The Eigenvalue Problem of Nonlinear Schrödinger Equation at Dirac Points of Honeycomb Lattice

Abstract

We give a rigorous deduction of the eigenvalue problem of the nonlinear Schrödinger equation (NLS) at Dirac Points for potential of honeycomb lattice symmetry. Based on a bootstrap method, we observe the bifurcation of the eigenfunctions into eight distinct modes from the two-dimensional degenerated eigenspace of the regressive linear Schrödinger equation. We give the existence, the way of construction, uniqueness in $H^2$ space and the $C^{\infty }$ continuity of these eigenfunctions.

28 pages