When linear stability does not exclude nonlinear instability.

We describe a mechanism that results in the nonlinear instability of stationary states even in the case where the stationary states are linearly stable. This instability is due to the nonlinearity-induced coupling of the linearization's internal modes of negative energy with the continuous spectrum. In a broad class of nonlinear Schrödinger equations considered, the presence of such internal modes guarantees the nonlinear instability of the stationary states in the evolution dynamics. To corroborate this idea, we explore three prototypical case examples: (a) an antisymmetric soliton in a double-well potential, (b) a twisted localized mode in a one-dimensional lattice with cubic nonlinearity, and (c) a discrete vortex in a two-dimensional saturable lattice. In all cases, we observe a weak nonlinear instability, despite the linear stability of the respective states.

Matter-wave bright solitons in spin-orbit coupled Bose-Einstein condensates.

We study matter-wave bright solitons in spin-orbit coupled Bose-Einstein condensates with attractive interactions. We use a multiscale expansion method to identify solution families for chemical potentials in the semi-infinite gap of the linear energy spectrum. Depending on the linear and spin-orbit coupling strengths, the solitons may present either a sech2-shaped or a modulated density profile reminiscent of the stripe phase of spin-orbit coupled repulsive Bose-Einstein condensates. Our numerical results are in excellent agreement with our analytical findings and demonstrate the potential robustness of solitons for experimentally relevant conditions.

Hamiltonian averaging for solitons with nonlinearity management.

We revisit the averaged equation, derived in Phys. Rev. Lett. 91, 240201 (2003)] from the nonlinear Schrödinger (NLS) equation with the nonlinearity management. We show that this averaged equation is valid only at the initial time interval, while a new Hamiltonian averaged NLS equation can be used at longer time intervals. Using the new averaged equation, we construct numerically matter-wave solitons in the context of the Bose-Einstein condensates under the Feshbach resonance management. We show that there is no threshold on the existence of dark solitons of large amplitudes, whereas such a threshold exists for bright solitons.

Averaging for solitons with nonlinearity management.

We develop an averaging method for solitons of the nonlinear Schrödinger equation with a periodically varying nonlinearity coefficient, which is used to effectively describe solitons in Bose-Einstein condensates, in the context of the recently proposed technique of Feshbach resonance management. Using the derived local averaged equation, we study matter-wave bright and dark solitons and demonstrate a very good agreement between solutions of the averaged and full equations.

Transverse instability of vector solitons and generation of dipole arrays.

We develop a theory of modulational instability of multiparameter solitary waves and analyze the transverse instability of composite (or vector) optical solitons in a saturable nonlinear medium. We demonstrate theoretically and experimentally that a soliton stripe breaks up into an array of ( 2+1)-dimensional dipole-mode vector solitons, thus confirming the robust nature of those solitons as fundamental composite structures of incoherently coupled fields.

Generation of soliton oscillations in nonlinear quadratic materials.

We show analytically and numerically that the generation of long-lasting soliton oscillations in resonant chi(2) optical materials possesses a threshold for the amplitude of a fundamental wave. The persistent oscillations of solitary waves reported by Etrich et al. [Phys. Rev. E 54, 4321 (1996)] are found to appear for finite values of the wave amplitude.