KEVREKIDIS , Panayotis G.

In this Chapter, we touch upon the wide topic of discrete breather (DB) formation with a special emphasis on the prototypical system of interest, namely the 4 model. We start by introducing the model and discussing some of the application areas/motivational aspects of exploring time periodic, spatially localized structures, such as the DBs....

The interplay of nonlinearity and topology results in many novel and emergent properties across a number of physical systems such as chiral magnets, nematic liquid crystals, Bose-Einstein condensates, photonics, high energy physics, etc. It also results in a wide variety of topological defects such as solitons, vortices, skyrmions, merons,...

In this paper, we examine a nonlinear model with an impurity emulating a bend. We justify the geometric interpretation of the model and connect it with earlier work on models including geometric effects. We focus on both the bifurcation and stability analysis of the modes that emerge as a function of the strength of the bend angle, but we also...

We study the resonances in the sine-Gordon equation driven by an ac force using a linear perturbation theory. We show that resonances take place when the driving frequency d is equal to half of the phonon modes' frequencies as has been shown numerically in our earlier work [ N. R. Quintero, A. Sanchez, and F. G. Mertens, Phys. Rev. E 62 , R60 (...

The aim of this work is to propose a method for testing the integrability of a model partial differential (PDE) and/or differential difference equation (DDE), by examining it in a finite but large domain. For monoparametric families of PDE/DDE's, that are known to possess isolated integrable points, we find that very special features occur in...

We examine the spatial modeling of the outbreak of COVID-19 in two regions: the autonomous community of Andalusia in Spain and the mainland of Greece. We start with a zero-dimensional (0D; ordinary-differential-equation-level) compartmental epidemiological model consisting of Susceptible, Exposed, Asymptomatic, (symptomatically) Infected,...

We prove nonexistence of breathers (spatially localized and time-periodic oscillations) for a class of Fermi-Pasta-Ulam lattices representing an uncompressed chain of beads interacting via Hertz's contact forces. We then consider the setting in which an additional on-site potential is present, motivated by the Newton's cradle under the effect...

We study localized waves in chains of oscillators coupled by Hertzian interactions and trapped in local potentials. This problem is originally motivated by Newton's cradle, a mechanical system consisting of a chain of touching beads subject to gravity and attached to inelastic strings. We consider an unusual setting with local oscillations and...

In the present work, we introduce a discrete formulation of the nonlinear Dirac equation in the form of a discretization of the Gross – Neveu model. The motivation for this discrete model proposal is both computational (near the continuum limit) and theoretical (using the understanding of the anti-con- tinuum limit of vanishing coupling)....

In this work, we explore a prototypical example of a genuine continuum breather (i.e., not a standing wave) and the conditions under which it can persist in a -symmetric medium. As our model of interest, we will explore the sine-Gordon equation in the presence of a -symmetric perturbation. Our main finding is that the breather of the...

We demonstrate the possibility for explicit construction in a discrete Hamiltonian model of an exact solution of the form exp⁡(−|n|), i.e., a discrete peakon. These discrete analogs of the well-known, continuum peakons of the Camassa–Holm equation [R. Camassa, D.D. Holm, Phys. Rev. Lett. 71 (1993) 1661] are found in a model different from the...

In the present work we explore the concept of solitary wave billiards. That is, instead of a point particle, we examine a solitary wave in an enclosed region and examine its collision with the boundaries and the resulting trajectories in cases which for particle billiards are known to be integrable and for cases that are known to be chaotic. A...

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The aim of the present work is to examine the role of discreteness in the interaction of both co-winding and counter-winding vortices in the context of the nonlinear Schrodinger equation. Contrary to the well-known ¨ rotation of same charge vortices, and translation of opposite charge vortices, we find that strong discreteness is able to halt...

Following the highly restrictive measures adopted by many countries for combating the current pandemic, the number of individuals infected by SARS-CoV-2 and the associated number of deaths steadily decreased. This fact, together with the impossibility of maintaining the lockdown indefinitely, raises the crucial question of whether it is...

As an extension of the class of nonlinear PT-symmetric models, we propose a system of sine-Gordon equations, with the PT symmetry represented by balanced gain and loss in them. The equations are coupled by sine-field terms and first-order derivatives. The sinusoidal coupling stems from local interaction between adjacent particles in coupled...

Following the highly restrictive measures adopted by many countries for combating the current pandemic, the number of individuals infected by SARS-CoV-2 and the associated number of deaths steadily decreased. This fact, together with the impossibility of maintaining the lockdown indefinitely, raises the crucial question of whether it is...

We demonstrate that time-periodic modulation of the nonlinearity coefficient in the discrete nonlinear Schrödinger equation strongly facilitates creation of traveling solitons in the lattice. We predict this possibility in a semiqualitative form analytically, and test it in direct numerical simulations. Systematic computations reveal several...

By applying a staggered driving force in a prototypical discrete model with a quartic nonlinearity, we demonstrate the spontaneous formation and destruction of discrete breathers with a selected frequency due to thermal fluctuations. The phenomenon exhibits the striking features of stochastic resonance (SR): a nonmonotonic behavior as noise is...

We introduce a two-dimensional discrete nonlinear Schrödinger (DNLS) equation with self-attractive cubic nonlinearity in a rotating reference frame. The model applies to a Bose-Einstein condensate stirred by a rotating strong optical lattice, or light propagation in a twisted bundle of nonlinear fibers. Two types of localized states are...

Guided by a rigorous mathematical result, we have earlier introduced a numerical algorithm, which using as input the cumulative number of deaths caused by COVID-19, can estimate the effect of easing of the lockdown conditions. Applying this algorithm to data from Greece, we extend it to the case of two subpopulations, namely, those consisting...

The existence of breather-type solutions, i.e., solutions that are periodic in time and exponentially localized in space, is a very unusual feature for continuum, nonlinear wave-type equations. Following an earlier work establishing a theorem for the existence of such structures, we bring to bear a combination of analysis-inspired numerical...

In this work, we explore a massless nonlinear Dirac equation, i.e., a nonlinear Weyl equation. We study the dynamics of its pulse solutions and find that a localized one-hump initial condition splits into a localized two hump pulse, while an associated phase structure emerges in suitable components of the spinor field. For times larger than a...