论文标题
在立方szego方程和相关系统的截断中湍流的级联
Turbulent cascades in a truncation of the cubic Szego equation and related systems
论文作者
论文摘要
从Gerard和Grellier的基础工作开始，已经研究了Cubic Szego方程作为确定性湍流的可集成模型。我们引入了该方程式的截断版本，其中消除了大多数傅立叶模式耦合，同时保留了模型的签名特征，即宽松对结构和有限维度动态不变的歧管的嵌套层次结构。尽管相互作用的结构贫穷，但我们新方程的动荡行为在适当的意义上比原始的立方szego方程更强。我们构建明确的分析解决方案，显示Sobolev规范的指数增长。此外，我们还引入了一个模型家族，该模型在我们的截短系统和原始的立方Szego方程之间插值，以及其他一些相关的变形。所有这些模型都具有松弛的对，不变的歧管，并显示了各种湍流级联。我们还提到数值证据，这些证据表明，在某些不同的，紧密相关的动态系统中，有限时间爆破的形式更强大。
The cubic Szego equation has been studied as an integrable model for deterministic turbulence, starting with the foundational work of Gerard and Grellier. We introduce a truncated version of this equation, wherein a majority of the Fourier mode couplings are eliminated while the signature features of the model are preserved, namely, a Lax-pair structure and a nested hierarchy of finite-dimensional dynamically invariant manifolds. Despite the impoverished structure of the interactions, the turbulent behaviors of our new equation are stronger in an appropriate sense than for the original cubic Szego equation. We construct explicit analytic solutions displaying exponential growth of Sobolev norms. We furthermore introduce a family of models that interpolate between our truncated system and the original cubic Szego equation, along with a few other related deformations. All of these models possess Lax pairs, invariant manifolds, and display a variety of turbulent cascades. We additionally mention numerical evidence that shows an even stronger type of turbulence in the form of a finite-time blow-up in some different, closely related dynamical systems.
