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We study linear stability analysis for spin-1 Bose-Einstein condensates (BEC). We show that all bounded solutions of this physical system are neutrally stable. In particular, all steady-state solutions of the physical system, and the associated discrete steady-state solutions are neutrally stable. Next, we consider the physical system without the affect of magnetic field. By exploiting the physical properties of both ferromagnetic and antiferromagnetic cases, we develop efficient multi-level pseudo-arclength continuation algorithms combined with a spectral collocation method for these two cases, respectively. When the magnetic field is imposed on the physical system, an additional multi-level continuation algorithm is described for the ferromagnetic case. Extensive numerical results for spin-1 BEC in a magnetic field, and in optical lattices are reported. [ABSTRACT FROM AUTHOR]

A contour integral termed $$\hbox {M}_{\mathrm{int}}$$ is presented for describing the fracture behavior of cracks passing through or terminating normally at a bimaterial interface. The $$\hbox {M}_{\mathrm{int}}$$ -integral is defined by performing the conventional M-integral along a contour enclosing the cracks, with the coordinate system properly originated for measure of the integration points. The presented formulation is considered to be feasible for problems with generally anisotropic elastic materials. Physically, the energy parameter $$\hbox {M}_{\mathrm{int}}$$ is shown to be equivalent to twice the released potential energy required for creation of the cracks. Such relation is exactly valid when the crack length is small compared with the size of the specimen, and approximately satisfied for problems containing relatively large cracks. Also, due to path-independence, the $$\hbox {M}_{\mathrm{int}}$$ -integral can be performed along an arbitrary outer contour, which is chosen to be far from the crack tips. With this property, the complicated singular stress field in the near-tip areas can be avoided so that a complicated finite element model around the crack tips is not required in the calculation. [ABSTRACT FROM AUTHOR]

We describe multi-parameter continuation methods combined with spectral collocation methods for computing numerical solutions of rotating two-component Bose–Einstein condensates (BECs), which are governed by the Gross–Pitaevskii equations (GPEs). Various types of orthogonal polynomials are used as the basis functions for the trial function space. A novel multi-parameter/multiscale continuation algorithm is proposed for computing the solutions of the governing GPEs, where the chemical potential of each component and angular velocity are treated as the continuation parameters simultaneously. The proposed algorithm can effectively compute numerical solutions with abundant physical phenomena. Numerical results on rotating two-component BECs are reported. [ABSTRACT FROM AUTHOR]