Klein Gordon Schrodinger System Research Articles

Uniform error estimates of a finite difference method for the Klein-Gordon-Schrödinger system in the nonrelativistic and massless limit regimes

We establish a uniform error estimate of a finite difference method for the Klein-Gordon-Schrodinger (KGS) equations with two dimensionless parameters \begin{document}$0 and \begin{document}$0 , which are the mass ratio and inversely proportional to the speed of light, respectively. In the simultaneously nonrelativistic and massless limit regimes, i.e., \begin{document}$γ\sim\varepsilon$\end{document} and \begin{document}$\varepsilon \to 0^+$\end{document} , the KGS equations converge singularly to the Schrodinger-Yukawa (SY) equations. When \begin{document}$0 , due to the perturbation of the wave operator and/or the incompatibility of the initial data, which is described by two parameters \begin{document}$α≥0$\end{document} and \begin{document}$β≥-1$\end{document} , the solution of the KGS equations oscillates in time with \begin{document}$O(\varepsilon)$\end{document} -wavelength, which requires harsh meshing strategy for classical numerical methods. We propose a uniformly accurate method based on two key points: (ⅰ) reformulating KGS system into an asymptotic consistent formulation, and (ⅱ) applying an integral approximation of the oscillatory term. Using the energy method and the limiting equation via the SY equations with an oscillatory potential, we establish two independent error bounds at \begin{document}$O(h^2+τ^2/\varepsilon)$\end{document} and \begin{document}$O(h^2+τ^2+τ\varepsilon^{α^*}+\varepsilon^{1+α^*})$\end{document} with \begin{document}$h$\end{document} mesh size, \begin{document}$τ$\end{document} time step and \begin{document}$α^* = \min\{1, α, 1+β\}$\end{document} . This implies that the method converges uniformly and optimally with quadratic convergence rate in space and uniformly in time at \begin{document}$O(τ^{4/3})$\end{document} and \begin{document}$O(τ^{1+\frac{α^*}{2+α^*}})$\end{document} for well-prepared ( \begin{document}$α^* = 1$\end{document} ) and ill-prepared ( \begin{document}$0≤α^* ) initial data, respectively. Thus the \begin{document}$\varepsilon$\end{document} -scalability of the method is \begin{document}$τ = O(1)$\end{document} and \begin{document}$h = O(1)$\end{document} for \begin{document}$0 , which is significantly better than classical methods. Numerical results are reported to confirm our error bounds. Finally, the method is applied to study the convergence rates of KGS equations to its limiting models in the simultaneously nonrelativistic and massless limit regimes.

Orbital instability of standing waves for the quadratic–cubic Klein-Gordon–Schrödinger system

We consider the Klein-Gordon–Schrodinger system with quadratic and cubic interactions. Smooth curves of periodic- and solitary-wave solutions are obtained via the implicit function theorem. Orbital instability of such waves is then established.

On existence and scattering theory for the Klein–Gordon–Schrödinger system in an infinite $$L^{2}$$ L 2 -norm setting

This paper is concerned with the initial value problem for the nonlinear Klein–Gordon–Schrodinger (KGS) system in $$\mathbb {R}^{n}\times \mathbb {R},\, n\ge 1$$ . We consider general polynomial nonlinearities that include in particular the classical Yukawa–KGS model. We show existence of local and global mild solutions for the KGS system with initial data in weak $$L^{r}$$ -spaces, which is an infinite $$L^{2}$$ -norm setting. Moreover, we obtain a persistence result in $$H^{s}$$ when the initial data belong to this class, which shows that the constructed data-solution map in weak- $$L^{r}$$ recovers $$H^{s} $$ -regularity. We also prove results of scattering and wave operators in that singular framework.