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.
Stationary solutions for the 2D critical Dirac equation with Kerr nonlinearity / Borrelli, W.. - In: JOURNAL OF DIFFERENTIAL EQUATIONS. - ISSN 0022-0396. - 263:11(2017), pp. 7941-7964. [10.1016/j.jde.2017.08.029]
Stationary solutions for the 2D critical Dirac equation with Kerr nonlinearity
William Borrelli
2017
Abstract
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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
