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

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...

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...

In this article we prove the existence of a new family of periodic solutions for discrete, nonlinear Schrodinger equations subject to spatially localized driving and damping. They provide an alternate description of the metastable behavior in such lattice systems which agrees with previous predictions for the evolution of metastable states while providing more accurate approximations to these states. We analyze the stability of these breathers, finding a very small positive e...

This work deals with soliton solutions of the nonlinear Schroedinger equation with cubic and quintic nonlinearities. We extend the procedure put forward in a recent Letter and we solve the equation in the presence of linear background, and cubic and quintic interactions which are modulated in space and time. As a result, we show how a simple parameter can be used to generate brightlike or darklike localized nonlinear waves which oscillate in several distinct ways, driven by t...

An unstable particle in quantum mechanics can be stabilized by frequent measurements, known as the quantum Zeno effect. A soliton with dissipation behaves like an unstable particle. Similar to the quantum Zeno effect, here we show that the soliton can be stabilized by modulating periodically dispersion, nonlinearity, or the external harmonic potential available in BEC. This can be obtained by analyzing a Painleve integrability condition, which results from the rigorous Painle...

We consider the problem of existence and stability of solitary traveling waves for the one dimensional discrete non linear Schroedinger equation (DNLS) with cubic nonlinearity, near the continuous limit.We construct a family of solutions close to the continuous traveling waves and prove their stability over long times. Applying a modulation method, we also show that we can describe the dynamics near these discrete traveling waves over long times.

The dynamics of soliton and quasisoliton solutions of cubic third order nonlinear Schr\"{o}dinger equation is studied. The regular solitons exist due to a balance between the nonlinear terms and (linear) third order dispersion; they are not important at small $\alpha_3$ ($\alpha_3$ is the coefficient in the third derivative term) and vanish at $\alpha_3 \to 0$. The most essential, at small $\alpha_3$, is a quasisoliton emitting resonant radiation (resonantly radiating soliton...

The existence, stability and other dynamical properties of a new type of multi-dimensional (2D or 3D) solitons supported by a transverse low-dimensional (1D or 2D, respectively) periodic potential in the nonlinear Schr\"{o}dinger equation with the self-defocusing cubic nonlinearity are studied. The equation describes propagation of light in a medium with normal group-velocity dispersion (GVD). Strictly speaking, solitons cannot exist in the model, as its spectrum does not sup...

We present a theoretical analysis of three-dimensional (3D) matter-wave solitons and their stability properties in coupled atomic and molecular Bose-Einstein condensates (BEC). The soliton solutions to the mean-field equations are obtained in an approximate analytical form by means of a variational approach. We investigate soliton stability within the parameter space described by the atom-molecule conversion coupling, atom-atom s-wave scattering, and the bare formation energy...

The two-dimensional cubic nonlinear Schrodinger equation admits a large family of one-dimensional bounded traveling-wave solutions. All such solutions may be written in terms of an amplitude and a phase. Solutions with piecewise constant phase have been well studied previously. Some of these solutions were found to be stable with respect to one-dimensional perturbations. No such solutions are stable with respect to two-dimensional perturbations. Here we consider stability of ...