Normal Form Method in Dispersive PDEs

• Normal form method is a systematic technique that reorders nonlinear dispersive dynamics by converting quadratic and higher nonlinearities into more manageable higher-order terms.
• It is crucial for analyzing the Intermediate Long Wave (ILW) equation, enabling precise control of low-regularity nonlinear interactions and convergence to KdV/BO limits.
• The method employs iterative integration by parts in time and frequency modulation decomposition to achieve regularity gains and handle singular limits in dispersive PDEs.

The normal form method is a systematic analytic technique in the paper of nonlinear dispersive equations designed to restructure quadratic and higher nonlinearities into terms with more favorable mapping properties, often by infinite or high-order integration by parts in time. In the context of the Intermediate Long Wave (ILW) equation and its singular limits, the normal form method is used to achieve fine control of nonlinear interactions at low regularity—especially to show convergence in critical spaces, and to propagate regularity. The approach has become essential in modern PDE analysis, complementing tools from harmonic analysis, conservation laws, and integrable structure.

1. Principles of the Normal Form Method

The normal form method is rooted in classical dynamical system theory, where it serves to eliminate non-resonant nonlinear terms via near-identity changes of variables (normal form transformations), rendering the dynamics more tractable. In dispersive PDE, it takes the form of infinite or iterative integration by parts in time, designed to exploit oscillations generated by the linear evolution and move the nonlinearity to higher order, hence gaining regularity or decay. The reduction heavily relies on the resonance structure of the equation, encoded in phase functions $\Omega$ associated with the dispersive operator.

For the ILW equation and related models, the normal form method involves:

• Decomposing the nonlinear Duhamel term into modulation regimes.
• Applying either finite or infinite iteration of normal form reductions to shift the nonlinearity to higher order while controlling remainder terms.
• Exploiting the analytic properties of the linear symbol, including non-resonance, to integrate by parts in time and extract smoothing and decay.

2. Implementation for ILW and Singular Limits

The normal form method for ILW is crucial in addressing singular convergence problems—such as the shallow-water (ILW → KdV) and deep-water (ILW → BO) limits, especially at low regularity. The method proceeds in two main steps:

Step 1: Frequency Modulation Decomposition.

• Decompose interactions into nearly-resonant and highly non-resonant regimes, based on the size of the phase function (modulation) $\Xi_\delta$.
• Treat highly non-resonant pieces by repeatedly integrating by parts in time.

Step 2: Infinite Iteration or Poincaré-Dulac Normal Forms.

• Implement an infinite normal form expansion ("infinite normal form method") to reduce the remainder to arbitrarily high order in the nonlinearity.
• Each stage induces multilinear terms involving higher and higher powers of the initial data, facilitating adequate summability and smoothing effects.
• At each generation, the process splits between nearly resonant and genuinely non-resonant interactions.

This strategy is critical for controlling the difference between solutions of the scaled ILW equation and its limiting model (e.g., KdV), especially in the regime where direct energy methods or finite-order reductions are insufficient due to the singularity in the linear symbol at shallow water ($\delta \to 0$).

3. Normal Form Method in the Shallow Water Limit of ILW

In the $L^2$-framework, the paper “Shallow-water convergence of the intermediate long wave equation in $L^2$” (Chapouto et al., 19 Nov 2025) provides a definitive application of the normal form method. The main convergence theorem asserts

$$\lim_{\delta\to0}\;\|\,v_\delta-v\,\|_{C([-T,T];L^2_x)} = 0$$

for solutions $v_\delta$ of the scaled ILW and $v$ of KdV, both on $\mathbb{R}$ or $\mathbb{T}$ and for arbitrary $L^2$ data.

The proof splits into high- and low-frequency analysis:

• High frequencies: Controlled uniformly in $\delta$ using equicontinuity properties furnished by integrable structure (perturbation determinants built from the Lax pair).
• Low frequencies: The difference equation (for $w := v_\delta - v$) is handled via an infinite sequence of normal form reductions, each step integrating by parts in time to gain factors of the resonance (phase gap) in the denominator. Crucially, these factors regularize the nonlinearity and compensate for the singularity as $\delta \to 0$.

The normal form expansions guarantee that, after localizing in frequency, errors from replacing ILW by KdV can be made arbitrarily small, overcoming the lack of uniform symbol bounds.

4. Quantitative Description of the Methodology

At each normal form step, the Duhamel term for the low frequency part is manipulated as: $$\int_0^t e^{i(t-\tau)\Xi_\delta}\mathcal{N}(w)(\tau) d\tau = \int_0^t \frac{1}{i\Xi_\delta} \partial_\tau \left[e^{i(t-\tau)\Xi_\delta}\mathcal{N}(w)(\tau)\right] d\tau - \int_0^t \frac{1}{i\Xi_\delta}e^{i(t-\tau)\Xi_\delta} \partial_\tau \mathcal{N}(w)(\tau)d\tau$$ This process can be iterated, yielding error terms with improved regularity (higher order in data) and inverse powers of the resonance. The symbol $\Xi_\delta$ encodes the phase mismatch between interacting modes; its non-vanishing yields the required smoothness.

The infinite iteration, akin to the Poincaré–Dulac procedure, ultimately gives:

• Remainders of arbitrarily high order in a localized norm,
• Sufficient decay or smallness to conclude convergence after summation over all frequencies.

5. Relation to Integrability and Other Techniques

The normal form method is used in conjunction with integrable-structure-based controls (notably, conservation of the perturbation determinant) for ILW. While energy methods or direct dispersive estimates provide partial information, normal form reductions allow for unconditional $L^2$ convergence results, even in singular scaling limits where symbol degeneracy precludes uniform energy estimates (Chapouto et al., 19 Nov 2025).

More broadly, this method is a modern tool for dispersive equations where resonance structure and nonlinear smoothing are critical, and is applied to a wide class of models beyond ILW, including Benjamin-Ono, KdV, and non-integrable equations.

6. Significance and Scope

The normal form method yields rigorous control in low-regularity regimes and enables sharp singular-limit results. In the context of ILW:

• It is a key ingredient in the now-complete $L^2$-well-posedness and convergence theory for both deep-water and shallow-water limits (Chapouto et al., 19 Nov 2025).
• It is used to control the non-smooth perturbative differences between the exact model (ILW) and its limiting integrable cases (KdV/BO).
• It facilitates unconditional convergence results, independent of initial regularity conditions aside from membership in $L^2$.
• It provides a robust framework for handling similar dispersive-difference problems across integrable and non-integrable systems.

7. Further Developments and Open Directions

Infinite normal form reduction in dispersive PDE continues to be active, particularly in the pursuit of results at critical or near-critical regularity, and for broader classes of equations where integrability cannot be fully exploited. Its synthesis with harmonic analysis, spectral invariants, and renormalization techniques is expected to remain at the forefront of modern analysis of dispersive systems (Chapouto et al., 19 Nov 2025, Li, 2022).

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Key reference:

• "Shallow-water convergence of the intermediate long wave equation in $L^2$" (Chapouto et al., 19 Nov 2025)
• See also the analysis of low-regularity convergence limits in "Deep-water and shallow-water limits of the intermediate long wave equation" (Li, 2022).