Intermediate Long Wave Equation

Updated 21 November 2025

ILW is a nonlocal nonlinear integrable PDE that models bidirectional internal gravity waves at the interface of two fluid layers, interpolating between the Benjamin–Ono and KdV equations.
It features a rich mathematical structure with a Hamiltonian formulation, Lax pair representation, and an infinite hierarchy of conservation laws, shedding light on soliton dynamics and invariant measures.
The equation underpins diverse applications, from rigorous well-posedness and asymptotic analyses in Sobolev spaces to connections with quantum integrable systems and gauge theories.

The Intermediate Long Wave (ILW) equation is a nonlocal nonlinear integrable partial differential equation modeling bidirectional internal waves at the interface between two immiscible, incompressible fluid layers of different densities, with finite lower-layer depth and an infinitely deep upper layer. The equation interpolates between the Benjamin–Ono (BO) equation (infinite depth) and the Korteweg–de Vries (KdV) equation (shallow water), and is central in both mathematical analysis and the theory of integrable systems.

1. Mathematical Structure and Hamiltonian Formulation

The ILW equation on the periodic circle $\mathbb{T}$ (or the line) with depth parameter $\delta > 0$ is given by

$\partial_t u + 2u \partial_x u + \mathcal{T}_\delta(\partial_x^2 u) = 0, \qquad u|_{t=0} = u_0,$

where the nonlocal operator $\mathcal{T}_\delta$ is defined on Fourier modes $(n \neq 0)$ by

$\widehat{\mathcal{T}_\delta f}(n) = -i \left[ \coth(\delta n) - \frac{1}{\delta n} \right] n \, \widehat{f}(n).$

Equivalently, define $G_\delta(n) = -i (\coth(\delta n) - 1/(\delta n))$ , then $\mathcal{T}_\delta(\partial_x^2 u)$ has symbol $G_\delta(n)$ .

The equation is Hamiltonian: $H_\delta(u) = \frac{1}{2} \int_{\mathbb{T}} u G_\delta(\partial_x u) \, dx + \frac{1}{k+1} \int_{\mathbb{T}} u^{k+1} dx,$ for the generalized ILW (gILW) with nonlinearity $u^k\partial_x u$ , $k \in 2\mathbb{N}+1$ (Li et al., 2022).

2. Physical Derivation and Limiting Behaviors

The ILW equation arises from asymptotic analysis of the two-layer fluid system, specifically modeling interfacial long internal gravity waves with a rigid lid at the surface and a flat impermeable bottom (Cullen et al., 2020, Ivanov et al., 11 Jun 2025). Key scaling parameters are the thickness of the lower layer $h$ and typical wavelength $L$ , with $\delta = h/L$ . The equation models finite-depth effects through $\delta$ and recovers:

Benjamin–Ono (BO) limit ( $\delta \to \infty$ ): $\coth(\delta n) \to \operatorname{sgn}(n)$ , so $G_\delta(n) \to -i\,\operatorname{sgn}(n)$ and the ILW reduces to the BO equation,

$\partial_t u + 2u\partial_x u + \mathcal{H}(\partial_x^2 u) = 0,$

where $\mathcal{H}$ is the Hilbert transform (Li et al., 2022, Cullen et al., 2020).

KdV (shallow water) limit ( $\delta \to 0$ ): one rescales $v(t,x) = \delta^{1-k} u(\delta t, \delta x)$ . In this limit $G_\delta(\partial_x) \sim \partial_x^3 / 2$ and the equation becomes KdV,

$\partial_t v + 6v\partial_x v + \partial_x^3 v = 0$

(Li et al., 2022, Li, 2022, Cullen et al., 2020).

In physical variables (internal wave context), the model incorporates nontrivial effects of background currents, density stratification, and the possible presence of variable bottom topography. The coefficients of nonlinearity and dispersion are explicit algebraic functions of such parameters (Ivanov et al., 11 Jun 2025).

3. Integrability, Hierarchies, and Exact Solutions

The ILW equation is completely integrable, admitting a Lax pair representation and an infinite hierarchy of polynomial conservation laws (Saut, 2018, Buryak et al., 2018, Li et al., 2022). The simplest forms of the hierarchy use difference-differential Lax operators, with all higher Hamiltonians constructed via residues of powers of an appropriate Lax operator (Buryak et al., 2018).

Soliton Solutions: The ILW admits rapidly decaying soliton solutions in physical space, contrasted with algebraically decaying solitons for BO. On the periodic domain, explicit elliptic-function solutions are available (Saut, 2018, Shiraishi et al., 2009).
Lax Pair Structure: The Lax representation underlies the inverse scattering transform for ILW with rapidly decaying data, similar to KdV and BO. On the periodic domain, generalized elliptic difference operators are used (Buryak et al., 2018, Shiraishi et al., 2009, Zabrodin et al., 2017). The Sato–Lax formalism provides the integrable discretization and connection to the quantum theory of Macdonald difference operators.
Gauge Theory and Quantum ILW: The $\mathfrak{gl}_N$ ILW hierarchy is realized as a quantum integrable system arising from the hydrodynamic limit of the elliptic Calogero–Moser model, and is quantized via 6d $\mathcal{N}=1$ supersymmetric $U(N)$ gauge theory. Quantum ILW Hamiltonians correspond to chiral ring observables of the gauge theory, providing links to AGT correspondences and $W$ -algebras (Bonelli et al., 2014).

4. Well-Posedness and Limiting Rigidity

The ILW equation is globally well-posed on the real line and torus in classical Sobolev spaces $H^s$ for $s\geq 0$ , and recent results extend this to $s>-\frac{1}{2}$ on the torus (Gassot et al., 5 Jun 2025), with $s=-\frac{1}{2}$ being critical and sharp for both local and global well-posedness (Chapouto et al., 2023). Below this threshold, explicit counterexamples show ill-posedness.

Deep-water limit theorems are established in both $L^2$ and low-regularity Sobolev spaces: global solutions to ILW converge strongly (unconditionally for $s > 0.1277$ ) to those of the BO equation as $\delta \to \infty$ , both on the line and on the torus (Forlano et al., 11 Mar 2024, Chapouto et al., 2023, Gassot et al., 5 Jun 2025). The shallow-water convergence to KdV is also established in $L^2$ uniformly over the circle and the line, exploiting the recent development of an $L^2$ -level Lax pair and a normal-form procedure (Chapouto et al., 19 Nov 2025).

The main mechanism in low regularity is that ILW can be analyzed as a smoothing perturbation of BO. A precise gauge transform and normal-form reductions are adapted from BO theory to ILW, yielding global a priori bounds and unconditional well-posedness at the same threshold (Forlano et al., 11 Mar 2024, Chapouto et al., 2023).

5. Invariant Gibbs Measures, Statistical Equilibria, and Probabilistic Dynamics

ILW admits Gibbs measures constructed from its Hamiltonian and higher conservation laws. For the defocusing gILW with $u^k\partial_x u$ nonlinearity ( $k$ odd), the formal Gibbs measure

$d\rho_\delta(u) = Z_\delta^{-1} \exp(- H_\delta(u)) du$

is made rigorous by introducing a frequency cutoff and Wick renormalization, resulting in well-defined truncated Gibbs measures which converge in total variation for $\delta > 0$ (Li et al., 2022).

Deep-water convergence: As $\delta \to \infty$ , the Gibbs measure for ILW converges in total variation to the corresponding Gibbs measure for BO (Li et al., 2022, Chapouto et al., 10 Sep 2024).
Shallow-water regime: Under natural rescaling, the Gibbs measures for the scaled ILW converge only weakly (not in total variation) to the Gibbs measure for KdV, due to mutual singularity of the base Gaussian measures (Li et al., 2022, Chapouto et al., 10 Sep 2024).
Higher-order dynamics: Invariant measures associated with higher conservation laws (generalized Gibbs measures) exhibit the same limiting behavior, with a remarkable “2-to-1 collapse” of the ILW invariants to those of KdV in the shallow-water limit (Chapouto et al., 10 Sep 2024).
Dynamical invariance: For $k \geq 3$ , the invariant Gibbs measures for ILW are invariant under the infinite-dimensional ILW dynamics, and their dynamics converge (in law) to those of BO and KdV under the relevant limits (Li et al., 2022, Chapouto et al., 10 Sep 2024).

The focusing case (even $k$ or negative sign) is non-normalizable beyond the $L^2$ -critical threshold, and invariant measures can only be constructed with an $L^2$ -cutoff, paralleling focusing NLS theory (Li et al., 2022).

6. Asymptotic Dynamics, Long-Time Behavior, and Open Problems

The long-time dispersive behavior of ILW displays several robust phenomena:

Decay in weighted spaces: Sharp persistence properties in weighted Sobolev spaces have been rigorously established, with decay of local energy and polynomially weighted norms for any $|x|^a u$ with $0 < a \leq s$ given $u_0 \in H^s$ (Linares et al., 27 Jun 2024).
Local mass decay: Virial-type identities show that the local $L^2$ mass in regions $|x| < t^b$ , $b < 2/3$ , must decay along a diverging sequence of times. Complete decay in the far-field right region $x > C_0 t$ is also proved, in analogy with KdV and BO (Muñoz et al., 2019, Linares et al., 27 Jun 2024).
No breathers or nontrivial time-periodic solutions: There are no “breather” solutions moving with sub-soliton speed in either ILW or generalizations (Muñoz et al., 2019).
Propagation of regularity: Localized smoothness is preserved and propagates ballistically under the ILW flow (Muñoz et al., 2019).

Long-time asymptotic completeness (soliton resolution) and the construction of an IST-based global asymptotic theory remain open. A full IST for the periodic setting is outstanding, despite the formal availability of a Lax pair and infinite hierarchy (Saut, 2018, Buryak et al., 2018). The dynamics in negative Sobolev spaces are understood up to $s = -1/2$ : the flow is not well-posed below this threshold due to lack of continuity for the initial-value map (Chapouto et al., 2023).

Nonchiral and multi-component systems: Nonchiral ILW models describe edge-wave interactions in quantum Hall systems, leading to coupled nonlocal and parity-symmetric systems with explicit N-soliton solutions controlled by hyperbolic or elliptic Calogero–Moser–Sutherland dynamics (Berntson et al., 2020).
Discrete Laplacian and elliptic deformations: Several generalizations replace the original kernel with elliptic functions or Weierstrass functions, yielding operators with discrete Laplacians and deep connections to quantum integrable models and Poisson algebras related to Macdonald operators (Shiraishi et al., 2009, Zabrodin et al., 2017).
Hydrodynamic origins and gauge theories: Quantum ILW arises as the hydrodynamic limit of the elliptic Calogero–Moser model and appears naturally in geometry and gauge theory, linking partition functions of supersymmetric gauge theories and $W$ -algebras (Bonelli et al., 2014).