TY - JOUR
T1 - Stationary solutions for the 2D critical Dirac equation with Kerr nonlinearity
AU - Borrelli, William
PY - 2017
Y1 - 2017
N2 - In this paper we prove the existence of an exponentially localized stationary solution for a two-dimensional cubic Dirac equation. It appears as an effective equation in the description of nonlinear waves for some Condensed Matter (Bose–Einstein condensates) and Nonlinear Optics (optical fibers) systems. The nonlinearity is of Kerr-type, that is of the form |ψ|2ψ and thus not Lorenz-invariant. We solve compactness issues related to the critical Sobolev embedding H[Formula presented](R2,C2)↪L4(R2,C4) thanks to a particular radial ansatz. Our proof is then based on elementary dynamical systems arguments.
AB - In this paper we prove the existence of an exponentially localized stationary solution for a two-dimensional cubic Dirac equation. It appears as an effective equation in the description of nonlinear waves for some Condensed Matter (Bose–Einstein condensates) and Nonlinear Optics (optical fibers) systems. The nonlinearity is of Kerr-type, that is of the form |ψ|2ψ and thus not Lorenz-invariant. We solve compactness issues related to the critical Sobolev embedding H[Formula presented](R2,C2)↪L4(R2,C4) thanks to a particular radial ansatz. Our proof is then based on elementary dynamical systems arguments.
KW - critical Dirac equations
KW - critical Dirac equations
UR - http://hdl.handle.net/10807/171316
U2 - 10.1016/j.jde.2017.08.029
DO - 10.1016/j.jde.2017.08.029
M3 - Article
SN - 0022-0396
VL - 263
SP - 7941
EP - 7964
JO - Journal of Differential Equations
JF - Journal of Differential Equations
ER -