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arxiv: 2202.06099 · v1 · pith:RWLAINE5 · submitted 2022-02-12 · math.AP · cond-mat.mtrl-sci· math-ph· math.MP· physics.optics· quant-ph

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

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classification math.AP cond-mat.mtrl-scimath-phmath.MPphysics.opticsquant-ph
keywords equationodingerschrdiraceigenfunctionseigenvaluegivehoneycomb
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We give a rigorous deduction of the eigenvalue problem of the nonlinear Schr\"odinger 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 Schr\"odinger equation. We give the existence, the way of construction, uniqueness in $H^2$ space and the $C^\infty$ continuity of these eigenfunctions.

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