Numerical analysis of parabolic p-Laplacian: Approximation of trajectories

The long time numerical approximation of the parabolic p-Laplacian problem with a time-independent forcing term and sufficiently smooth initial data is studied. Convergence and stability results which are uniform for t is an element of [0, infinity) are established in the L-2, W-1,W-p norms for the backward Euler and the Crank-Nicholson schemes with the finite element method (FEM). This result extends the existing uniform convergence results for exponentially contractive semigroups generated by some semilinear systems to nonexponentially contractive semigroups generated by some quasilinear systems

Instability of Bose-Einstein condensates in tilted lattices with time-periodical modulation

We study the dynamical stability of Bose-Einstein condensates in an optical lattice with a time-periodic modulation potential and a constant acceleration force simultaneously. We derive the explicit expressions of quasienergies and obtain the stability diagrams in the parameter space of the interaction strength and the modulation amplitude. The ratio of the acceleration force to the modulation frequency characterizes two cases: integer and non-integer resonances. For integer resonances, the critical interaction strength $g_{\mathrm{c}}$ shows an alternate behavior where the completely unstable regions correspond to the negative effective tunneling strength. Among non-integer resonances, we observe that $g_{\mathrm{c}}$ peaks are centered around half-integer resonances for which the completely unstable regions disappear, accompanied with a whole displacement of $g_{\mathrm{c}}$. Compared with integer and half-integer resonances, the crossovers between them show no explicit dependence of $g_{\mathrm{c}}$ on the modulation amplitude