We study matter-wave bright solitons in spin-orbit (SO) coupled Bose-Einstein condensates (BECs) with attractive interatomic 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 resemble either standard bright nonlinear Schrödinger solitons or exhibit a modulated density profile, reminiscent of the stripe phase of SO-coupled repulsive BECs. Our numerical results are in excellent agreement with our analytical findings, and demonstrate the potential robustness of such solitons for experimentally relevant conditions through stability analysis and direct numerical simulations. [12]. While the above studies refer to BECs with repulsive interactions, to the best of our knowledge, SO-coupled BECs with attractive interactions have not been studied so far. The latter, is the theme of the present work.As it is known, attractive BECs can become themselves matter-wave bright solitons [13], i.e., self-trapped and highly localized mesoscopic quantum systems that can find a variety of applications [14]. Here, we demonstrate the existence, stability and dynamics of matter-wave bright solitons in SOcoupled attractive BECs. In particular, starting from the corresponding mean-field model, we consider the nonlinear waves emerging in the semi-infinite gap of the linear spectrum. Similarly to the repulsive interaction case of Ref.[7], we find three distinct states having: (a) zero momentum, (b) finite momentum, +k 0 or −k 0 , and (c) stripe densities formed by the interference of the modes with ±k 0 momentum. We analytically identify these branches, in very good agreement with our numerical computations, and determine their spin polarizations. We also analyze the stability of these solutions, illustrating that branches (a) and (c) are generically stable, while branch (b) is stable for sufficiently small atom numbers. Hence, these newly emerging matter-wave solitons in SO-coupled BECs

Motivated by recent experimental results, we present a systematic theoretical analysis of dark-bright-soliton interactions and multiple-dark-bright-soliton complexes in atomic two-component Bose-Einstein condensates. We study analytically the interactions between two dark-bright solitons in a homogeneous condensate and then extend our considerations to the presence of the trap. We illustrate the existence of robust stationary dark-bright-soliton "molecules," composed of two or more solitons, which are formed due to the competition of the interaction forces between the dark-and bright-soliton components and the trap force. Our analysis is based on an effective equation of motion, derived for the distance between two dark-bright solitons. This equation provides equilibrium positions and characteristic oscillation frequencies of the solitons, which are found to be in good agreement with the eigenfrequencies of the anomalous modes of the system.

We explore the stability and dynamics of dark-bright solitons in two-component elongated BoseEinstein condensates by developing effective 1D vector equations as well as solving the corresponding 3D Gross-Pitaevskii equations. A strong dependence of the oscillation frequency and of the stability of the dark-bright (DB) soliton on the atom number of its components is found. Spontaneous symmetry breaking leads to oscillatory dynamics in the transverse degrees of freedom for a large occupation of the component supporting the dark soliton. Moreover, the interactions of two DB solitons are investigated with special emphasis on the importance of their relative phases. Experimental results showcasing dark-bright soliton dynamics and collisions in a BEC consisting of two hyperfine states of 87 Rb confined in an elongated optical dipole trap are presented.Introduction. Multi-component systems of nonlinear waves are a fascinating topic with a rich and diverse history spanning a variety of areas, including Bose-Einstein condensates (BECs) in atomic physics [1], optical fibers and crystals in nonlinear optics [2], and integrable systems in mathematical physics [3]. Of particular interest are the so-called "symbiotic solitons", namely structures that would not otherwise exist in one-component settings, but can be supported by the interaction between the optical or atomic species components. A prototypical example of such a structure is the dark-bright (DB) soliton in self-defocusing, two-component systems, whereby the dark soliton (density dip) which typically arises in self-defocusing media [1][2][3][4] creates, through nonlinearity, a trapping mechanism that localizes a density hump (bright soliton) in the second component.

Motivated by recent experimental results, we study beating dark-dark solitons as a prototypical coherent structure that emerges in two-component Bose-Einstein condensates. We showcase their connection to darkbright solitons via SO(2) rotation, and infer from it both their intrinsic beating frequency and their frequency of oscillation inside a parabolic trap. We identify them as exact periodic orbits in the Manakov limit of equal interand intra-species nonlinearity strengths with and without the trap and showcase the persistence of such states upon weak deviations from this limit. We also consider large deviations from the Manakov limit illustrating that this breathing state may be broken apart into dark-antidark soliton states. Finally, we consider the dynamics and interactions of two beating dark-dark solitons in the absence and in the presence of the trap, inferring their typically repulsive interaction.

We consider nonlinear analogs of parity-time-(PT -) symmetric linear systems exhibiting defocusing nonlinearities. We study the ground state and odd excited states (dark solitons and vortices) of the system and report the following remarkable features. For relatively weak values of the parameter ε controlling the strength of the PT -symmetric potential, excited states undergo (analytically tractable) spontaneous symmetry breaking; as ε is further increased, the ground state and first excited state, as well as branches of higher multisoliton (multivortex) states, collide in pairs and disappear in blue-sky bifurcations, in a way which is strongly reminiscent of the linear PT phase transition-thus termed the nonlinear PT phase transition. Past this critical point, initialization of, e.g., the former ground state, leads to spontaneously emerging solitons and vortices.