We construct a new Evans function for quasi-periodic solutions to the linearisation of the sine-Gordon equation about a periodic travelling wave. This Evans function is written in terms of fundamental solutions to a Hill’s equation. Applying the Evans-Krein function theory of Kollár and Miller (SIAM Rev 56(1):73–123, 2014) to our Evans function, we provide a new method for computing the Krein signatures of simple characteristic values of the linearised sine-Gordon equation. By varying the Floquet exponent parametrising the quasi-periodic solutions, we compute the linearised spectra of periodic travelling wave solutions of the sine-Gordon equation and track dynamical Hamiltonian–Hopf bifurcations via the Krein signature. Finally, we show that our new Evans function can be readily applied to the general case of the nonlinear Klein–Gordon equation with a non-periodic potential.

我们构造了一个新的Evans函数，用于对周期传播的正弦-Gordon方程线性化的准周期解。该Evans函数是根据希尔方程的基本解来编写的。将Kollár和Miller的Evans-Krein函数理论（SIAM Rev 56（1）：73–123，2014）应用到我们的Evans函数中，我们提供了一种计算线性正弦-Gordon简单特征值的Kerin签名的新方法方程。通过改变准周期解的Floquet指数参数，我们计算了正弦-Gordon方程的周期行波解的线性化谱，并通过Kerin签名跟踪了动力学哈密顿量-霍普夫分叉。最后，