Vassilis M. Rothos

Travelling waves in Hamiltonian systems on 2D lattices with nearest neighbour interactions

We study travelling waves on a two-dimensional lattice with linear and nonlinear coupling between nearest particles and a periodic nonlinear substrate potential. Such a discrete system can model molecules adsorbed on a substrate crystal surface. We show the existence of both uniform sliding states and periodic travelling waves as well in a two-dimensional sine-Gordon lattice equation using topological and variational methods.

Travelling waves of discrete nonlinear Schrödinger equations with nonlocal interactions

Existence and bifurcation results are derived for quasi periodic travelling waves of discrete nonlinear Schrödinger equations with nonlocal interactions and with polynomial-type potentials. Variational tools are used. Several concrete nonlocal interactions are studied as well.

Multibreathers in Klein–Gordon chains with interactions beyond nearest neighbors

Abstract We study the existence and stability of multibreathers in Klein–Gordon chains with interactions that are not restricted to nearest neighbors. We provide a general framework where such long range effects can be taken into consideration for arbitrarily varying (as a function of the node distance) linear couplings between arbitrary sets of neighbors in the chain. By examining special case examples such as three-site breathers with next-nearest-neighbors, we find crucial modifications to the nearest-neighbor picture of one-dimensional oscillators being excited either in- or anti-phase. Configurations with nontrivial phase profiles emerge from or collide with the ones with standard ( 0 or π ) phase difference profiles, through supercritical or subcritical bifurcations respectively. Similar bifurcations emerge when examining four-site breathers with either next-nearest-neighbor or even interactions with the three-nearest one-dimensional neighbors. The latter setting can be thought of as a prototype for the two-dimensional building block, namely a square of lattice nodes, which is also examined.

Full-time dynamics of modulational instability in spinor Bose-Einstein condensates

We describe the full-time dynamics of modulational instability in F = 1 spinor Bose-Einstein condensates for the case of the integrable three-component model associated with the matrix nonlinear Schrodinger equation. We obtain an exact homoclinic solution of this model by employing the dressing method which we generalize to the case of the higher-rank projectors. This homoclinic solution describes the development of modulational instability beyond the linear regime, and we show that the modulational instability demonstrates the reversal property when the growth of the modulated amplitude is changed by its exponential decay. Spinor Bose-Einstein condensate BEC with an optical confinement represents a unique macroscopic system with the spin degrees of freedom 1,2. The interplay between the mean-field effective nonlinearities of three-component mat- ter waves and their spin properties produce many interesting phenomena such as spin mixing 2, as well as the formation of spin domains 3,4 and spin textures 5,6. Various prop- erties of the spinor BEC have been analyzed theoretically 7-9. The ground state of the spinor BEC with the hyperfine spin F = 1 can be either ferromagnetic maximum spin pro- jection or polar zero spin projection. It was shown in Ref. 10 within the linear stability analysis of the spinor conden- sate model that the ferromagnetic phase of the condensate can experience instability for large enough densities of at- oms, while the polar phase remains always modulationally stable. Wadati and co-authors 11 demonstrated that the three- component nonlinear equations describing the evolution of the F = 1 BEC can be reduced, under special constraints im- posed on the condensate parameters, to the completely inte- grable matrix nonlinear Schrodinger NLS equation 12. Both bright and dark three-component BEC solitons have been found in the framework of this model 13-17. As regards the linear stability analysis presented in Ref. 10, only an initial linear stage of the perturbation devel- opment can be explored by this method which predicts the exponential growth of the modulation frequency sidebands for some conditions; i.e., it describes the conditions of modu- lational instability MI. A physical mechanism behind the MI is the parametric coupling between the spin degrees of freedom which leads to a population transfer between the spin components. To study the long-time evolution of insta- bilities, numerical methods are used as a rule. For the scalar NLS equation, the problem of the long-time evolution of the MI was studied by the truncation of the original model to a finite number of modes as usually, the three-mode approxi- mation18. More complete analysis of the long-time MI dynamics 19,20 is based on a linear constraint imposed on the real and imaginary parts of solutions of the scalar NLS equation, and it allows one to find a class of three-parameter

Travelling Waves in Nonlinear Magnetic Metamaterials

In this article, a model of one-dimensional metamaterial formed by a discrete array of nonlinear resonators is considered. The existence and uniqueness results of periodic and asymptotic travelling waves of the system are presented. The existence and the stability of asymptotic waves are also computed and discussed numerically.

Periodically Forced Nonlinear Oscillatory Acoustic Vacuum

In this work, we study the in-plane oscillations of a finite lattice of particles coupled by linear springs under distributed harmonic excitation. Melnikov-type analysis is applied for the persistence of periodic oscillations of a reduced system.

Stability of Discrete Breathers in Magnetic Metamaterials

We consider the discrete Klein–Gordon equation for magnetic metamaterials derived by Lazarides, Eleftheriou, and Tsironis Phys Rev Lett 97:157406, 2006). We obtain a general criterion for spectral stability of multi-site breathers for a small coupling constant. We show how this criterion differs from the one derived in the standard discrete Klein–Gordon equation (Koukouloyannis and Kevrekidis, Nonlinearity 22:2269–2285, 2009; Pelinovsky and Sakovich, Nonlinearity 25:3423–3451, 2012).