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We give a rigorous deduction of the eigenvalue problem of the nonlinearSchr\"odinger equation (NLS) at Dirac Points for potential of honeycomb latticesymmetry. Based on a bootstrap method, we observe the bifurcation of theeigenfunctions into eight distinct modes from the two-dimensional degeneratedeigenspace of the regressive linear Schr\"odinger equation. We give theexistence, the way of construction, uniqueness in $H^2$ space and the$C^\infty$ continuity of these eigenfunctions.

The Eigenvalue Problem of Nonlinear Schr\"odinger Equation at Dirac Points of Honeycomb Lattice