On a Schrödinger-Landau-Lifshitz System: Variational Structure and Numerical Methods

Isotropic Landau-Lifshitz Equation

By using the geometric concept of PDEs with prescribed curvature representation, we prove that the 1+2 dimensional isotropic Landau-Lifshitz equation is gauge equivalent to a 1+2 dimensional nonlinear Schrödinger-type system. From the nonlinear Schrödinger-type system, we construct blowing up H3(R2)-solutions to the Landau-Lifshitz equation, which reveals the blow up phenomenon of the equation.

High-frequency limit of the Maxwell-Landau-Lifshitz system in the diffractive optics regime

We study semilinear Maxwell-Landau-Lifshitz systems in one space dimension. For highly oscillatory and prepared initial data, we construct WKB approximate solutions over long times O(1/ε). The leading terms of the WKB solutions solve cubic Schrödinger equations. We show that the nonlinear normal form method of Joly, Métivier and Rauch [5] applies to this context. This implies that the Schröding...

On a class of nonlinear fractional Schrödinger-Poisson systems

In this paper, we are concerned with the following fractional Schrödinger-Poisson system:    (−∆s)u + V (x)u + φu = m(x)|u|q−2|u|+ f(x,u), x ∈ Ω, (−∆t)φ = u2, x ∈ Ω, u = φ = 0, x ∈ ∂Ω, where s,t ∈ (0,1], 2t + 4s > 3, 1 < q < 2 and Ω is a bounded smooth domain of R3, and f(x,u) is linearly bounded in u at infinity. Under some assumptions on m, V and f we obtain the existence of non-trivial so...

Global Well-posedness of the Landau-lifshitz-gilbert Equation for Initial Data in Morrey Space

We establish the global well-posedness of the Landau-Lifshitz-Gilbert equation in Rn for any initial data m0 ∈ H1 ∗(R, S2) whose gradient belongs to the Morrey space M2,2(Rn) with small norm ‖∇m0‖M2,2(Rn). The method is based on priori estimates of a dissipative Schrödinger equation of GinzburgLandau types obtained from the Landau-Lifshitz-Gilbert equation by the moving frame technique.

Existence of infinitely many solutions for coupled system of Schrödinger-Maxwell's equations