We look for bounded periodic solutions for a parametric fractional problem involving a continuous nonlinearity with subcritical growth. By using a variant of Caffarelli and Silvestre extension method adapted to the periodic case and variational tools we prove the existence of at least three bounded periodic solutions when the parameter varies in an appropriate range.

Conical diffraction in honeycomb lattices is analyzed. This phenomenon arises in nonlinear Schrödinger equations with honeycomb lattice potentials. In the tight-binding approximation the wave envelope is governed by a nonlinear classical Dirac equation. Numerical simulations show that the Dirac equation and the lattice equation have the same conical diffraction properties. Similar conical diffraction occurs in both the linear and nonlinear regimes. The Dirac system reveals the underlying mechanism for the existence of conical diffraction in honeycomb lattices.

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