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

TL;DR

This paper rigorously analyzes the eigenvalue problem of the nonlinear Schrödinger equation at Dirac points in honeycomb lattices, revealing bifurcation phenomena and constructing eigenfunctions with proven uniqueness and smoothness.

Contribution

It provides a rigorous derivation of eigenfunctions bifurcating from degenerate linear eigenstates at Dirac points, including their existence, construction, and regularity.

Findings

Eigenfunctions bifurcate into eight modes from degenerate eigenspace

Existence and uniqueness of eigenfunctions in H^2 space

Eigenfunctions are infinitely smooth (C^)

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.