KAM-Type Theorem for Reversible Infinite-D Systems

• KAM-type theorem is a framework that guarantees the persistence of invariant tori and quasi-periodic motions in perturbed dynamical systems under specific non-resonance conditions.
• It employs rigorous analytic regularity, Diophantine inequalities, and an infinite-dimensional Newton–KAM iterative scheme to manage small divisors and ensure convergence.
• The theorem’s applications include nonlinear PDEs and reversible coupled non-linear Schrödinger systems, providing a robust explanation for the long-term spectral stability of quasi-periodic solutions.

The Kolmogorov–Arnold–Moser (KAM) type theorem describes the persistence of invariant tori and quasi-periodic motions in dynamical systems subject to small perturbations. While the classical KAM framework addresses Hamiltonian (symplectic, volume-preserving) systems, various extensions have been established for infinite-dimensional, reversible, and even dissipative or quantum contexts. Rigorous KAM results derive precise non-resonance (Diophantine) conditions, analytic and parameter regularity requirements, and explicit measure estimates on the surviving set of frequencies or actions. This family of theorems is essential for understanding the long-term stability structure in nonlinear partial differential equations (PDEs), statistical mechanics, quantum many-body systems, and celestial mechanics.

1. Foundations and Analytical Framework

A KAM-type theorem begins with an unperturbed integrable system in which dynamics are confined to invariant tori carrying quasi-periodic motion. For infinite-dimensional systems, the phase space involves both finite "tangential" and infinite "normal" modes, typically realized as weighted sequence spaces such as

$$\ell^2_p = \left\{ z = (z_j)_{j\in\mathbb Z^d} : \|z\|_p^2 = \sum_{j\in\mathbb Z^d}(1+|j|^2)^p |z_j|^2 < \infty \right\}.$$

The general setting introduces action–angle variables on a finite set of modes and complex coordinates on an infinite complement: $$\mathcal{P}_p = \mathbb{T}^n \times \mathbb{T}^m \times \mathbb{R}^n \times \mathbb{R}^m \times \ell^2_p \times \ell^2_p \times \ell^2_p \times \ell^2_p.$$

Reversibility plays a crucial role in these generalizations. The system admits an involution

$$S : (\theta, \phi, I, J, z, \bar{z}, w, \bar{w}) \mapsto (-\theta, -\phi, I, J, \bar{z}, z, \bar{w}, w),$$

and one considers vector fields $X$ satisfying reversibility: $DS\circ X + X\circ S = 0.$ The unperturbed normal form comprises linear flows in the actions and normal coordinates: $$N + A = \sum_{b=1}^{n} \omega_b(\xi) \partial_{\theta_b} + \cdots + \sum_{j\in L^c} (i\,\lambda_j(\xi) z_j\,\partial_{z_j} - i\,\lambda_j(\xi)\bar{z}_j\,\partial_{\bar{z}_j}) + \cdots$$ where the frequencies $\lambda_j(\xi)$ and $\mu_j(\xi)$ obey asymptotic expansions with small corrections.

2. Hypotheses and Non-resonance Conditions

KAM-type theorems require a hierarchy of structural assumptions:

• Non-degeneracy: The frequency map $\xi \mapsto (\omega(\xi), \Omega(\xi))$ must be a $C^W$-diffeomorphism onto its image.
• Spectral Asymptotics: Normal frequencies have the expansion

$$\lambda_j(\xi) = |j|^2 + \theta_j(\xi), \qquad \|\theta\|_{C^W} \leq L \ll 1,$$

ensuring a cluster structure supporting perturbative analysis.

• Diophantine Non-resonance: Frequencies satisfy inequalities of the form

$$|\langle k, \omega \rangle + \langle K, \Omega \rangle| \geq \frac{\gamma}{(|k| + |K|)^\tau}, \qquad \tau > d^{-1},$$

for nonzero integer multi-indices $(k, K)$ with bounded length, and for coupling with both tangential and normal frequencies:

$$|\langle k, \omega \rangle + \langle K, \Omega \rangle + \lambda_i | \geq \frac{\gamma}{(|k|+|K|)^\tau}, \quad \text{etc.}$$

• Analyticity and Regularity: The perturbing vector field $P$ is required to be real-analytic and small in a weighted analytic norm.
• Momentum Conservation: For PDEs on tori, $P$ must commute with generators of spatial translations,

$$M_l =\sum_{j\in\mathbb Z^d}j_l(z_j\partial_{z_j}-\bar z_j\partial_{\bar z_j} + w_j\partial_{w_j}-\bar w_j\partial_{\bar w_j}),$$

reflecting invariance under continuous symmetries.

• Töplitz–Lipschitz Property: High-frequency components of $P$ exhibit uniform tail estimates with exponential mode separation decay.

These hypotheses ensure that only finitely many resonances arise at each iteration and that measure-theoretic loss is controlled.

3. Infinite-Dimensional Newton–KAM Scheme

The proof employs a Newton–KAM iterative scheme extending classical finite-dimensional constructions:

1. Truncation: Decompose the perturbation as $P = R + Q$, isolating low-order Fourier modes in $R$.
2. Homological Equation: Find a generator $F$ solving

$[N + A, F] + R = [R],$

where $[R]$ is the average part of $R$. The equations to be solved have the form

$$\langle k, \omega \rangle + \langle K, \Omega \rangle \pm \lambda_i \pm \lambda_j \neq 0,$$

with small divisors controlled by Diophantine inequalities.

1. Canonical Coordinate Change: The time-1 flow of $F$,

$\Phi_F^1,$

is $S$-invariant and eliminates the leading low-order terms in the perturbation.

1. Iteration and Convergence: The process is iterated with geometrically decreasing parameters $(\gamma_v, \varepsilon_v, r_v, s_v)$, ensuring contraction of the perturbation size and convergence of the coordinate maps $\Psi_v$ to an analytic embedding.

The method exploits the Töplitz–Lipschitz regularity and momentum conservation to restrict the accumulation of resonances, maintaining a Cantor-like structure for the set of admissible parameter values.

4. Reversible Coupled Nonlinear Schrödinger Systems

A central application is to reversible coupled nonlinear Schrödinger (NLS) equations on the $d$-dimensional torus $\mathbb{T}^d$: $$\begin{cases} u_t - \Delta u + M_\varepsilon u + G_1(|u|^2, |v|^2) = 0 \ v_t - \Delta v + M_\varepsilon v + G_2(|u|^2, |v|^2) = 0 \end{cases}$$ where $G_i = o((|u| + |v|)^3)$ are real-analytic nonlinearities and $M_\varepsilon$ is a generic parameter.

Writing mode expansions for $u$ and $v$, and introducing action-angle variables for tangential modes and complex coordinates for normal modes, one recasts the system in the reversible normal form structure required by the abstract theorem. Each assumption (A1)–(A6) is verified: translation invariance (for momentum conservation), multilinear convolution structure (for Töplitz–Lipschitz), and generic parameters (for Diophantine non-resonance).

For a "Cantor family" of $\varepsilon$ of asymptotic density one, the result produces small-amplitude, quasi-periodic solutions: $$u(t, x) = \sum_{b=1}^n \sqrt{I_b}e^{i(\theta_b^0 + \widetilde{\omega}_b(\varepsilon)t + n^{(b)} \cdot x)} + O(s),$$

$$v(t, x) = \sum_{b=1}^m \sqrt{J_b}e^{i(\phi_b^0 + \widetilde{\Omega}_b(\varepsilon)t + m^{(b)} \cdot x)} + O(s),$$

with frequency corrections $O(s^2)$. The full analytic description of the invariant tori and the linear stability of these quasi-periodic orbits (all Lyapunov exponents in tangential directions equal to zero) are established. The spectrum of the linearized system is purely imaginary about each invariant torus.

5. Measure Estimates and Parameter Sets

The KAM iteration excludes only a small measure of "resonant" parameter values at each step. The final set

$\mathcal{O}_\gamma \subset \mathcal{O}$

(from which the quasi-periodic tori persist) satisfies

$$\mathrm{meas\,}(\mathcal{O} \setminus \mathcal{O}_\gamma) = O(\gamma^{1/4}),$$

so as $\gamma \to 0$, almost all parameter values are admissible. This "Cantor set" retains asymptotic density one as the perturbation amplitude tends to zero. The persistent tori are embedded as real-analytic maps, and the flow on each torus is conjugate to a linear rotation with slightly shifted frequencies.

6. Extensions and Significance

The KAM-type theorem for reversible infinite-dimensional systems generalizes the classical Hamiltonian KAM paradigm to a broader class of evolution equations, relevant for nonlinear Schrödinger equations and related models. The analysis is robust with respect to the presence or absence of Hamiltonian structure (i.e., applies to reversible systems that are not symplectic), and does not require the perturbation to possess extra symmetry beyond momentum conservation and the Töplitz–Lipschitz property. Linear stability (zero Lyapunov exponents) and an explicit, analytic description of the surviving tori are major features.

This mechanism explains the persistence and typical stability of quasi-periodic structures in large classes of PDEs and provides a rigorous foundation for spectral stability in reversible nonlinear dispersive systems, with direct implications for the theory of nonlinear waves on multi-dimensional tori and the fine structure of solution sets for nonlinear Schrödinger equations (Sun et al., 2019).