ILW Perturbation Determinant

• ILW perturbation determinant is a spectral invariant derived from operator and scattering theory, linking the logarithm of the transmission coefficient to infinite conservation laws.
• Its operator-theoretic formulation uses a renormalized series and Hilbert–Schmidt estimates, ensuring analytic convergence under appropriate smallness conditions.
• The determinant is pivotal for establishing uniform well-posedness and high-frequency control, which underpin convergence in the shallow-water regime.

The perturbation determinant for the intermediate long wave (ILW) equation is a spectral invariant constructed via operator-theoretic and scattering-theoretic approaches, playing a central role in the analysis of integrability, conservation laws, and high-frequency control for the ILW dynamics. It encodes the "logarithm of the transmission coefficient" associated with the ILW Lax pair, generating infinitely many conservation quantities and enabling rigorous arguments for $L^2$ convergence in shallow-water regimes. The perturbation determinant formalism, following the construction by Harrop-Griffiths, Killip, and Vișan (2025), underlies the recent advances in establishing uniform well-posedness, equicontinuity, and convergence results for the ILW equation on both the real line and the circle (Chapouto et al., 19 Nov 2025).

1. Lax Pair Formulation and Scattering Theory

The ILW equation on $M = \mathbb{R}$ or $\mathbb{T}$ with depth parameter $\delta > 0$,

$u_t - \mathcal{G}_\delta^2 u_x = (u^2)_x,$

admits a Lax pair representation with spectral parameter $\lambda > 0$. The Lax operator is

$$L_u(\lambda) = -i\partial_x + \tfrac{1}{2}(e^{2i\lambda D} - 1) + u(x), \quad D = -i\partial_x.$$

The associated Lax equation,

$$\frac{d}{dt}L_{u(t)}(\lambda) = [P_{u(t)}(\lambda), L_{u(t)}(\lambda)],$$

implies isospectrality of $L_{u(t)}(\lambda)$ under the ILW flow.

Within the scattering formalism, $L_u(\lambda) \psi = \lambda\psi$ is solved with asymptotics $\psi \sim e^{\pm ix\xi}$, leading to a $2 \times 2$ scattering matrix $S(\lambda)$, whose $(1,1)$-element defines the transmission coefficient $T(\lambda)$. The Fredholm theory yields

$$\frac{\det(L_u(\lambda) - \lambda)}{\det(L_0(\lambda) - \lambda)} = T(\lambda)^{-1},$$

and thus the logarithmic perturbation determinant,

$$\log\det(L_0(\lambda)^{-1}(L_0(\lambda) + u)) = -\log T(\lambda),$$

which equates the spectral shift formalism to the transmission coefficient.

2. Operator-Theoretic Definition

The modern approach, following Harrop-Griffiths, Killip, and Vișan, defines the free resolvent

$R_\delta(\lambda) = (L_0(\lambda) + \lambda)^{-1}.$

On the Fourier side with $D \mapsto \xi$,

$$a_\delta(\xi) = \xi + \tfrac{1}{2}(e^{-2\xi\delta} - 1),$$

yields

$$R_\delta(\lambda) = \left(-i\xi + \tfrac{1}{2}(e^{2i\lambda\xi} - 1) + \lambda\right)^{-1}.$$

Defining $$A(\lambda;u) = \sqrt{R_\delta(\lambda)}\, u\, \sqrt{R_\delta(\lambda)}$$, which is Hilbert–Schmidt under suitable bounds on $u$, the renormalized logarithmic perturbation determinant is

$$\Xi(\lambda;u) = \sum_{j=2}^\infty \frac{(-1)^{j-1}}{j} \operatorname{tr}[A(\lambda;u)^j].$$

This series matches (up to convention) $-\log T(\lambda)$, robustly linking spectral and integrability structures.

3. Analytic and Spectral Properties

The $j=2$ term of $\Xi(\lambda;u)$ is given by

$$\frac{1}{2} \operatorname{tr}[A(\lambda;u)^2] = \frac{1}{2}\|A(\lambda;u)\|_{I_2}^2,$$

where

$$\|A(\lambda;u)\|_{I_2}^2 = \int_M F(\xi;\lambda) |u(\xi)|^2 d\xi,$$

and

$$F(\xi;\lambda) = \frac{1}{2\pi} \int_M \frac{d\eta}{(a_\delta(\eta) + \lambda)(a_\delta(\xi+\eta) + \lambda)}$$

is a positive weight multiplier with two-sided asymptotics,

$$F(\xi;\lambda) \sim \frac{\sqrt{1+\lambda}/\lambda + \log(1 + |\xi|/(1+\lambda))}{\xi^2/(1+|\xi|) + \lambda},$$

and $\lim_{\lambda \to 0} F(\xi;\lambda) = 1/\xi^2$ for each $\xi$. Each term $\operatorname{tr}[A(\lambda;u)^j]$ is analytic in $\lambda > 0$ and convergent when $\|A(\lambda;u)\|_{I_2} < 1$, facilitated by appropriate $H^s$-norm smallness conditions on $u$ and sufficiently large $\lambda$ domains.

4. Conservation and High-Frequency Control

If $\|A(\lambda;u(0))\|_{I_2}^2 < 1/36$, then $\Xi(\lambda;u(t))$ is conserved under the ILW evolution. Since the leading term is a weighted $L^2$ mass, integrating in $\lambda$ over dyadic subsets,

$$\int_{|\xi|>N} |\hat u(\xi)|^2 d\xi \lesssim \int_{\lambda_1}^{\lambda_2} \operatorname{tr}[A(\lambda;u)^2]\,w(\lambda)\,d\lambda,$$

controls the high-frequency portion of $u$. The resulting "weakly uniform" equicontinuity bound,

$$\sup_{0 < \delta \leq \delta_0} \sup_{t \in \mathbb{R}} \|P_{>N} u_\delta(t)\|_{L^2_x} \to 0 \quad(N \to \infty),$$

where $P_{>N}$ is the high-frequency projection, ensures uniform control of high-frequency tails for all solutions $u_\delta$ in the vanishing-depth ILW system. This property is pivotal for establishing shallow-water limit convergence in $L^2$ (Chapouto et al., 19 Nov 2025).

5. Relation to Conserved Quantities and Integrability

Because $\Xi(\lambda;u)$ is conserved for each $\lambda$ within the admissible range, its Laurent (or Taylor) expansion,

$\Xi(\lambda;u) = \sum_{n \geq 0} c_n(u) \lambda^n,$

generates the full sequence of ILW conservation laws. Specifically, the leading $j=2$ term recovers a weighted $L^2$ norm, while higher $j \geq 3$ encode nonlinear conserved quantities, matching the hierarchy of ILW Hamiltonians (mass, momentum, energy, etc.).

6. Summary of Definition and Mathematical Role

In the ILW equation, the perturbation determinant is most succinctly defined as

$$\Xi(\lambda;u) = \sum_{j=2}^\infty \frac{(-1)^{j-1}}{j}\; \operatorname{tr} \left[\left(\sqrt{R_\delta(\lambda)}\,u\,\sqrt{R_\delta(\lambda)}\right)^j\right],$$

with

$$R_\delta(\lambda) = \left(-i\partial_x + \frac{1}{2}(e^{2i\lambda \partial_x} - 1) + \lambda\right)^{-1}.$$

This operator coincides with the logarithm of the transmission coefficient defined via Fredholm determinant and Jost solutions. It is a time-invariant under the ILW flow, enables a priori bounds on high-frequency modes, and is instrumental in proving well-posedness, high-frequency equicontinuity, and precise $L^2$ convergence from ILW to KdV in the shallow-water regime (Chapouto et al., 19 Nov 2025).