Michael I. Weinstein

Honeycomb lattice potentials and Dirac points

We prove that the two-dimensional Schroedinger operator with a potential having the symmetry of a honeycomb structure has dispersion surfaces with conical singularities (Dirac points) at the vertices of its Brillouin zone. No assumptions are made on the size of the potential. We then prove the robustness of such conical singularities to a restrictive class of perturbations, which break the honeycomb lattice symmetry. General small perturbations of potentials with Dirac points do not have Dirac points; their dispersion surfaces are smooth. The presence of Dirac points in honeycomb structures is associated with many novel electronic and optical properties of materials such as graphene.

Geometric Analysis of Bifurcation and Symmetry Breaking in a Gross—Pitaevskii Equation

AbstractGross–Pitaevskii and nonlinear Hartree equations are equations of nonlinear Schrödinger type that play an important role in the theory of Bose–Einstein condensation. Recent results of Aschbacher et al.(3) demonstrate, for a class of 3-dimensional models, that for large boson number (squared L2norm),

Wave Packets in Honeycomb Structures and Two-Dimensional Dirac Equations

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Band-Edge Solitons, Nonlinear Schrödinger/Gross–Pitaevskii Equations, and Effective Media

, the ground state does not have the symmetry properties of the ground state at small

Degenerate dispersive equations arising in the study of magma dynamics

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Asymptotic Stability of Ascending Solitary Magma Waves

. We present a detailed global study of the symmetry breaking bifurcation for a 1-dimensional model Gross–Pitaevskii equation, in which the external potential (boson trap) is an attractive double-well, consisting of two attractive Dirac delta functions concentrated at distinct points. Using dynamical systems methods, we present a geometric analysis of the symmetry breaking bifurcation of an asymmetric ground state and the exchange of dynamical stability from the symmetric branch to the asymmetric branch at the bifurcation point.