We investigate 1D and 2D radial domain-wall (DW) states in the system of two nonlinear-Schr{o}dinger/Gross-Pitaevskii equations, which are coupled by the linear mixing and by the nonlinear XPM (cross-phase-modulation). The system has straightforward
applications to two-component Bose-Einstein condensates, and to the bimodal light propagation in nonlinear optics. In the former case, the two components represent different hyperfine atomic states, while in the latter setting they correspond to orthogonal polarizations of light. Conditions guaranteeing the stability of flat continuous wave (CW) asymmetric bimodal states are established, followed by the study of families of the corresponding DW patterns. Approximate analytical solutions for the DWs are found near the point of the symmetry-breaking bifurcation of the CW states. An exact DW solution is produced for ratio 3:1 of the XPM and SPM coefficients. The DWs between flat asymmetric states, which are mirror images to each other, are completely stable, and all other species of the DWs, with zero crossings in one or two components, are fully unstable. Interactions between two DWs are considered too, and an effective potential accounting for the attraction between them is derived analytically. Direct simulations demonstrate merger and annihilation of the interacting DWs. The analysis is extended for the system including single- and double-peak external potentials. Generic solutions for trapped DWs are obtained in a numerical form, and their stability is investigated. An exact stable solution is found for the DW trapped by a single-peak potential. In the 2D geometry, stable two-component vortices are found, with topological charges s=1,2,3. Radial oscillations of annular DW-shaped pulsons, with s=0,1,2, are studied too. A linear relation between the period of the oscillations and the mean radius of the DW ring is derived analytically.
Nonlinear periodic systems, such as photonic crystals and Bose-Einstein condensates (BECs) loaded into optical lattices, are often described by the nonlinear Schrodinger/Gross-Pitaevskii equation with a sinusoidal potential. Here, we consider a model
based on such a periodic potential, with the nonlinearity (attractive or repulsive) concentrated either at a single point or at a symmetric set of two points, which are represented, respectively, by a single {delta}-function or a combination of two {delta}-functions. This model gives rise to ordinary solitons or gap solitons (GSs), which reside, respectively, in the semi-infinite or finite gaps of the systems linear spectrum, being pinned to the {delta}-functions. Physical realizations of these systems are possible in optics and BEC, using diverse variants of the nonlinearity management. First, we demonstrate that the single {delta}-function multiplying the nonlinear term supports families of stable regular solitons in the self-attractive case, while a family of solitons supported by the attractive {delta}-function in the absence of the periodic potential is completely unstable. We also show that the {delta}-function can support stable GSs in the first finite gap in both the self-attractive and repulsive models. The stability analysis for the GSs in the second finite gap is reported too, for both signs of the nonlinearity. Alongside the numerical analysis, analytical approximations are developed for the solitons in the semi-infinite and first two finite gaps, with the single {delta}-function positioned at a minimum or maximum of the periodic potential. In the model with the symmetric set of two {delta}-functions, we study the effect of the spontaneous symmetry breaking of the pinned solitons. Two configurations are considered, with the {delta}-functions set symmetrically with respect to the minimum or maximum of the potential.
It is well known that the two-dimensional (2D) nonlinear Schrodinger equation (NLSE) with the cubic-quintic (CQ) nonlinearity supports a family of stable fundamental solitons, as well as solitary vortices (alias vortex rings), which are stable for su
fficiently large values of the norm. We study stationary localized modes in a symmetric linearly coupled system of two such equations, focusing on asymmetric states. The model may describe optical bullets in dual-core nonlinear optical waveguides (including spatiotemporal vortices that were not discussed before), or a Bose-Einstein condensate (BEC) loaded into a dual-pancake trap. Each family of solutions in the single-component model has two different counterparts in the coupled system, one symmetric and one asymmetric. Similarly to the earlier studied coupled 1D system with the CQ nonlinearity, the present model features bifurcation loops, for fundamental and vortex solitons alike: with the increase of the total energy (norm), the symmetric solitons become unstable at a point of the direct bifurcation, which is followed, at larger values of the energy, by the reverse bifurcation restabilizing the symmetric solitons. However, on the contrary to the 1D system, the system may demonstrate a double bistability for the fundamental solitons. The stability of the solitons is investigated via the computation of instability growth rates for small perturbations. Vortex rings, which we study for two values of the spin, s = 1 and 2, may be subject to the azimuthal instability, like in the single-component model. We also develop a quasi-analytical approach to the description of the bifurcations diagrams, based on the variational approximation. Splitting of asymmetric vortices, induced by the azimuthal instability, is studied by means of direct simulations. Interactions between initially quiescent solitons of different types are studied too.
