Stabilization of solitons of the multidimensional nonlinear Schrodinger equation: Matter-wave breathers

The nonlinear Schr{\"o}dinger equation with derivative cubic nonlinearity admits a family of solitons, which are orbitally stable in the energy space. In this work, we prove the orbital stability of multi-solitons configurations in the energy space, under suitable assumptions on the speeds and frequencies of the composing solitons. The main ingredients of the proof are modulation theory, energy coercivity and monotonicity properties.

All stationary solutions to the one-dimensional nonlinear Schroedinger equation under box and periodic boundary conditions are presented in analytic form. We consider the case of repulsive nonlinearity; in a companion paper we treat the attractive case. Our solutions take the form of stationary trains of dark or grey density-notch solitons. Real stationary states are in one-to-one correspondence with those of the linear Schr\"odinger equation. Complex stationary states are un...

We present new solutions in terms of elementary functions of the multi-component nonlinear Schr\"odinger equations and known solutions of the Davey-Stewartson equations such as multi-soliton, breather, dromion and lump solutions. These solutions are given in a simple determinantal form and are obtained as limiting cases in suitable degenerations of previously derived algebro-geometric solutions. In particular we present for the first time breather and rational breather soluti...

To study controlled evolution of non-autonomous matter-wave solitons in spinor Bose-Einstein condensates with spatiotemporal modulation, we focus on a system of three coupled Gross-Pitaevskii (GP) equations with space-time-dependent external potentials and temporally modulated gain/loss distributions. An integrability condition and a non-isospectral Lax pair for the coupled GP equations are obtained. Using it, we derive an infinite set of dynamical invariants, the first two o...

We investigate the stability properties of breather soliton trains in a three-dimensional Bose-Einstein Condensate with Feshbach Resonance Management of the scattering length. This is done so as to generate both attractive and repulsive interaction. The condensate is con ned only by a one dimensional optical lattice and we consider both strong, moderate, and weak con nement. By strong con nement we mean a situation in which a quasi two dimensional soliton is created. Moderate...

Using similarity transformations we construct explicit nontrivial solutions of nonlinear Schr\"odinger equations with potentials and nonlinearities depending on time and on the spatial coordinates. We present the general theory and use it to calculate explicitly non-trivial solutions such as periodic (breathers), resonant or quasiperiodically oscillating solitons. Some implications to the field of matter-waves are also discussed.

Solitary waves in a general nonlinear lattice are discussed, employing as a model the nonlinear Schr\"odinger equation with a spatially periodic nonlinear coefficient. An asymptotic theory is developed for long solitary waves, that span a large number of lattice periods. In this limit, the allowed positions of solitary waves relative to the lattice, as well as their linear stability properties, hinge upon a certain recurrence relation which contains information beyond all ord...

In this paper we investigate the emergence of time-periodic and and time-quasiperiodic (sometimes infinitely long lived and sometimes very long lived or metastable) solutions of discrete nonlinear wave equations: discrete sine Gordon, discrete $\phi^4$ and discrete nonlinear Schr\"odinger. The solutions we consider are periodic oscillations on a kink or standing wave breather background. The origin of these oscillations is the presence of internal modes, associated with the s...

We numerically investigate the long time dynamics of spatially periodic breather solutions of the 1-D nonlinear Schr\"odinger equation under parametric forcing of the form $f(x)=f_0 \exp(iKx)$ along with dissipation. In the absence of dissipation, robust soliton-like excitations are observed that travel with constant amplitude and velocity. With dissipation, these solitons lose energy (and amplitude) yet gain speed - a characteristic not observed in an ordinary soliton. Moreo...

Results concerning the existence and spectral stability and instability of multiple periodic wave solutions for the nonlinear Schr\"odinger system with \textit{dnoidal} and \textit{cnoidal} profile will be determined in this manuscript. The spectral analysis for the corresponding linearized operator is established by using the comparison theorem and tools of Floquet theory. The main results are determined by applying the spectral stability theory in \cite{KapitulaKevrekidisSa...