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Energy cascades and condensation via coherent dynamics in Hamiltonian systems

Anxo Biasi, Patrick Gérard

TL;DR

This work provides analytic, deterministic descriptions of turbulence-like energy transfer in fully resonant Hamiltonian systems. It identifies two solvable families, the Z- and Y-Hamiltonians, that exhibit distinct finite-time cascades: a $-3/2$-cascade with coherent condensation to the ground mode and a dual transfer of energy to high modes, and a $-5/2$-cascade with a different spectral power law and structure formation. Central to the analysis is an invariant manifold reduction that collapses the dynamics to three complex amplitudes $(b,c,p)$ coupled to a generating function $F(t)$, enabling explicit expressions for spectra, Sobolev-norm blow-up, and position-space profiles, including a point-like spike and a cusp. The results connect to the cubic Szegő equation and its deformations, offering a rigorous framework to benchmark energy cascades and coherent-structure formation in nonlinear wave systems with potential relevance to NLS-type models and optics. The findings lay groundwork for generalizing to broader Hamiltonian families and motivate numerical studies to explore the prevalence and physical realizations of such cascades.

Abstract

This work makes analytic progress in the deterministic study of turbulence in Hamiltonian systems by identifying two types of energy cascade solutions and the corresponding large- and small-scale structures they generate. The first cascade represents condensate formation via a highly coherent process recently uncovered, while the second cascade, which has not been previously observed, leads to the formation of other large-scale structures. The concentration of energy at small scales is characterized in both cases by the development of a power-law spectrum in finite time, causing the blow-up of Sobolev norms and the formation of coherent structures at small scales. These structures approach two different types of singularities: a point discontinuity in one case and a cusp in the other. The results are fully analytic and explicit, based on two solvable families of Hamiltonian systems identified in this study.

Energy cascades and condensation via coherent dynamics in Hamiltonian systems

TL;DR

This work provides analytic, deterministic descriptions of turbulence-like energy transfer in fully resonant Hamiltonian systems. It identifies two solvable families, the Z- and Y-Hamiltonians, that exhibit distinct finite-time cascades: a -cascade with coherent condensation to the ground mode and a dual transfer of energy to high modes, and a -cascade with a different spectral power law and structure formation. Central to the analysis is an invariant manifold reduction that collapses the dynamics to three complex amplitudes coupled to a generating function , enabling explicit expressions for spectra, Sobolev-norm blow-up, and position-space profiles, including a point-like spike and a cusp. The results connect to the cubic Szegő equation and its deformations, offering a rigorous framework to benchmark energy cascades and coherent-structure formation in nonlinear wave systems with potential relevance to NLS-type models and optics. The findings lay groundwork for generalizing to broader Hamiltonian families and motivate numerical studies to explore the prevalence and physical realizations of such cascades.

Abstract

This work makes analytic progress in the deterministic study of turbulence in Hamiltonian systems by identifying two types of energy cascade solutions and the corresponding large- and small-scale structures they generate. The first cascade represents condensate formation via a highly coherent process recently uncovered, while the second cascade, which has not been previously observed, leads to the formation of other large-scale structures. The concentration of energy at small scales is characterized in both cases by the development of a power-law spectrum in finite time, causing the blow-up of Sobolev norms and the formation of coherent structures at small scales. These structures approach two different types of singularities: a point discontinuity in one case and a cusp in the other. The results are fully analytic and explicit, based on two solvable families of Hamiltonian systems identified in this study.

Paper Structure

This paper contains 25 sections, 3 theorems, 106 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Fully resonant Hamiltonian systems
  3. Energy cascades
  4. Condensation
  5. Main results
  6. Results on $\mathbf{-3/2}$-cascades and coherent condensation
  7. Results on $\mathbf{-5/2}$-cascades and structure formation
  8. The Z-Hamiltonian systems: $\mathbf{-3/2}$-cascades & coherent condensation
  9. Local well-posedness
  10. An invariant manifold
  11. General strategy to obtain explicit solutions
  12. Coherent condensation and dual-cascade behavior
  13. Explicit solutions representing the formation of condensates
  14. Dual cascade behavior and blow-up of Sobolev norms
  15. Structure formation in position space
  16. ...and 10 more sections

Key Result

Proposition 3.1

Let $\xi >3/2$ and $(\alpha_{n,0})_{n\geq 0}$ be a sequence of complex numbers such that There exists $\tau =\tau (R)>0$ and a unique sequence $(\alpha_n(t))_{n\geq 0}$ of $C^1$ functions on the interval $[-\tau ,\tau ]$ satisfying $\alpha_n(0)=\alpha_{n,0}$, the estimate and Equation eq:Resonant_Equation with eq:C_nmij_equation. Furthermore, the conservation laws eq:Conserved_equatities_N_E hol

Figures (4)

  • Figure 1: Analytic solution representing a $-3/2$-cascade in finite time $T$, and specifically, a process of coherent condensation. (a): Evolution of the first elements of the amplitude spectrum, observing $|\alpha_n(t)|^2 \to N \delta_{0,n}$. (b): Development of the power law $|\alpha_{n\gg 1}|^2\sim |c|^2 n^{-3/2}$, where $|c|^2$ decays to zero. (c, d): Time-evolution of the amount of conserved quantities, $N$ (c) and $E$ (d), stored at the range of modes indicated in the subscript. (e): Position space representation of the condensation process through $|u(t,\theta)|$ at several times. The spike-like structure has finite amplitude but narrows to a point $\theta^*$. Plot (e.6) shows a more detailed view around that point.
  • Figure 2: Analytic solution representing a $-5/2$-cascade in finite time $T$. (a): Evolution of the first elements of the amplitude spectrum. (b): Development of the power law $|\alpha_{n\gg 1}|^2\sim n^{-5/2}$. (c, d): Time-evolution of the amount of $N$ (c) and $E$ (d) stored at the range of modes indicated in the subscript. (e): Position space representation $|u(t,\theta)|^2$ at several times. A sharpening pointed corner arises at a point $\theta^*$. Plot (e.6) shows a more detailed view around that point. Note the different nature between the singularities emerging here and in Fig. \ref{['fig:All_Type_1']}(e.6).
  • Figure 3: Shapes of $V(F)$ for the motions present in the invariant manifold (\ref{['eq:invariant_manifold']}). The shaded areas represent values of $F$ out of $[0,F_c)$ (i.e., $x$ out of $[0,x_c)$). $F_{\text{min}}$ ($F_{\text{max}}$) represents the minimum (maximum) value of $F(t)$.
  • Figure 4: $E$-$S$ region where energy cascades take place (white area and black solid lines). In the white area $V(F_c)<0$, while $V(F_c)=0$ on the borders. Solid and dashed lines indicate $V'(F_c)>0$ and $V'(F_c)<0$, respectively. At black dots $V'(F_c)=0$. The red dots mark the values $(E,S)$ associated with the shapes of the potential presented on the right. We fixed $s=N=2$.

Theorems & Definitions (5)

  • Proposition 3.1
  • proof
  • Lemma 3.1
  • Proposition 4.1
  • proof