Periodic solutions to Klein-Gordon systems with linear couplings

TL;DR

This paper investigates the existence, regularity, and asymptotic behavior of time-periodic solutions to linearly coupled Klein-Gordon systems, showing convergence to uncoupled solutions as the coupling diminishes.

Contribution

It introduces a variational approach to find periodic solutions for coupled Klein-Gordon systems and characterizes their behavior as the coupling parameter approaches zero.

Findings

Existence of time-periodic solutions for small coupling constants.

Solutions converge to uncoupled wave solutions as coupling tends to zero.

Higher regularity properties of solutions are established.

Abstract

In this paper, we study the nonlinear Klein-Gordon systems arising from relativistic physics and quantum field theories $$\left\{\begin{array}{lll} u_{tt}- u_{xx} +bu + \varepsilon v + f(t,x,u) =0,\; v_{tt}- v_{xx} +bv + \varepsilon u + g(t,x,v) =0 \end{array}\right. $$ where $u,v$ satisfy the Dirichlet boundary conditions on spatial interval $[0, \pi]$, $b>0$ and $f$, $g$ are $2\pi$-periodic in $t$. We are concerned with the existence, regularity and asymptotic behavior of time-periodic solutions to the linearly coupled problem as $\varepsilon$ goes to 0. Firstly, under some superlinear growth and monotonicity assumptions on $f$ and $g$, we obtain the solutions $(u_\varepsilon, v_\varepsilon)$ with time-period $2\pi$ for the problem as the linear coupling constant $\varepsilon$ is sufficiently small, by constructing critical points of an indefinite functional via variational methods. Secondly, we give precise characterization for the asymptotic behavior of these solutions, and show that as $\varepsilon\rightarrow 0$, $(u_\varepsilon, v_\varepsilon)$ converge to the solutions of the wave equations without the coupling terms. Finally, by careful analysis which are quite different from the elliptic regularity theory, we obtain some interesting results concerning the higher regularity of the periodic solutions.