXiv2025

NLS approximation and dark solitons for the Adlam-Allen model of cold collisionless plasmas

The present work extends earlier considerations on a quintessential model of cold, collisionless plasmas, namely the Adlam-Allen model. Previously, an analysis of homoclinic solutions around a non-vanishing background (associated with a saddle equilibrium) led to a Korteweg-de Vries reduction. Here, we consider a different equilibrium of the co-traveling frame model, namely a center, and expanding around it, by means of a multiple-scale methodology and suitable scalings, leads to an effective defocusing nonlinear Schr{ö}dinger equation. Leveraging the latter, we construct, for the first time to our knowledge, physically realistic dark soliton waveforms of the Adlam-Allen model. We subsequently test the numerical evolution of such coherent structures, identifying them as long-lived waveforms of the full model.

nlin.PS↗

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↗

arXiv2024

The discrete nonlinear Schrödinger equation with linear gain and nonlinear loss: the infinite lattice with nonzero boundary conditions and its finite dimensional approximations

The study of nonlinear Schrödinger-type equations with nonzero boundary conditions define challenging problems both for the continuous (partial differential equation) or the discrete (lattice) counterparts. They are associated with fascinating dynamics emerging by the ubiquitous phenomenon of modulation instability. In this work, we consider the discrete nonlinear Schrödinger equation with linear gain and nonlinear loss. For the infinite lattice supplemented with nonzero boundary conditions which describe solutions decaying on the top of a finite background, we give a rigorous proof that for the corresponding initial-boundary value problem, solutions exist for any initial condition, if and only if, the amplitude of the background has a precise value $A_*$ defined by the gain-loss parameters. We argue that this essential property of this infinite lattice can't be captured by finite lattice approximations of the problem. Commonly, such approximations are defined by lattices with periodic boundary conditions or as it is shown herein, by a modified problem closed with Dirichlet boundary conditions. For the finite dimensional dynamical system defined by the periodic lattice, the dynamics for all initial conditions are captured by a global attractor. Analytical arguments corroborated by numerical simulations show that the global attractor is trivial, defined by a plane wave of amplitude $A_*$. Thus, any instability effects or localized phenomena simulated by the finite system can be only transient prior the convergence to this trivial attractor. Aiming to simulate the dynamics of the infinite lattice as accurately as possible, we study the dynamics of localized initial conditions on the constant background and investigate the potential impact of the global asymptotic stability of the background with amplitude $A_*$ in the long-time evolution of the system.

math-ph↗

arXiv2022

Interactions of solitary waves in the Adlam-Allen model

We study the interactions of two or more solitary waves in the Adlam-Allen model describing the evolution of a (cold) plasma of positive and negative charges, in the presence of electric and transverse magnetic fields. In order to show that the interactions feature an exponentially repulsive nature, we elaborate two distinct approaches: (a) using energetic considerations and the Hamiltonian structure of the model; (b) using the so-called Manton's method. We compare these findings with results of direct simulations, and identify adjustments necessary to achieve a quantitative match between them. Additional connections are made, such as with solitons of the Korteweg--de Vries equation. New challenges are identified in connection to this model and its solitary waves.

nlin.PS↗

arXiv2020

Solitary and Periodic Waves in Collisionless Plasmas: The Adlam-Allen Model Revisited

We consider the Adlam-Allen (AA) system of partial differential equations which, arguably, is the first model that was introduced to describe solitary waves in the context of propagation of hydrodynamic disturbances in collisionless plasmas. Here, we identify the solitary waves of the model by implementing a dynamical systems approach. The latter suggests that the model also possesses periodic wave solutions --which reduce to the solitary wave in the limiting case of infinite period-- as well as rational solutions which are obtained herein. In addition, employing a long-wave approximation via a relevant multiscale expansion method, we establish the asymptotic reduction of the AA system to the Korteweg-de Vries equation. Such a reduction, is not only another justification for the above solitary wave dynamics, but also may offer additional insights for the emergence of other possible plasma waves. Direct numerical simulations are performed for the study of multiple solitary waves and their pairwise interactions. The stability of solitary waves is discussed in terms of potentially relevant criteria, while the robustness of spatially periodic wave solutions is touched upon by our numerical experiments.

nlin.PS↗

arXiv2017

A Hamiltonian Bifurcation Perspective on Two Interacting Vortex Pairs: From Symmetric to Asymmetric Leapfrogging, Period Doubling and Chaos

In this work we study the dynamical behavior of two interacting vortex pairs, each one of them consisting of two point vortices with opposite circulation in the 2d plane. The vortices are considered as effective particles and their interaction can be desribed in classical mechanics terms. We first construct a Poincaré section, for a typical value of the energy, in order to acquire a picture of the structure of the phase space of the system. We divide the phase space in different regions which correspond to qualitatively distinct motions and we demonstrate its different temporal evolution in the "real" vortex-space. Our main emphasis is on the leapfrogging periodic orbit, around which we identify a region that we term the "leapfrogging envelope" which involves mostly regular motions, such as higher order periodic and quasi-periodic solutions. We also identify the chaotic region of the phase plane surrounding the leapfrogging envelope as well as the so-called walkabout and braiding motions. Varying the energy as our control parameter, we construct a bifurcation tree of the main leapfrogging solution and its instabilities, as well as the instabilities of its daughter branches. We identify the symmetry-breaking instability of the leapfrogging solution (in line with earlier works), and also obtain the corresponding asymmetric branches of periodic solutions. We then characterize their own instabilities (including period doubling ones) and bifurcations in an effort to provide a more systematic perspective towards the types of motions available to this dynamical system.

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↗

arXiv2014

Dynamics of Three Non-co-rotating Vortices in Bose-Einstein Condensates

In this work we use standard Hamiltonian-system techniques in order to study the dynamics of three vortices with alternating charges in a confined Bose-Einstein condensate. In addition to being motivated by recent experiments, this system offers a natural vehicle for the exploration of the transition of the vortex dynamics from ordered to progressively chaotic behavior. In particular, it possesses two integrals of motion, the {\it energy} (which is expressed through the Hamiltonian $H$) and the {\it angular momentum} $L$ of the system. By using the integral of the angular momentum, we reduce the system to a two degree-of-freedom one with $L$ as a parameter and reveal the topology of the phase space through the method of Poincaré surfaces of section. We categorize the various motions that appear in the different regions of the sections and we study the major bifurcations that occur to the families of periodic motions of the system. Finally, we correspond the orbits on the surfaces of section to the real space motion of the vortices in the plane.

nlin.CD↗

arXiv2013

Non-Existence of phase-shift breathers in one-dimensional Klein-Gordon lattices with nearest-neighbor interactions

It is well known that one-dimensional Klein-Gordon lattices with nearest-neighbor interactions can support multibreathers with phase differences between the successive "central" oscillators $ϕ_i=0\ \mbox{or}\ π$ (standard configurations). In this paper we prove that in this kind of systems, the standard configurations are the only possible ones, so phase-shift breathers (configurations with $ϕ_i\neq0,\, π$) cannot be supported. This fact also determines the linear stability of the existing multibreathers.

nlin.PS↗

arXiv2009

On the stability of multibreathers in Klein-Gordon chains

In the present paper, a theorem, which determines the linear stability of multibreathers in Klein-Gordon chains, is proven. Specifically, it is shown that for soft nonlinearities, and positive inter-site coupling, only structures with adjacent sites excited out-of-phase may be stable, while only in-phase ones may be stable for negative coupling. The situation is reversed for hard nonlinearities. This theorem can be applied in $n$-site breathers, where $n$ is any finite number and provides an $\cal{O}(\sqrtε)$ estimation of the characteristic exponents of the solution. To complement the analysis, we perform numerical simulations and establish that the results are in excellent agreement with the theoretical predictions, at least for small values of the coupling constant $ε$.

nlin.PS↗