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The results of calculations of the dynamics of a field in a cavity that consists of a stationary mirror and a periodically displacing mirror are presented. The initial distribution of the field corresponds to the equilibrium thermal (Planck) distribution. It is shown that, if conditions of a resonant excitation are implemented, pulses can be formed the degree of unipolarity of which increases from the initial zero degree to an almost maximum (unit) degree. [ABSTRACT FROM AUTHOR]

We generalize a recently proposed model based on the cubic complex Ginzburg-Landau (CGL) equation, which gives rise to stable dissipative solitons supported by localized gain applied at a “hot spot” (HS), in the presence of the linear loss in the bulk. We introduce a model with the Kerr nonlinearity concentrated at the HS, together with the local gain and, possibly, with the local nonlinear loss. The model, which may be implemented in laser cavities based on planar waveguides, gives rise to exact solutions for pinned dissipative solitons. In the case when the HS does not include the localized nonlinear loss, numerical tests demonstrate that these solitons are stable/unstable if the localized nonlinearity is self-defocusing/focusing. Another new setting considered in this work is a pair of two symmetric HSs. We find exact asymmetric solutions for it, although they are unstable. Numerical simulations demonstrate that stable modes supported by the HS pair tend to be symmetric. An unexpected conclusion is that the interaction between breathers pinned to two broad HSs, which are the only stable modes in isolation in that case, transforms them into a static symmetric mode. [ABSTRACT FROM AUTHOR]

We consider self-trapping of 2D solitons in the model based on the Gross-Pitaevskii/nonlinear Schrödinger equation with the self-attractive cubic nonlinearity and a periodic potential of the optical-lattice (OL) type. It is known that this model may suppress the collapse, giving rise to a family of stable fundamental solitons. Here, we report essential dynamical features of self-trapping of the fundamental solitons from input configurations of two types, with vorticity 0 or 1. We identify regions in the respective parameter spaces corresponding to the formation of the soliton, collapse, and decay. A noteworthy result is the self-trapping of stable fundamental solitons in cases when the input norm essentially exceeds the collapse threshold. We also compare predictions of the dynamical variational approximation with direct numerical simulations. [ABSTRACT FROM AUTHOR]

It is known that optical-lattice (OL) potentials can stabilize solitons and solitary vortices against the critical collapse, generated by cubic attractive nonlinearity in the 2D geometry. We demonstrate that OLs can also stabilize various species of fundamental and vortical solitons against the supercritical collapse, driven by the double-attractive cubic-quintic nonlinearity (however, solitons remain unstable in the case of the pure quintic nonlinearity). Two types of OLs are considered, producing similar results: the 2D Kronig-Penney “checkerboard”, and the sinusoidal potential. Soliton families are obtained by means of a variational approximation, and as numerical solutions. The stability of all families, which include fundamental and multi-humped solitons, vortices of oblique and straight types, vortices built of quadrupoles, and supervortices, strictly obeys the Vakhitov-Kolokolov criterion. The model applies to optical media and BEC in “pancake” traps. [ABSTRACT FROM AUTHOR]

We study the effect of an optical lattice (OL) on the ground-state properties of one-dimensional ultracold bosons with three-body attractive interactions and two-body repulsive interactions, which are described by a cubic-quintic Gross-Pitaevskii equation with a periodic potential. Without the optical lattice and with a vanishing two-body interaction term, normalizable soliton solutions of the Townes type are possible only at a critical value of the interaction strength, at which an infinite degeneracy of the ground state occurs; a repulsive two-body interaction makes such localized solutions unstable. We show that the OL opens a stability window around the critical point when the strength of the periodic potential is above a critical threshold. We also consider the effect of an external parabolic trap, studying how the stability properties depend on the matching between minima of the periodic potential and the minimum of the parabolic trap. [ABSTRACT FROM AUTHOR]

The evolution of two-frequency solitons in an optical fiber, as well as the practically important special case of absence of the second-harmonic wave, in the presence of a longitudinal nonuniformity of the coefficients characterizing the propagation nonlinearity are considered. The solitons found for media with constant values of the nonlinearity coefficients are used as initial distributions for media with a periodic dependence of the nonlinearity coefficients on the longitudinal coordinate. Modulation of the coefficient of cubic or quadratic nonlinearity is shown to result in oscillations of the peak intensity of the solitons (in both their components if two-color solitons are considered). In the case of a weak modulation of the nonlinearity coefficients, oscillations of the peak intensity occur at the frequency coinciding with the frequency of modulation of the nonlinearity coefficients. Under the weak influence of a periodically modulated cubic nonlinearity, parameters of quadratic solitons also oscillate upon the propagation. Regions of stability of solitons in the space of the modulation parameters are established. [ABSTRACT FROM AUTHOR]

We present one-dimensional (1D) stability analysis of a recently proposed method to filter and control localized states of the Bose–Einstein condensate (BEC), based on novel trapping techniques that allow one to conceive methods to select a particular BEC shape by controlling and manipulating the external potential well in the three-dimensional (3D) Gross–Pitaevskii equation (GPE). Within the framework of this method, under suitable conditions, the GPE can be exactly decomposed into a pair of coupled equations: a transverse two-dimensional (2D) linear Schrödinger equation and a one-dimensional (1D) longitudinal nonlinear Schrödinger equation (NLSE) with, in a general case, a time-dependent nonlinear coupling coefficient. We review the general idea how to filter and control localized solutions of the GPE. Then, the 1D longitudinal NLSE is numerically solved with suitable non-ideal controlling potentials that differ from the ideal one so as to introduce relatively small errors in the designed spatial profile. It is shown that a BEC with an asymmetric initial position in the confining potential exhibits breather-like oscillations in the longitudinal direction but, nevertheless, the BEC state remains confined within the potential well for a long time. In particular, while the condensate remains essentially stable, preserving its longitudinal soliton-like shape, only a small part is lost into “radiation”. [ABSTRACT FROM AUTHOR]

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 Schrödinger 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 support a true bandgap. Nevertheless, the variational approximation (VA) and numerical computations reveal stable solutions that seem as completely localized ones, an explanation to which is given. The solutions are of the gap-soliton type in the transverse direction(s), in which the periodic potential acts in combination with the diffraction and self-defocusing nonlinearity. Simultaneously, in the longitudinal (temporal) direction these are ordinary solitons, supported by the balance of the normal GVD and defocusing nonlinearity. Stability of the solitons is predicted by the VA, and corroborated by direct simulations. [ABSTRACT FROM AUTHOR]

We consider spatial solitons in a channel waveguide or in a periodic array of rectangular potential wells (the Kronig-Penney (KP) model) in the presence of the uniform cubic-quintic (CQ) nonlinearity. Using the variational approximation and numerical methods, we. nd two branches of fundamental (single-humped) soliton solutions. The soliton characteristics, in the form of the integral power Q (or width w) vs. the propagation constant k, reveal a strong bistability with two different solutions found for a given k. Violating the known Vakhitov-Kolokolov criterion, the solution branches with dQ/dk > 0 and dQ/dk < 0 are simultaneously stable. Various families of higher-order solitons are also found in the KP version of the model: symmetric and antisymmetric double-humped solitons, three-peak solitons with and without the phase shift π between the peaks, etc. In a relatively shallow KP lattice, all the solitons belong to the semi-infinite gap beneath the linear band structure of the KP potential, while finite gaps between the bands remain empty (solitons in the finite gaps can be found if the lattice is much deeper). But in contrast to the situation known for the model combining a periodic potential and the self-focusing Kerr nonlinearity, the fundamental solitons fill only a finite zone near the top of the semi-infinite gap, which is a manifestation of the saturable character of the CQ nonlinearity. [ABSTRACT FROM AUTHOR]

We introduce a model of three parallel-coupled nonlinear waveguiding cores equipped with Bragg gratings (BGs), which form an equilateral triangle. The most promising way to create multi-core BG configuration is to use inverted gratings, written on internal surfaces of relatively broad holes embedded in a photonic-crystal-fiber matrix. The objective of the work is to investigate solitons and their stability in this system. New results are also obtained for the earlier investigated dual-core system. Families of symmetric and antisymmetric solutions are found analytically, extending beyond the spectral gap in both the dual- and tri-core systems. Moreover, these families persist in the case (strong coupling between the cores) when there is no gap in the system’s linear spectrum. Three different types of asymmetric solitons are found (by means of the variational approach and numerical methods) in the tri-core system. They exist only inside the spectral gap, but asymmetric solitons with nonvanishing tails are found outside the gap as well. Stability of the solitons is explored by direct simulations, and, for symmetric solitons, in a more rigorous way too, by computation of eigenvalues for small perturbations. The symmetric solitons are stable up to points at which two types of asymmetric solitons bifurcate from them. Beyond the bifurcation, one type of the asymmetric solitons is stable, and the other is not. Then, they swap their stability. Asymmetric solitons of the third type are always unstable. When the symmetric solitons are unstable, their instability is oscillatory, and, in most cases, it transforms them into stable breathers. In both the dual- and tri-core systems, the stability region of the symmetric solitons extends far beyond the gap, persisting in the case when the system has no gap at all. The whole stability region of antisymmetric solitons (a new type of solutions in the tri-core system) is located outside the gap. Thus, solitons in multi-core BGs can be observed experimentally in a much broader frequency band than in the single-core one, and in a wider parameter range than it could be expected. Asymmetric delocalized solitons, found outside the spectral gap, can be stable too. [ABSTRACT FROM AUTHOR]

We produce several families of solutions for two-component nonlinear Schrödinger/Gross-Pitaevskii equations. These include domain walls and the first example of an antidark or gray soliton in one component, bound to a bright or dark soliton in the other. Most of these solutions are linearly stable in their entire domain of existence. Some of them are relevant to nonlinear optics, and all to Bose-Einstein condensates (BECs). In the latter context, we demonstrate robustness of the structures in the presence of parabolic and periodic potentials (corresponding, respectively, to the magnetic trap and optical lattices in BECs). [ABSTRACT FROM AUTHOR]