在本文中，我们建立了一个全面的数学工具箱，用于构造和谱稳定性分析实直线上具有空间周期系数的一般半线性演化问题的稳态多重前沿解与脉冲解。从一组 N 个具有匹配周期末态的非退化基本前沿解出发，只要前沿界面之间的距离足够大，我们就能在这些 N 个基本前沿解的形式拼接附近实现多重前沿解。此外，我们证明了非退化基本脉冲解伴随着大空间周期的周期脉冲解。我们证明，底层基本前沿解或脉冲解的谱（不）稳定性性质会被分岔产生的多重前沿解或周期脉冲解所继承。存在性与谱分析依赖于压缩映射论证和 Evans 函数技巧，并利用指数二分性来刻画可逆性与 Fredholm 性质。为了证明我们方法的适用性，我们分析了一些基准模型中多重前沿解和周期脉冲解的存在性与稳定性，例如具有周期势的 Gross-Pitaevskii 方程和 Klausmeier 反应-扩散-平流系统，从而识别出新的（稳定）解类。特别地，我们的方法首次给出了具有周期势的 Gross-Pitaevskii 方程中周期波的谱稳定性和轨道稳定性结果，以及该方程多脉冲解的新的不稳定性判据。
英文摘要：
In this paper, we develop a comprehensive mathematical toolbox for the construction and spectral stability analysis of stationary multiple front and pulse solutions to general semilinear evolution problems on the real line with spatially periodic coefficients. Starting from a collection of $N$ nondegenerate primary front solutions with matching periodic end states, we realize multifront solutions near a formal concatenation of these $N$ primary fronts, provided the distances between the front interfaces is sufficiently large. Moreover, we prove that nondegenerate primary pulses are accompanied by periodic pulse solutions of large spatial period. We show that spectral (in)stability properties of the underlying primary fronts or pulses are inherited by the bifurcating multifronts or periodic pulse solutions. The existence and spectral analyses rely on contraction-mapping arguments and Evans-function techniques, leveraging exponential dichotomies to characterize invertibility and Fredholm properties. To demonstrate the applicability of our methods, we analyze the existence and stability of multifronts and periodic pulse solutions in some benchmark models, such as the Gross-Pitaevskii equation with periodic potential and a Klausmeier reaction-diffusion-advection system, thereby identifying novel classes of (stable) solutions. In particular, our methods yield the first spectral and orbital stability result of periodic waves in the Gross-Pitaevskii equation with periodic potential, as well as new instability criteria for multipulse solutions to this equation.
