Inviscid Stationary Internal Wave Operator

• The inviscid stationary internal wave operator is a differential/pseudodifferential operator that governs steady-state internal wave fields in stratified, inviscid fluids with complex boundary conditions.
• It incorporates spectral analysis, Hamiltonian formulations, and Dirichlet–Neumann operator theory to address linear and nonlinear dynamics in layered fluid systems.
• Its study underpins reductions to classical models like KdV, Benjamin–Ono, and ILWE, offering insights into resonance, scattering, and stability in geophysical applications.

An inviscid stationary internal wave operator is a linear or nonlinear differential (or pseudodifferential) operator that governs steady-state (time-independent) internal wave fields in stratified, incompressible, inviscid fluids—often in a multilayer configuration—subject to appropriate physical boundary conditions. The structure, spectral theory, and bifurcation properties of these operators are central in the existence theory and qualitative dynamics of internal waves in oceans, laboratory tanks, and geophysical models. Such operators arise in Hamiltonian formulations of the Euler equations, in stationary limits of Boussinesq-type models, and as boundary value problems in domains with complex geometries or variable topography.

1. Mathematical Formulation in Layered Inviscid Fluids

In stratified systems composed of two immiscible, incompressible, and inviscid fluid layers, one typically works with the steady Euler equations under gravity and possibly interfacial tension, in a moving frame with constant horizontal speed. The velocity potentials $\phi_1, \phi_2$ in the upper and lower layers solve Laplace equations in their respective domains, bounded by rigid, flat, or variable boundaries and by an internal interface at $y=\eta(\mathbf{x})$, where $\mathbf{x}=(x_1, ..., x_{d-1})$: $$\Delta \phi_1 = 0,\quad \eta(\mathbf{x}) < y < h_1, \qquad \Delta \phi_2 = 0,\quad -h_2 < y < \eta(\mathbf{x})$$ Boundary conditions include rigid-lid or free-surface conditions at top/bottom, kinematic and dynamic (Bernoulli) matching at the interface:

• Kinematic: continuity of normal velocity,
• Dynamic: pressure continuity (with interfacial tension, surface tension terms appear as curvature operators).

The full steady problem is an infinite-dimensional Hamiltonian system on a phase space of interface elevation and layer potentials, with a symplectic structure and a Hamiltonian given by the total energy (including surface tension) (Nilsson, 2018).

2. Linearization and Spectral Analysis of the Stationary Operator

Linearization about the quiescent flat-interface equilibrium yields a stationary internal wave operator: $L u = \lambda u,$ where $u$ collects interface and potential perturbations. For spatial dynamics in an unbounded coordinate (e.g., $x$), the linearized operator $L$ may have the structure:

• Differential/pseudodifferential formulation in $x$ (evolutionary in $x$ as a "time-like" variable),
• Operator coefficients depending on physical parameters (density ratio $\rho_1/\rho_2$, depths $h_1, h_2$, gravity $g$, interfacial tension $\sigma$, propagation angles).

In multi-dimensional settings, $L$ typically couples the interface elevation to layer potentials through Dirichlet–Neumann operators, leading to a nonlocal (Fourier-multiplier) character (Nilsson, 2018, Ivanov et al., 11 Jun 2025).

The spectrum of $L$ is determined by a dispersion relation involving spatial eigenvalues and Fourier modes. The intersection structure of lines $Q_k$ (determined by wavenumber and propagation angles) with the dispersion curve $D=0$ delineates the possible bifurcation scenarios and thus the nature of stationary wave solutions.

3. Hamiltonian and Dirichlet–Neumann Operator Structure

The dynamics are governed by a Hamiltonian functional $H[\eta, \xi]$, where $\eta$ is the interfacial elevation and $\xi$ a canonically conjugate momentum built from interface traces of layer potentials: $$H[\eta, \xi] = \frac{1}{2}\int_\mathbb{R} \xi\, G B^{-1} G_1\,\xi\,dx + \frac{g(\rho-\rho_1)}{2} \int \eta^2 dx + \cdots,$$ with $G, G_1$ the Dirichlet–Neumann operators for lower and upper layers (self-adjoint, positive definite, nonlocal Fourier multipliers), and $B = \rho_1 G + \rho G_1$ (Ivanov et al., 11 Jun 2025, Compelli et al., 2018). The stationary operator arises as the functional derivative $\delta H/\delta \eta$ set to zero. For weakly nonlinear, unidirectional regimes, this procedure yields integrable or near-integrable models such as the Intermediate Long Wave Equation (ILWE), Benjamin–Ono (BO), and Korteweg–de Vries (KdV), where the operator is a nonlocal pseudodifferential operator encoding the internal-wave dispersion and stratification.

4. The Stationary Internal Wave Operator in Channel and Geometric Settings

For internal waves in 2D or 3D channels with variable or subcritical topography, the stationary internal wave operator arises as a second-order differential operator: $$L u = \bigl[ (1-\lambda^2) \partial_{x_2}^2 - \lambda^2 \partial_{x_1}^2 \bigr] u,$$ with Dirichlet boundary conditions. The outgoing radiation condition is characterized by microlocal frequency projections onto outgoing/incoming modes at the channel ends. The operator's resolvent structure and spectral properties (absolutely continuous spectrum, C¹ density of the spectral measure) are proved via limiting absorption principles and boundary-layer reductions (Li et al., 2024, Li et al., 24 Sep 2025).

The associated scattering matrix is constructed via geometric optics and billiard maps, and its difference from the pure pullback operator encodes diffractive and non-geometric corrections through a smoothing operator.

5. Bifurcation, Center Manifolds, and Nonlinear Analysis

In three-dimensional settings with periodicity and/or nontrivial topology, the stationary internal wave operator admits center-manifold reductions at bifurcation points associated with resonances (e.g., $00(\mathrm{i}s)(\mathrm{i}\kappa_0)$ resonance, Hamiltonian–Hopf bifurcation), giving rise to reduced finite-dimensional Hamiltonian systems (Nilsson, 2018). Center-manifold theory and normal-form analysis establish the existence of families of stationary solutions:

• Doubly periodic waves (spatially periodic in two directions),
• Homoclinic or multipulse solutions (solitary waves with localized profiles in one direction and periodic in another).

The dimension of the center manifold is dictated by the multiplicity of imaginary-axis eigenvalues in the linearized spectrum.

6. Operator Theory, Fredholm Properties, and Physical Implications

The inviscid stationary internal wave operator exhibits key mathematical properties:

• Self-adjointness (or, for certain Dirichlet–Neumann reductions, nearly self-adjoint structure),
• Ellipticity on suitably deformed (e.g., complex-scaled) domains when Morse–Smale billiard dynamics hold,
• Fredholm index zero in natural function spaces; invertibility established by explicit boundary-layer analysis and scattering theory (Li et al., 2024, Jézéquel et al., 12 Jan 2026).

Under Morse–Smale dynamical assumptions for billiard reflections in bounded domains, uniform invertibility and resonance-freeness for these operators are established for frequencies near the physical forcing frequency, both in the inviscid limit and with vanishing viscosity (Jézéquel et al., 12 Jan 2026).

These properties guarantee well-posedness of stationary wave problems, dictate the stability of internal-wave attractors, and underlie the spectral and long-time behavior of forced internal wave fields.

7. Asymptotic Regimes and Reductions to Classical Models

In long-wave and shallow/deep regimes, the inviscid stationary internal wave operator reduces, after appropriate scaling and expansion of Dirichlet–Neumann operators, to classical integrable models:

• Benjamin–Ono operator: Appears in the deep-lower-layer, long-wave regime as a pseudodifferential operator with $|D|$ (Hilbert transform) structure (Compelli et al., 2018, Ivanov et al., 11 Jun 2025).
• KdV operator: Emerges in the shallow water regime from the expansion of nonlocal multipliers.
• ILWE operator: Governs intermediate long waves and interpolates between the above limits.

Parametric dependence on stratification, depth, and bottom topography is explicit, and variable-coefficient generalizations are derived for slowly varying bathymetry. The precise operator structure determines dispersion, nonlinearity, and solitonic properties of stationary internal wave solutions in these asymptotic frameworks (Ivanov et al., 11 Jun 2025).