Bright–dark soliton dynamics and interaction for the variable coefficient three-coupled nonlinear Schrödinger equations

Under investigation in this paper is the variable coefficient three-coupled nonlinear Schrödinger (CNLS) equations, which govern the dynamics of solitonic excitations along three-spine [Formula: see text]-helical protein with inhomogeneous effect. Via the Hirota method and symbolic computation, the exact two-bright-one-dark (TBD) and one-bright-two-dark (BTD) soliton solutions are constructed analytically. The propagation properties are discussed for TBD and BTD solitons when the variable coefficient is a hyperbolic secant function. Figures are plotted to reveal the following interactions of TBD and BTD two solitons: (1) Evolution without interactions of double-parabola-shaped solitons, of double-[Formula: see text]-shaped solitons and of parabola-[Formula: see text]-shaped solitons; (2) Evolution with periodic interaction of double-parabola-shaped solitons and of parabola-[Formula: see text]-shaped solitons; (3) Collision of double-[Formula: see text]-shaped solitons and of parabola-[Formula: see text]-shaped solitons.

Vector periodic solitons for three-coupled nonlinear Schrödinger equations

Under investigation in this paper are the three-coupled nonlinear Schrödinger (CNLS) equations, which govern the dynamics of solitonic excitations along three-spine [Formula: see text]-helical protein with inhomogeneous effect. The vector periodic solitons (VPSs) of three-CNLS equations have been obtained by a transformation that transforms the corresponding equations to the integrable mixed three-CNLS equations. The VPSs getting from two-bright-one-dark solitons are different to the ones from two-dark-one-bright solitons. The latter propagates on periodic wave background and the interspine interaction coefficient adjusts the oscillation period. Moreover, the different-type VPSs are reported through manipulating external inhomogeneity potential. And it shows that VPSs can transform into bright–bright–bright solitons in the special conditions. Dynamic behaviors of those solitons are discussed through graphic simulation.

Darboux transformation and solitons for an integrable nonautonomous nonlinear integro-differential Schrödinger equation

By modifying the scheme for an isospectral problem, the non-isospectral Ablowitz–Kaup–Newell–Segur (AKNS) hierarchy is constructed via allowing the time varying spectrum. In this paper, we consider an integrable nonautonomous nonlinear integro-differential Schrödinger equation discussed before in “Multi-soliton management by the integrable nonautonomous nonlinear integro-differential Schrödinger equation” [Y. J. Zhang, D. Zhao and H. G. Luo, Ann. Phys. 350 (2014) 112]. We first analyze the integrability conditions and identify the model. Second, we modify the existing Darboux transformation (DT) for such a non-isospectral problem. Third, the nonautonomous soliton solutions are obtained via the resulting DT and basic properties of these solutions in the inhomogeneous media are discussed graphically to illustrate the influences of the variable coefficients. In the process, a technique by selecting appropriate spectral parameters instead of the variable inhomogeneities is employed to realize a different type of one-soliton management. Several novel optical solitons are constructed and their features are shown by some specific figures. In addition, four kinds of the special localized two-soliton solutions are obtained. The solitonic excitations localized both in space and time, which exhibit the feature of the so-called rogue waves but with a zero background, are discussed.

Brownian motion of solitons in a Bose–Einstein condensate

We observed and controlled the Brownian motion of solitons. We launched solitonic excitations in highly elongatedRb87Bose–Einstein condensates (BECs) and showed that a dilute background of impurity atoms in a different internal state dramatically affects the soliton. With no impurities and in one dimension (1D), these solitons would have an infinite lifetime, a consequence of integrability. In our experiment, the added impurities scatter off the much larger soliton, contributing to its Brownian motion and decreasing its lifetime. We describe the soliton’s diffusive behavior using a quasi-1D scattering theory of impurity atoms interacting with a soliton, giving diffusion coefficients consistent with experiment.