Compacton existence and spin-orbit density dependence in Bose-Einstein condensates.

We demonstrate the existence of compactons matter waves in binary mixtures of Bose-Einstein condensates (BEC) trapped in deep optical lattices (OL) subjected to equal contributions of intraspecies Rashba and Dresselhaus spin-orbit coupling (SOC) under periodic time modulations of the intraspecies scattering length. We show that these modulations lead to a rescaling of the SOC parameters that involves the density imbalance of the two components. This gives rise to density dependent SOC parameters that strongly influence the existence and the stability of compacton matter waves. The stability of SOC-compactons is investigated both by linear stability analysis and by time integrations of the coupled Gross-Pitaevskii equations. We find that SOC restricts the parameter ranges for stable stationary SOC-compacton existence but, on the other side, it gives a more stringent signature of their occurrence. In particular, SOC-compactons should appear when the intraspecies interactions and the number of atoms in the two components are perfectly balanced (or close to being balanced for the metastable case). The possibility to use SOC-compactons as a tool for indirect measurements of the number of atoms and/or the intraspecies interactions is also suggested.

Matter waves in atomic-molecular condensates with Feshbach resonance management.

The dynamics of matter waves in the atomic to molecular condensate transition with a time-modulated atomic scattering length is investigated. Both the cases of rapid and slow modulations are studied. In the case of rapid modulations, the average over oscillations for the system is derived. The corresponding conditions for dynamical suppression of the association of atoms into the molecular field, or of second-harmonic generation in nonlinear optical systems, are obtained. For the case of slow modulations, we find resonant enhancement in the molecular field. We then illustrate chaos in the atomic-molecular BEC system. We suggest a sequential application of the two types of modulations, slow and rapid, when producing molecules.

Zeno effect and switching of solitons in nonlinear couplers.

The Zeno effect is investigated for soliton type pulses in a nonlinear directional coupler with dissipation. The effect consists in increase of the coupler transparency with increase of the dissipative losses in one of the arms. It is shown that localized dissipation can lead to switching of solitons between the arms. Power losses accompanying the switching can be fully compensated by using a combination of dissipative and active (in particular, parity-time-symmetric) segments.

Compactons in nonlinear Schrödinger lattices with strong nonlinearity management.

The existence of compactons in the discrete nonlinear Schrödinger equation in the presence of fast periodic time modulations of the nonlinearity is demonstrated. In the averaged discrete nonlinear Schrödinger equation, the resulting effective interwell tunneling depends on the modulation parameters and on the field amplitude. This introduces nonlinear dispersion in the system and can lead to a prototypical realization of single- or multisite stable discrete compactons in nonlinear optical waveguide and Bose-Einstein condensate arrays. These structures can dynamically arise out of Gaussian or compactly supported initial data.

Dissipative periodic waves, solitons, and breathers of the nonlinear Schrödinger equation with complex potentials.

Exact solutions for the generalized nonlinear Schrödinger equation with inhomogeneous complex linear and nonlinear potentials are found. We have found localized and periodic solutions for a wide class of localized and periodic modulations in the space of complex potentials and nonlinearity coefficients. Examples of stable and unstable solutions are given. We also demonstrated numerically the existence of stable dissipative breathers in the presence of an additional parabolic trap.

Generalized neighbor-interaction models induced by nonlinear lattices.

It is shown that the tight-binding approximation of the nonlinear Schrödinger equation with a periodic linear potential and periodic in space nonlinearity coefficient gives rise to a number of nonlinear lattices with complex, both linear and nonlinear, neighbor interactions. The obtained lattices present nonstandard possibilities, among which we mention a quasilinear regime, where the pulse dynamics obeys essentially the linear Schrödinger equation. We analyze the properties of such models both in connection to their modulational stability, as well as in regard to the existence and stability of their localized solitary wave solutions.

Ising-Bloch transition for the parametric Ginzburg-Landau equation with rapidly varying perturbations.

We study the effects of rapid periodic and stochastic modulations of parameters in systems described by the complex parametric Ginzburg-Landau equation. Amplitude equations, which govern the dynamics of the field averaged over the rapid modulations, are derived. For temporal modulations of the linear detuning the threshold for the transition from Ising to Bloch walls is shifted depending on the strength of the perturbation. In contrast to this, rapid perturbations of the linear gain lead only to a decrease of the amplitude of both wall types leaving the bifurcation point of the Ising-Bloch transition unchanged. Stochastic perturbations of the detuning lead to a Brownian motion of the Bloch wall beyond bifurcation where the velocity is given analytically. All theoretical predictions are confirmed by numerical simulations of the full stochastic Ginzburg-Landau equation.

Adiabatic dynamics of periodic waves in Bose-Einstein condensates with time dependent atomic scattering length.

Evolution of periodic matter waves in one-dimensional Bose-Einstein condensates with time-dependent scattering length is described. It is shown that variation of the effective nonlinearity is a powerful tool for controlled generation of bright and dark solitons starting with periodic waves.

Interaction of pulses in the nonlinear Schrödinger model.

The interaction of two rectangular pulses in the nonlinear Schrödinger model is studied by solving the appropriate Zakharov-Shabat system. It is shown that two real pulses may result in an appearance of moving solitons. Different limiting cases, such as a single pulse with a phase jump, a single chirped pulse, in-phase and out-of-phase pulses, and pulses with frequency separation, are analyzed. The thresholds of creation of new solitons and multisoliton states are found.

Soliton dynamics in a random Toda chain.

This paper addresses the soliton dynamics in a Toda lattice with a randomly distributed chain of masses. Applying the inverse scattering transform, we derive effective equations for the decay of the soliton amplitude that take into account radiative losses. It is shown that the soliton energy decays as approximately N(-3/2) for small-amplitude solitons and approximately exp(-N) for large-amplitude solitons. The decay rate does not depend on the incoming energy for large-amplitude soliton. An important feature is the generation of a soliton gas consisting of a large collection of small solitons (a number of the order of epsilon (-2) of solitons with momenta of the order of epsilon (2), where epsilon is the strength of fluctuations). The soliton gas plays an important role in that the changes in the conservation equations cannot be correctly understood if the soliton production is neglected. The role of the correlation length of fluctuations on the soliton decay is discussed. It is shown that in the presence of long-range correlation the Toda soliton is not backscattered, but progressively converted into forward-going radiation. The analytical predictions are confirmed by full numerical simulations.