Self trapping transition for a nonlinear impurity within a linear chain

In the present work we revisit the issue of the self-trapping dynamical transition at a nonlinear impurity embedded in an otherwise linear lattice. For our Schr\"odinger chain example, we present rigorous arguments that establish necessary conditions and corresponding parametric bounds for the transition between linear decay and nonlinear persistence of a defect mode. The proofs combine a contraction mapping approach applied in the fully dynamical problem in the case of a 3D-lattice, together with variational arguments for the derivation of parametric bounds for the creation of stationary states associated with the expected fate of the self-trapping dynamical transition. The results are relevant for both power law nonlinearities and saturable ones. The analytical results are corroborated by numerical computations.Comment: 16 pages, 7 figures. To be published in Journal of Mathematical Physic

Conservation laws, exact travelling waves and modulation instability for an extended nonlinear Schr\"odinger equation

We study various properties of solutions of an extended nonlinear Schr\"{o}dinger (ENLS) equation, which arises in the context of geometric evolution problems -- including vortex filament dynamics -- and governs propagation of short pulses in optical fibers and nonlinear metamaterials. For the periodic initial-boundary value problem, we derive conservation laws satisfied by local in time, weak $H^2$ (distributional) solutions, and establish global existence of such weak solutions. The derivation is obtained by a regularization scheme under a balance condition on the coefficients of the linear and nonlinear terms -- namely, the Hirota limit of the considered ENLS model. Next, we investigate conditions for the existence of traveling wave solutions, focusing on the case of bright and dark solitons. The balance condition on the coefficients is found to be essential for the existence of exact analytical soliton solutions; furthermore, we obtain conditions which define parameter regimes for the existence of traveling solitons for various linear dispersion strengths. Finally, we study the modulational instability of plane waves of the ENLS equation, and identify important differences between the ENLS case and the corresponding NLS counterpart. The analytical results are corroborated by numerical simulations, which reveal notable differences between the bright and the dark soliton propagation dynamics, and are in excellent agreement with the analytical predictions of the modulation instability analysis.Comment: 27 pages, 5 figures. To be published in Journal of Physics A: Mathematical and Theoretica

Lower and upper estimates on the excitation threshold for breathers in DNLS lattices

We propose analytical lower and upper estimates on the excitation threshold for breathers (in the form of spatially localized and time periodic solutions) in DNLS lattices with power nonlinearity. The estimation depending explicitly on the lattice parameters, is derived by a combination of a comparison argument on appropriate lower bounds depending on the frequency of each solution with a simple and justified heuristic argument. The numerical studies verify that the analytical estimates can be of particular usefulness, as a simple analytical detection of the activation energy for breathers in DNLS lattices.Comment: 10 pages, 3 figure

Kuznetsov-Ma breather-like solutions in the Salerno model

The Salerno model is a discrete variant of the celebrated nonlinear Schr\"odinger (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.Comment: 9 pages, 4 figure

Stationary states of a nonlinear Schrödinger lattice with a harmonic trap

We study a discrete nonlinear Schrödinger lattice with a parabolic trapping potential. The model, describing, e.g., an array of repulsive Bose-Einstein condensate droplets confined in the wells of an optical lattice, is analytically and numerically investigated. Starting from the linear limit of the problem, we use global bifurcation theory to rigorously prove that – in the discrete regime – all linear states lead to nonlinear generalizations thereof, which assume the form of a chain of discrete dark solitons (as the density increases). The stability of the ensuing nonlinear states is studied and it is found that the ground state is stable, while the excited states feature a chain of stability/instability bands. We illustrate the mechanisms under which discreteness destabilizes the dark-soliton configurations, which become stable only in the continuum regime. Continuation from the anti-continuum limit is also considered, and a rich bifurcation structure is revealed

Propagation of periodic wave trains along the magnetic field in a collision-free plasma

In this work, a systematic study, examining the propagation of periodic and solitary wave along the magnetic field in a cold collision-free plasma, is presented. Employing the quasi-neutral approximation and the conservation of momentum flux and energy flux in the frame co-traveling with the wave, the exact analytical solution of the stationary solitary pulse is found analytically in terms of particle densities, parallel and transverse velocities, as well as transverse magnetic fields. Subsequently, this solution is generalized in the form of periodic waveforms represented by cnoidal-type waves. These considerations are fully analytical in the case where the total angular momentum flux $L$, due to the ion and electron motion together with the contribution due to the Maxwell stresses, vanishes. A graphical representation of all associated fields is also provided