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

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arxiv 2202.06099 v1 pith:RWLAINE5 submitted 2022-02-12 math.AP cond-mat.mtrl-scimath-phmath.MPphysics.opticsquant-ph

Yejia Chen , Ruihan Peng , Qidong Fu , Fangwei Ye , Weidong Luo This is my paper · ORCID
classification math.APcond-mat.mtrl-scimath-phmath.MPphysics.opticsquant-ph
keywords equationodingerschrdiraceigenfunctionseigenvaluegivehoneycomb
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abstract

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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