Global well-posedness of the fractional Klein-Gordon-Schrödinger system with rough initial data

Abstract

We investigate the low regularity local and global well-posedness of the Cauchy problem for the coupled Klein-Gordon-Schrödinger system with fractional Laplacian in the Schrödinger equation in R 1+1 . We use Bourgain space method to study this problem and prove that this system is locally well-posed for Schrödinger data in H s 1 and wave data in H s 2 × H s 2 −1 for 3/4 − α < s 1 0 and −1/2 < s 2 < 3/2, where α is the fractional power of Laplacian which satisfies 3/4 < α 1. Based on this local well-posedness result, we also obtain the global well-posedness of this system for s 1 = 0 and −1/2 < s 2 < 1/2 by using the conservation law for the L 2 norm of u.

On the L∞ convergence of a conservative Fourier pseudo‐spectral method for the space fractional nonlinear Schrödinger equation

“…Then, we can prove that the conservative Fourier pseudo‐spectral method is unconditionally convergent with order of O ( τ 2 + N α /2 − r ) in the discrete L ∞ norm. The method of error analysis in the discrete L ∞ for the presented conservative Fourier pseudo‐spectral scheme can be extended to other PDEs involving fractional Laplacian, for example, the space fractional Allen‐Cahn Equation [36] and the Klein‐Gordon‐Schrödinger Equation [37].…”

Optimal error estimate of a conservative Fourier pseudo-spectral method for the space fractional nonlinear Schrödinger equation

“…Furthermore, The method of error analysis in the discrete L ∞ for the presented conservative Fourier pseudo-spectral scheme can be extended to other PDEs involving fractional Laplacian, for example, the space fractional Allen-Cahn equation [26] and the Klein-Gordon-Schrödinger equation [13].…”

Structure-preserving algorithms for the two-dimensional fractional Klein-Gordon-Schrödinger equation

“…As well known, many continuous systems possess some physical quantities that naturally arise from the physical context, such as energy, momentum and mass. For system (1.1)-(1.2) with periodic boundary conditions, Guo et al [5,8] derived that the system has fractional mass and energy conservation laws M(t) = M(0), E(t) = E(0), (1.5) where the mass has the form M(t) := Ω |ϕ| 2 dx, (1.6) and the energy is defined as…”