The Schrödinger and the Dirac Equation - Linear Nonlinear and Integrodifferential

Mario Schönberg

L. H. Haddad, Lincoln D. Carr

We present relativistic linear stability equations (RLSE) for quasi-relativistic cold atoms in a honeycomb optical lattice. These equations are derived from first principles and provide a method for computing stabilities of arbitrary localized solutions of the nonlinear Dirac equation (NLDE), a relativistic generalization of the nonlinear Schrödinger equation. We present a variety of such localized solutions: skyrmions, solitons, vortices, and half-quantum vortices, and study their stabilities via the RLSE. When applied to a uniform background, our calculations reveal an experimentally observable effect in the form of Cherenkov radiation.

Heinz Otto Cordes

Wilhelm Fushchych

• Discovery of the Schrodinger equation • Derivation (uniqueness) of the Schrodinger equation • High order equation of the Schrodinger type • Nonlocal symmetry of the Schrodinger equation [2, 7] • High-order evolution equations. Dependence of mass on velocity in the nonlocal Galilei-invariant theory [6, 19] • Galilei relativity principle and nonlinear Schrodinger-type equations [2, 5, 10–17] • Nonlocal symmetry of the linear Schrodinger–de Broglie–Klein–Gordon–Fock– Kudar–de Donder–Van Dunger [3, 18, 8] • Nonlocal Galilei symmetry of a relativistic equation [8, 9] • Nonlocal Galilei symmetry of the Dirac equation [7] • Galilei symmetry of a relativistic equation.

Lincoln D. Carr, Laith H. Haddad, Christopher M. Weaver +1 more

Bose-Einstein condensates (BECs) are a macroscopic state of quantum matter at temperatures nine orders of magnitude colder than outer space. BECs can be trapped in crystals of light: when the crystal takes the form of a honeycomb optical lattice, Dirac physics arises at the Brillouin zone edges, just like in graphene. The weakly interacting quantum gas comprising the BEC generates an effective nonlinearity, giving rise to the nonlinear Dirac equation (NLDE) [1], as we will show. This equation, together with the relativistic linear stability equations (RLSE) [2], comprise a relativistic generalization of the famous nonlinear Schrodinger equation and Boguliubov-de Gennes equations familiar from, for example, propagation of light in Kerr nonlinear media. We present a zoo of exotic vortex [3], skyrmion, and soliton [4,5] solutions to the NLDE+RLSE system, and explain the practical realization of these concepts in present experiments on BECs in the optical equivalent of 2D graphene and 1D armchair and zigzag graphene nanoribbons.

Spyros Efthimiades