Continuous families of solitary waves in non-symmetric complex potentials: A Melnikov theory approach

The existence of stationary solitary waves in symmetric and non-symmetric complex potentials is studied by means of Melnikov's perturbation method. The latter provides analytical conditions for the existence of such waves that bifurcate from the homogeneous nonlinear modes of the system and are located at specific positions with respect to the underlying potential. It is shown that the necessary conditions for the existence of continuous families of stationary solitary waves, as they arise from Melnikov theory, provide general constraints for the real and imaginary part of the potential, that are not restricted to symmetry conditions or specific types of potentials. Direct simulations are used to compare numerical results with the analytical predictions, as well as to investigate the propagation dynamics of the solitary waves.Comment: 19 pages, 14 figure

Interactions and scattering of quantum vortices in a polariton fluid

Quantum vortices, the quantized version of classical vortices, play a prominent role in superfluid and superconductor phase transitions. However, their exploration at a particle level in open quantum systems has gained considerable attention only recently. Here we study vortex pair interactions in a resonant polariton fluid created in a solid-state microcavity. By tracking the vortices on picosecond time scales, we reveal the role of nonlinearity, as well as of density and phase gradients, in driving their rotational dynamics. Such effects are also responsible for the split of composite spin-vortex molecules into elementary half-vortices, when seeding opposite vorticity between the two spinorial components. Remarkably, we also observe that vortices placed in close proximity experience a pull-push scenario leading to unusual scattering-like events that can be described by a tunable effective potential. Understanding vortex interactions can be useful in quantum hydrodynamics and in the development of vortex-based lattices, gyroscopes, and logic devices.Comment: 12 pages, 7 figures, Supplementary Material and 5 movies included in arXi

Energy criterion for the spectral stability of discrete breathers

Discrete breathers are ubiquitous structures in nonlinear anharmonic models ranging from the prototypical example of the Fermi-Pasta-Ulam model to Klein-Gordon nonlinear lattices, among many others. We propose a general criterion for the emergence of instabilities of discrete breathers analogous to the well-established Vakhitov-Kolokolov criterion for solitary waves. The criterion involves the change of monotonicity of the discrete breather's energy as a function of the breather frequency. Our analysis suggests and numerical results corroborate that breathers with increasing (decreasing) energy-frequency dependence are generically unstable in soft (hard) nonlinear potentials.Comment: 5 pages, 3 figures. Includes Supplementary Materia

A PT-symmetric dual-core system with the sine-Gordon nonlinearity and derivative coupling

As an extension of the class of nonlinear PT -symmetric models, we propose a systemof 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 couplingstems from local interaction between adjacent particles in coupled Frenkel-Kontorova (FK) chains,while the cross-derivative coupling, which was not considered before, is induced by three-particleinteractions, provided that the particles in the parallel FK chains move in different directions.Nonlinear modes are then studied in this system. In particular, kink-kink (KK) and kink-antikink(KA) complexes are explored by means of analytical and numerical methods. It is predictedanalytically and confirmed numerically that the complexes are unstable for one sign of the sinusoidalcoupling, and stable for another. Stability regions are delineated in the underlying parameter space.Unstable complexes split into free kinks and antikinks, that may propagate or become quiescent,depending on whether they are subject to gain or loss, respectively

The closeness of localised structures between the Ablowitz-Ladik lattice and Discrete Nonlinear Schr\"odinger equations II: Generalised AL and DNLS systems

The Ablowitz-Ladik system, being one of the few integrable nonlinear lattices, admits a wide class of analytical solutions, ranging from exact spatially localised solitons to rational solutions in the form of the spatiotemporally localised discrete Peregrine soliton. Proving a closeness result between the solutions of the Ablowitz-Ladik and a wide class of Discrete Nonlinear Schr\"odinger systems in a sense of a continuous dependence on their initial data, we establish that such small amplitude waveforms may be supported in the nonintegrable lattices, for significant large times. The nonintegrable systems exhibiting such behavior include a generalisation of the Ablowitz-Ladik system with a power-law nonlinearity and the Discrete Nonlinear Schr\"odinger with power-law and saturable nonlinearities. The outcome of numerical simulations illustrates in an excellent agreement with the analytical results the persistence of small amplitude Ablowitz-Ladik analytical solutions in all the nonintegrable systems considered in this work, with the most striking example being that of the Peregine soliton.Comment: arXiv admin note: text overlap with arXiv:2102.0533

Rogue Waves in Ultracold Bosonic Seas

In this work, we numerically consider the initial value problem for nonlinear Schrodinger (NLS)type models arising in the physics of ultracold boson gases, with generic Gaussian wavepacketinitial data. The corresponding Gaussian\u27s width and, wherever relevant also its amplitude, serveas control parameters. First we explore the one-dimensional, standard NLS equation with generalpower law nonlinearity, in which large amplitude excitations reminiscent of Peregrine solitons orregular solitons appear to form, as the width of the relevant Gaussian is varied. Furthermore, thevariation of the nonlinearity exponent aims at a rst glimpse of the interplay between rogue orsoliton formation and collapse features. The robustness of the main features to noise in the initialdata is also conrmed. To better connect our study with the physics of atomic condensates, andexplore the role of dimensionality eects, we also consider the nonpolynomial Schrodinger equation(NPSE), as well as the full three-dimensional NLS equation, and examine the degree to whichrelevant considerations generalize

SO(2)-induced breathing patterns in multi-component Bose-Einstein condensates

In this work, we employ the SO(2)-rotations of a two-component, one-, two- and three-dimensionalnonlinear Schrodinger system at and near the Manakov limit, to construct vector solitons and vortexstructures. This way, stable stationary dark-bright solitons and their higher-dimensional siblingsare transformed into robust oscillatory dark-dark solitons (and generalizations thereof), with andwithout a harmonic connement. By analogy to the one-dimensional case, vector higher-dimensionalstructures take the form of vortex-vortex states in two dimensions and, e.g., vortex ring-vortex ringones in three dimensions. We consider the eects of unequal (self- and cross-) interaction strengths,where the SO(2) symmetry is only approximately satised, showing the dark-dark soliton oscillationis generally robust. Similar features are found in higher dimensions too, although our case examplessuggest that phenomena such as phase separation may contribute to the associated dynamics. Theseresults, in connection with the experimental realization of one-dimensional variants of such statesin optics and Bose-Einstein condensates (BECs), suggest the potential observation of the higherdimensionalbound states proposed herein