We consider the one-dimensional Swift-Hohenberg equation coupled to a conservation law. As a parameter increases the system undergoes a Turing bifurcation. We study the dynamics near this bifurcation. First, we show that stationary, periodic solutions bifurcate from a homogeneous ground state. Second, we construct modulating traveling fronts which model an invasion of the unstable ground state by the periodic solutions. This provides a mechanism of pattern formation for the studied system. The existence proof uses center manifold theory for a reduction to a finite-dimensional problem. This is possible despite the presence of infinitely many imaginary eigenvalues for vanishing bifurcation parameter since the eigenvalues leave the imaginary axis with different velocities if the parameter increases. Furthermore, compared to non-conservative systems, we address new difficulties arising from an additional neutral mode at Fourier wave number $k=0$ by exploiting that the amplitude of the conserved variable is small compared to the other variables.

中文翻译：

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在附加守恒定律的情况下调制 Swift-Hohenberg 方程的行进前沿

我们考虑与守恒定律耦合的一维 Swift-Hohenberg 方程。随着参数的增加，系统会经历图灵分叉。我们研究了这个分岔附近的动力学。首先，我们展示了平稳的、周期性的解决方案从齐次基态分叉。其次，我们构建了调制行进前沿，通过周期解来模拟不稳定基态的入侵。这为所研究的系统提供了模式形成的机制。存在性证明使用中心流形理论来简化为有限维问题。尽管对于消失的分岔参数存在无限多个虚特征值，但这是可能的，因为如果参数增加，特征值以不同的速度离开虚轴。此外，与非保守系统相比，