In this study we investigate for the first time the formation of dynamical energy cascades in higher order KdV-type equations. In the beginning we recall what is known about the dynamic cascades for the classical KdV (quadratic) and mKdV (cubic) equations. Then, we investigate further the mKdV case by considering a richer set of initial perturbations in order to check the validity and persistence of various facts previously established for the narrow-banded perturbations. Afterwards we focus on higher order nonlinearities (quartic and quintic) which are found to be quite different in many respects from the mKdV equation. Throughout this study we consider both the direct and double energy cascades. It was found that the dynamic cascade is always formed, but its formation is not necessarily accompanied by the nonlinear stage of the modulational instability. The direct cascade structure remains invariant regardless of the size of the spectral domain. In contrast, the double cascade shape can depend on the size of the spectral domain, even if the total number of cascading modes remains invariant. The results obtained in this study can be potentially applied to plasmas, free surface and internal wave hydrodynamics.

中文翻译：

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四次和五次广义 KdV 方程中动态能量级联的形成

在这项研究中，我们首次研究了高阶 KdV 型方程中动态能量级联的形成。一开始，我们回忆一下关于经典 KdV（二次）和 mKdV（三次）方程的动态级联的已知内容。然后，我们通过考虑一组更丰富的初始扰动来进一步研究 mKdV 情况，以检查先前为窄带扰动建立的各种事实的有效性和持久性。之后，我们关注高阶非线性（四次和五次），发现它们在许多方面与 mKdV 方程完全不同。在整个研究中，我们考虑了直接和双能量级联。发现总是形成动态级联，但它的形成并不一定伴随着调制不稳定性的非线性阶段。无论谱域的大小如何，直接级联结构都保持不变。相比之下，双级联形状可能取决于谱域的大小，即使级联模式的总数保持不变。在这项研究中获得的结果可以潜在地应用于等离子体、自由表面和内波流体动力学。