In the paper, we study the Prandtl system with initial data admitting non-degenerate critical points. For any index \sigma\in [3/2, 2], we obtain the local in time well-posedness in the space of Gevrey class \sigma\in [3/2, 2], in the tangential variable and Sobolev class in the normal variable so that the monotonicity condition on the tangential velocity is not needed to overcome the loss of tangential derivative. This answers the open question raised in the paper of D. Grard-Varet and N. Masmoudi [{\it Ann. Sci. c. Norm. Supr}.(4) 48 (2015), no. 6, 1273-1325], in which the case \sigma\in [3/2, 2], is solved.

The linear spectrum and corresponding Bloch modes of shallow honeycomb lattices near Dirac points are investigated. Via perturbation theory, the dispersion relation is found to have threefold degeneracy at leading order with eigenvalue splitting at the following two orders; i.e., the threefold eigenvalue splits into single and double values. Multiscale perturbation methods are employed to describe the nonlinear dynamics of the associated wave envelopes. The dynamics of the envelope depends on different asymptotic balances whereupon a three-level nonlinear Dirac-type equation or a two-level nonlinear Dirac equation is derived. The analysis agrees well with direct numerical simulations.

No abstract uploaded!

No abstract uploaded!

No abstract uploaded!