连续谱哈密顿系统中的斑图形成一个正规型单波模型
Pattern formation in Hamiltonian systems with continuous spectra; a normal-form single-wave model
摘要：
生物、化学和物理问题中的斑图形成已受到相当大的关注，其中耗散系统受到了很大的关注。例如，金兹伯格-朗道方程是一种规范形式，它描述了由于在各种耗散问题中出现单一模式的不稳定性而导致的斑图形成。类似地，某种“单波模型”出现在许多具有共同模式形成行为的物理环境中。这些系统具有哈密顿结构，单波模型是一种描述相空间中形成的模式的哈密顿平均场论。单波模型最初是在非线性等离子体理论的背景下推导出来的，它描述了阈值附近的行为和随后的非线性演化的不稳定等离子体波。然而，单波模型也出现在流体力学中，特别是剪切流和涡旋动力学，星系动力学，凝聚态物理的XY和Potts模型，以及其他以平均场相互作用为特征的哈密顿理论。我们证明，通过一个合适的渐近分析，如何出现的单波模型从一个大类的非线性对流传输理论。简化的一个基本要素是，在线性稳定性问题中，哈密顿系统具有连续谱，其不是由无限空间域引起的，而是由沿相空间中的曲线的奇异共振引起的，在相空间中，波速与材料速度相匹配（等离子体问题中的波粒共振，或流体问题中的临界水平）。当系统处于稳态时，连续谱的动力学表现为朗道阻尼现象。这种动力学现象在不同的背景下被重新发现，鉴于单波模型的规范形式特征，这并不奇怪。
英文摘要：
Pattern formation in biological, chemical and physical problems has received considerable attention, with much attention paid to dissipative systems. For example, the Ginzburg--Landau equation is a normal form that describes pattern formation due to the appearance of a single mode of instability in a wide variety of dissipative problems. In a similar vein, a certain "single-wave model" arises in many physical contexts that share common pattern forming behavior. These systems have Hamiltonian structure, and the single-wave model is a kind of Hamiltonian mean-field theory describing the patterns that form in phase space. The single-wave model was originally derived in the context of nonlinear plasma theory, where it describes the behavior near threshold and subsequent nonlinear evolution of unstable plasma waves. However, the single-wave model also arises in fluid mechanics, specifically shear-flow and vortex dynamics, galactic dynamics, the XY and Potts models of condensed matter physics, and other Hamiltonian theories characterized by mean field interaction. We demonstrate, by a suitable asymptotic analysis, how the single-wave model emerges from a large class of nonlinear advection-transport theories. An essential ingredient for the reduction is that the Hamiltonian system has a continuous spectrum in the linear stability problem, arising not from an infinite spatial domain but from singular resonances along curves in phase space whereat wavespeeds match material speeds (wave-particle resonances in the plasma problem, or critical levels in fluid problems). The dynamics of the continuous spectrum is manifest as the phenomenon of Landau damping when the system is ... Such dynamical phenomena have been rediscovered in different contexts, which is unsurprising in view of the normal-form character of the single-wave model.
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