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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

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We analyze the existence, stability, and internal modes of gap solitons in nonlinear periodic systems described by the nonlinear Schrödinger equation with a sinusoidal potential, such as photonic crystals, waveguide arrays, optically-induced photonic lattices, and Bose-Einstein condensates loaded onto an optical lattice. We study bifurcations of gap solitons from the band edges of the Floquet-Bloch spectrum, and show that gap solitons can appear near all lower or upper band edges of the spectrum, for focusing or defocusing nonlinearity, respectively. We show that, in general, two types of gap solitons can bifurcate from each band edge, and one of those two is always unstable. A gap soliton corresponding to a given band edge is shown to possess a number of internal modes that bifurcate from all band edges of the same polarity. We demonstrate that stability of gap solitons is determined by location of the internal modes with respect to the spectral bands of the inverted spectrum and, when they overlap, complex eigenvalues give rise to oscillatory instabilities of gap solitons.