Pure Capillary Solitary Waves

Updated 8 February 2026

Pure capillary solitary waves are spatially localized traveling solutions where surface tension is the only restoring force, omitting gravity.
They do not exist in classical irrotational Euler flows but arise when modifications such as constant vorticity, compressibility, or nonlocal dispersion are introduced.
Analytical techniques like conformal mapping, Hamiltonian reduction, and spectral analysis elucidate the critical role of dispersion-modifying mechanisms in wave formation.

Pure capillary solitary waves are spatially localized, traveling solutions to fluid interface equations in which surface tension is the sole restoring force and gravity is neglected ( $g=0$ ). They arise as a theoretical question in the classification of nonlinear water waves but display a sharp dichotomy across different physical regimes: nonexistence in classical two-dimensional Euler flows under irrotational conditions, but existence when the governing equations or physical setting are modified—such as allowing for constant vorticity, considering compressible models, or employing nonlocal dispersive approximations like the Whitham equation.

1. Mathematical Formulation in Eulerian and Holomorphic Variables

The classical setting is a two-dimensional, incompressible, inviscid, irrotational fluid, with either infinite or finite depth, bounded above by a free interface. The fields of concern are the free-surface elevation $\eta(x,t)$ and the velocity potential $\phi(x,y,t)$ , harmonic in the fluid domain. Surface tension $\sigma>0$ replaces gravity as the only restoring force, with the governing Eulerian system:

$\partial_t \eta = G(\eta)\, \psi$

$\partial_t \psi + \frac{1}{2} \left(\phi_x^2 + \phi_y^2\right)_{y=\eta} - \sigma\, \mathcal{H}(\eta) = 0$

where $\psi(x,t) = \phi(x, \eta(x,t),t)$ , $G(\eta)$ is the Dirichlet–Neumann operator, and $\mathcal{H}(\eta) = \partial_x \left( \frac{\eta_x}{\sqrt{1+\eta_x^2}} \right)$ is the free-surface curvature (Ifrim et al., 2018, Ifrim et al., 2021).

For traveling-wave (solitary-wave) solutions of the form $\eta(x,t) = \eta(\xi)$ , $\psi(x,t) = \Psi(\xi)$ with $\xi = x - ct$ and asymptotic decay as $|\xi| \to \infty$ , a nonlocal, fully nonlinear ODE system must be satisfied.

A powerful reformulation utilizes conformal mapping to holomorphic (complex) coordinates, introducing a variable $W(\alpha) = Z(\alpha) - \alpha$ such that the interface is parametrized via $Z(\alpha)$ (with $W(\alpha)$ holomorphic in $\Im \alpha < 0$ ). In this setting, solitary-wave existence reduces to the solvability of a capillary Babenko-type equation, sometimes further reduced via logarithmic transformations for analytic treatment.

2. Nonexistence in Irrotational Euler Water Waves

In both infinite and finite depth, rigorous proofs establish nonexistence of nontrivial, localized, irrotational pure capillary solitary waves in two dimensions. The principal results are:

Infinite Depth (Irrotational): Every decaying solution to the steady capillary water-wave equation reduces (via a holomorphic/commutator estimate centered on the Babenko equation) to the trivial flat interface $\eta \equiv 0$ (Ifrim et al., 2018). The core proof uses a commutator estimate (Coifman–Meyer) to show that any decaying solution to the key equation must vanish identically.
Finite Depth (Irrotational): By extending the conformal mapping and employing the “Tilbert transform” (the finite-depth analog of the Hilbert transform), Ifrim, Pineau, Tataru, and Taylor derived a Pohozaev-type identity (energy-multipliers with large-scale cutoffs). This again forces $U \equiv 0$ (in logarithmic variables) for decaying solutions, yielding nonexistence (Ifrim et al., 2021). The methodology elegantly closes the existence/nonexistence problem for solitary waves in 2D Euler capillary flows (without gravity) for both depths.

These results contrast with the situation for gravity–capillary waves ( $g>0$ , $\sigma>0$ ), pure gravity waves in finite depth, or in higher dimensions, where existence can arise under distinct mechanisms.

3. Existence under Constant Vorticity

The addition of constant vorticity ( $\gamma \neq 0$ ) fundamentally alters the spectral properties and allows for solitary wave families even in the pure capillarity regime:

Finite Depth: For the two-dimensional, finite-depth Euler equations with constant vorticity, the spatial-dynamics/Centre manifold methodology (Hamiltonian reduction and normal-form expansion) yields a KdV-type profile equation on a two-dimensional reversible center manifold. When $\gamma d/c \approx 1$ and $\sigma/(c^2d) > 1/3$ , there exists a unique (up to translation) $C^2$ solitary-wave profile with $\sech^2$ leading-order shape. This is the precise parameter regime in which solitary pure capillary vorticity enables solitary waves (Hsiao et al., 1 Feb 2026).
Infinite Depth: For constant vorticity in infinite depth, the (generalized) Babenko equation incorporates quadratic and cubic vorticity-induced terms. Asymptotic reduction near a critical carrier velocity $c_*$ translates to a focusing cubic NLS profile, admitting a unique family of small-amplitude envelope-carrier solitary waves. The leading order envelope is given by a $\sech$ function, modulated by carrier frequency $\omega = -(2\gamma^2/\sigma)^{1/3}$ (Rowan et al., 2024).
Physical Mechanism: Vorticity shifts the linear dispersion relation so that a bifurcation to localized nonlinear solitary solutions becomes possible. For small-amplitude waves near the critical curve in parameter space, this bifurcation is governed by reversible Hamiltonian ODE dynamics (KdV or NLS normal form), with the solitary envelope shrinking and oscillating more rapidly with increasing $|\gamma|$ .

4. Pure Capillary Solitary Waves in Non-Eulerian Models

Compressible Fluids and Euler–Korteweg Model

The Euler–Korteweg system, modeling compressible capillary fluids, admits 1D solitary waves (solitons) as homoclinic orbits to a Hamiltonian ODE for the density profile, provided a speed regime (“saddle-point condition”) is met. These solitary waves are nontrivial classical solutions localized in the primary direction (Paddick, 2015). However, they are subject to nonlinear transverse instability: Small $y$ -dependent perturbations with a critical range of transverse wavenumbers grow exponentially, eventually destroying the soliton profile in the transverse direction.

Nonlocal Dispersive Approximations: Whitham Equation

For the (nonlocal, weakly nonlinear) capillary Whitham equation with surface tension and zero gravity ( $g=0,\, T>0$ ):

$u_t + \partial_x M_T u + \tfrac{1}{2} u \partial_x u = 0$

where $M_T$ is a Fourier multiplier operator with symbol $m_T(k) = \sqrt{(\tanh k)/k \cdot (1 + Tk^2)}$ , constructive computer-assisted proof yields spectrally stable solitary waves for a range of parameters (Cadiot, 2024). Existence follows via radii-polynomial/Newton-Kantorovich iteration around numerically computed (finite-mode) approximates, with full validation of analytic and spectral properties.

5. Summary Table: Existence and Nonexistence Results

Model / Regime	Solitary Wave Existence?	Key Reference
2D Euler, irrotational, $g=0$ , infinite depth	No	(Ifrim et al., 2018)
2D Euler, irrotational, $g=0$ , finite depth	No	(Ifrim et al., 2021)
2D Euler, constant vorticity, finite depth	Yes, for $\gamma$ near critical, $\sigma/(c^2d)>1/3$	(Hsiao et al., 1 Feb 2026)
2D Euler, constant vorticity, infinite depth	Yes, NLS regime	(Rowan et al., 2024)
Euler–Korteweg model (compressible)	Yes, for each $c$ in certain regime	(Paddick, 2015)
Capillary Whitham equation ( $T>0$ )	Yes, unique, stable	(Cadiot, 2024)

Existence refers to genuine, spatially decaying solitary/traveling-wave solutions. No indicates rigorous nonexistence under stated regularity and decay assumptions.

Several unresolved avenues and extensions are highlighted in the literature:

Three-Dimensional Flows: Existence/nonexistence of pure capillary solitary waves in 3D remains unresolved; conformal mapping tools are not directly applicable (Ifrim et al., 2021).
Nontrivial Vorticity Distributions: The generality of the existence mechanism for variable (non-constant) vorticity is not completely classified.
Alternative Surface Energies: Modifications to the surface energy functional or boundary conditions may alter existence theory.
General Regularity/Decay: Whether weakening regularity or decay hypotheses yields “almost-solitary” or localized nontrivial profiles.
Stability: Spectral and nonlinear stability properties of the constructed solutions, especially outside 1D or allowing strong vorticity.

7. Comparative Context and Physical Implications

The landscape of solitary waves in the pure capillarity regime demonstrates a sharp sensitivity to physical and mathematical constraints:

In classical irrotational Euler water waves, both infinite and finite depth cases prohibit pure capillary solitary waves, in stark contrast to the well-known solitary waves for pure gravity or gravity-capillary regimes.
The existence with constant vorticity underscores the critical role of background shear in modifying the spectral and Hamiltonian landscape, allowing for persistent localized structures.
Non-Eulerian (e.g., Whitham, Euler–Korteweg) and approximate models accommodate solitary profiles, but may lack some physical constraints of full water-wave theory.
In all existence cases, solution amplitude and width exhibit precise scalings with small parameters near bifurcation curves; for instance, in the constant vorticity regime, $c-c_*\propto A^2$ , envelope widths scale as $1/\epsilon$ , and carrier frequencies are set by combinations of $\gamma$ and $\sigma$ .

Pure capillary solitary waves thus exist only via dispersion-modifying mechanisms or in extended model classes, with the classical 2D irrotational Euler system admitting none. The analytical methods developed (commutator estimates, Hamiltonian reduction, implicit function constructions) have established a rigorous and nearly complete classification in two dimensions.