Whitham Equation: Nonlocal Water-Wave Model

• The Whitham equation is a nonlocal dispersive wave equation that combines quadratic nonlinearity and the full linear dispersion relation of gravity water waves.
• Its formulation uses a Fourier multiplier to reproduce the exact phase speed, enabling analysis of traveling waves, cusped highest waves, and wave breaking.
• The model underpins studies on bifurcation, modulational instability, and extensions to capillary-gravity and bidirectional systems in dispersive PDE research.

Searching arXiv for recent and foundational papers on the Whitham equation to ground the article in cited sources. The Whitham equation is a nonlocal dispersive wave equation introduced as a model for unidirectional surface water waves that combines the quadratic nonlinearity of shallow-water theory with the full linear dispersion relation of the water-wave problem. In one standard form it is written as

$u_t + \mathcal{K}*u_x + \frac{3}{2}uu_x = 0,$

with Fourier multiplier symbol

$$\widehat{\mathcal{K}}(k)=\sqrt{\frac{\tanh(k)}{k}},$$

or, in closely related normalizations, as

$u_t + (K*u+u^2)_x=0$

with $m(\xi)=\frac{\tanh \xi}{\xi}$ in the traveling-wave formulation (Moldabayev et al., 2014, Truong et al., 2020). Its central mathematical feature is the replacement of local dispersive terms such as $\eta_{xxx}$ by a nonlocal convolution operator whose symbol reproduces the exact unidirectional linear phase speed for gravity waves. This construction places the equation between asymptotic long-wave models such as KdV and the full Euler equations: it retains a simple quadratic steepening mechanism while admitting phenomena associated with weaker high-frequency dispersion, including cusped highest waves, wave breaking, and a rich global bifurcation structure (Ehrnstrom et al., 2016, Hur, 2015).

1. Definition and model structure

The Whitham equation was proposed as an alternate model equation for the simplified description of uni-directional wave motion at the surface of an inviscid fluid (Moldabayev et al., 2014). In dimensionless variables one widely used form is

$u_t + \mathcal{K}*u_x + \frac{3}{2}uu_x = 0,$

where the dispersive operator is a Fourier multiplier with symbol

$$\widehat{\mathcal{K}}(k)=\sqrt{\frac{\tanh(k)}{k}}.$$

The same equation is also written as

$u_t + 2uu_x + Lu_x = 0,\qquad Lf=K*f,$

with kernel

$$K(x)=\frac{1}{2\pi}\int_{-\infty}^{\infty}m(\xi)e^{ix\xi}\,d\xi,\qquad m(\xi)=\sqrt{\frac{\tanh \xi}{\xi}},$$

and, in a solitary-wave setting after integration,

$c\varphi-K*\varphi-\varphi^2=0$

or equivalently

$$\widehat{\mathcal{K}}(k)=\sqrt{\frac{\tanh(k)}{k}},$$0

depending on normalization (Ehrnstrom et al., 2016, Truong et al., 2020).

The defining modeling principle is that the equation combines the quadratic flux nonlinearity of the Korteweg–de Vries equation with the full linear dispersion relation of uni-directional gravity water waves (Binswanger et al., 2020). In the derivation from Hamiltonian theory, the unidirectional Whitham equation takes the form

$$\widehat{\mathcal{K}}(k)=\sqrt{\frac{\tanh(k)}{k}},$$1

with Fourier symbol

$$\widehat{\mathcal{K}}(k)=\sqrt{\frac{\tanh(k)}{k}},$$2

so the exact linear dispersion relation of the surface-water problem is retained while the nonlinearity remains quadratic (Moldabayev et al., 2014).

This nonlocality is analytically decisive. The kernel is singular near the origin, with

$$\widehat{\mathcal{K}}(k)=\sqrt{\frac{\tanh(k)}{k}},$$3

and in the periodic highest-wave analysis the integral kernel corresponding to the symbol $$\widehat{\mathcal{K}}(k)=\sqrt{\frac{\tanh(k)}{k}},$$4 is completely monotone on $$\widehat{\mathcal{K}}(k)=\sqrt{\frac{\tanh(k)}{k}},$$5 (Ehrnstrom et al., 2016, Hur, 2015). These properties underlie the appearance of non-smooth traveling waves and the delicate Fredholm, regularity, and compactness arguments that distinguish the Whitham equation from local dispersive models.

2. Derivation, scaling, and comparison with KdV and BBM

A Hamiltonian derivation identifies a scaling regime in which the Whitham equation can be obtained from the Hamiltonian theory of surface water waves (Moldabayev et al., 2014). In that framework, the free-surface problem is rewritten in terms of the Dirichlet–Neumann operator, and truncation in a regime adapted to Whitham solitary waves yields a bidirectional Whitham system. Restriction to right-propagating waves then leads to the one-way Whitham equation (Moldabayev et al., 2014).

The contrast with KdV is structural. KdV employs a local third-derivative dispersive correction and only approximates the water-wave dispersion for long waves, whereas the Whitham equation retains the entire dispersion symbol. The data explicitly state that this is thought to provide a more faithful description of shorter waves of small amplitude than traditional long-wave models such as the KdV equation (Moldabayev et al., 2014). Numerical comparisons reported there show that, in a wide parameter range of amplitudes and wavelengths, the Whitham equation performs on par with or better than both the KdV and BBM equations (Moldabayev et al., 2014).

For capillary-gravity waves the same modeling principle leads to the capillary-gravity Whitham equation

$$\widehat{\mathcal{K}}(k)=\sqrt{\frac{\tanh(k)}{k}},$$6

with multiplier

$$\widehat{\mathcal{K}}(k)=\sqrt{\frac{\tanh(k)}{k}},$$7

or, equivalently in another normalization,

$$\widehat{\mathcal{K}}(k)=\sqrt{\frac{\tanh(k)}{k}},$$8

This preserves the exact capillary-gravity dispersion relation and produces a much richer periodic bifurcation diagram than in the pure-gravity case (Charalampidis et al., 2021, Remonato et al., 2016).

A plausible implication is that the Whitham equation should be viewed not as a single asymptotic truncation comparable to KdV term-by-term, but as a model class whose accuracy is concentrated in the choice of linear symbol. The sources support this interpretation indirectly by emphasizing that the linear water-wave dispersion is retained exactly while the nonlinear closure remains shallow-water and weakly nonlinear (Moldabayev et al., 2014, Dinvay et al., 2020).

3. Traveling waves, bifurcation, and highest waves

Traveling waves are among the central objects in Whitham theory. For the $$\widehat{\mathcal{K}}(k)=\sqrt{\frac{\tanh(k)}{k}},$$9-periodic problem, the normalized profile equation is

$u_t + (K*u+u^2)_x=0$0

and global bifurcation for smooth periodic traveling waves was established along a branch emerging from the trivial state (Ehrnstrom et al., 2013). For each integer $u_t + (K*u+u^2)_x=0$1, the local bifurcation point is

$u_t + (K*u+u^2)_x=0$2

and there exists an analytic local curve of nontrivial $u_t + (K*u+u^2)_x=0$3-periodic even solutions with leading term $u_t + (K*u+u^2)_x=0$4 (Ehrnstrom et al., 2013).

The periodic global theory shows that each local branch extends globally and that any subset of solutions in the global branch contains a sequence converging uniformly to a solution in $u_t + (K*u+u^2)_x=0$5, $u_t + (K*u+u^2)_x=0$6 (Ehrnstrom et al., 2013). Numerical results along the principal branch exhibit a turning point and suggest convergence to a highest cusped wave (Ehrnstrom et al., 2013). That conjectural picture was later resolved in the periodic case: the highest periodic traveling-wave solution exists as a limiting case at the end of the main bifurcation curve of $u_t + (K*u+u^2)_x=0$7-periodic solutions and has optimal $u_t + (K*u+u^2)_x=0$8-regularity at its crests (Ehrnstrom et al., 2016). Near a crest,

$u_t + (K*u+u^2)_x=0$9

and more precisely

$m(\xi)=\frac{\tanh \xi}{\xi}$0

for small $m(\xi)=\frac{\tanh \xi}{\xi}$1 (Ehrnstrom et al., 2016).

The solitary-wave problem is subtler because the small-amplitude limit is singular and standard Crandall–Rabinowitz theory does not apply (Truong et al., 2020). In that setting the traveling-wave equation is

$m(\xi)=\frac{\tanh \xi}{\xi}$2

for decaying profiles $m(\xi)=\frac{\tanh \xi}{\xi}$3 (Truong et al., 2020). The local theory uses a nonlocal center manifold theorem adapted from Faye and Scheel, together with direct verification of Fredholm properties, to construct a unique local curve of nontrivial, even solitary waves emanating from $m(\xi)=\frac{\tanh \xi}{\xi}$4 (Truong et al., 2020). Near small amplitude, the reduced equation is a perturbed KdV-type ODE,

$m(\xi)=\frac{\tanh \xi}{\xi}$5

which admits a unique even positive exponentially decaying solution for each small $m(\xi)=\frac{\tanh \xi}{\xi}$6 (Truong et al., 2020).

Global analytic bifurcation then continues this local solitary-wave branch into a connected global curve $m(\xi)=\frac{\tanh \xi}{\xi}$7 (Truong et al., 2020). The large-amplitude analysis must confront possible translational loss of compactness, which is ruled out using qualitative properties of the equation and the integral identity

$m(\xi)=\frac{\tanh \xi}{\xi}$8

The only remaining global alternative is that

$m(\xi)=\frac{\tanh \xi}{\xi}$9

and simultaneously $\eta_{xxx}$0 (Truong et al., 2020). The limiting profile is an extreme solitary wave that is even, bounded, continuous, exponentially decaying, smooth except at the crest, and satisfies

$\eta_{xxx}$1

Unlike the highest Stokes wave of the classical water-wave problem, whose crest is Lipschitz with a corner, the extreme Whitham solitary wave is continuous with a square-root cusp (Truong et al., 2020).

4. Kernel analysis and regularity mechanisms

The kernel corresponding to the symbol $\eta_{xxx}$2 plays a central role in the regularity theory (Ehrnstrom et al., 2016). One of the key periodic highest-wave results is that this kernel is completely monotone on $\eta_{xxx}$3, meaning

$\eta_{xxx}$4

The paper also provides an explicit representation formula,

$\eta_{xxx}$5

and shows exponential decay as $\eta_{xxx}$6 (Ehrnstrom et al., 2016). Near zero the asymptotic behavior

$\eta_{xxx}$7

is precisely what drives the $\eta_{xxx}$8-regularity threshold at highest crests (Ehrnstrom et al., 2016).

In the solitary-wave theory the principal linear operator is

$\eta_{xxx}$9

and its Fredholm properties are established using pseudodifferential operator theory (Truong et al., 2020). The data specify that this analysis involves subtle symbol and index calculations in the sense of Grushin, and that compactness is recovered by combining symmetry, monotonicity, Arzelà–Ascoli, bootstrap regularity, and the Whitham-specific integral identity (Truong et al., 2020).

These kernel facts also feed into maximum-principle-type arguments in the periodic theory. The complete monotonicity and positivity properties support touching principles and monotonicity of traveling-wave profiles (Ehrnstrom et al., 2016). In the capillary-gravity problem, a related threshold appears: the integral kernel corresponding to the linear dispersion operator is completely monotone if and only if

$u_t + \mathcal{K}*u_x + \frac{3}{2}uu_x = 0,$0

which is distinct from the classical Bond number threshold $u_t + \mathcal{K}*u_x + \frac{3}{2}uu_x = 0,$1 (Ehrnström et al., 2019). That distinction governs how far positivity and convexity arguments can be pushed in the presence of surface tension.

A plausible implication is that the singular crest regularity is not an incidental pathology but a direct manifestation of the kernel singularity encoded by exact water-wave dispersion. The sources support this in the precise sense that the square-root cusp is traced to the jump in kernel regularity at zero and is proved by detailed asymptotic analysis rather than by formal scaling alone (Ehrnstrom et al., 2016, Truong et al., 2020).

5. Instability, modulational theory, and wave breaking

The Whitham equation supports both dispersive traveling waves and finite-time wave breaking. Wave breaking was proved for smooth initial data with sufficiently negative slope: solutions remain bounded while

$u_t + \mathcal{K}*u_x + \frac{3}{2}uu_x = 0,$2

for some finite $u_t + \mathcal{K}*u_x + \frac{3}{2}uu_x = 0,$3 (Hur, 2015). The proof follows characteristics

$u_t + \mathcal{K}*u_x + \frac{3}{2}uu_x = 0,$4

and derives an ODE for the slope along characteristics,

$u_t + \mathcal{K}*u_x + \frac{3}{2}uu_x = 0,$5

with careful control of the nonlocal forcing term $u_t + \mathcal{K}*u_x + \frac{3}{2}uu_x = 0,$6 using the kernel singularity (Hur, 2015). The result resolves a Whitham conjecture and extends to KdV-type equations with fractional dispersion for $u_t + \mathcal{K}*u_x + \frac{3}{2}uu_x = 0,$7 (Hur, 2015).

For periodic waves, modulational instability and superharmonic instability provide a complementary picture of large-time dynamics. In the standard Whitham equation, small-amplitude periodic solutions become modulationally unstable when $u_t + \mathcal{K}*u_x + \frac{3}{2}uu_x = 0,$8 (Carter et al., 2021). In the cubic-vortical Whitham equation, the zero-vorticity small-amplitude cutoff is

$u_t + \mathcal{K}*u_x + \frac{3}{2}uu_x = 0,$9

which is closer to the full water-wave threshold $$\widehat{\mathcal{K}}(k)=\sqrt{\frac{\tanh(k)}{k}}.$$0 than the classical Whitham value (Carter et al., 2021).

A general modulational framework is available for generalized Whitham equations

$$\widehat{\mathcal{K}}(k)=\sqrt{\frac{\tanh(k)}{k}}.$$1

with general nonlinear flux $$\widehat{\mathcal{K}}(k)=\sqrt{\frac{\tanh(k)}{k}}.$$2 and general dispersion relation $$\widehat{\mathcal{K}}(k)=\sqrt{\frac{\tanh(k)}{k}}.$$3 (Binswanger et al., 2020). Formal Whitham modulation theory yields a system of three conservation laws for mean, quadratic momentum, and wave number, and in the weakly nonlinear regime the strict hyperbolicity criterion is

$$\widehat{\mathcal{K}}(k)=\sqrt{\frac{\tanh(k)}{k}}.$$4

where

$$\widehat{\mathcal{K}}(k)=\sqrt{\frac{\tanh(k)}{k}}.$$5

The modulational instability criterion is the sign reversal

$$\widehat{\mathcal{K}}(k)=\sqrt{\frac{\tanh(k)}{k}}.$$6

(Binswanger et al., 2020). This formal criterion was rigorously justified for equations of Whitham-type by spectral perturbation theory: weak hyperbolicity of the Whitham system is necessary for modulational stability, and strict hyperbolicity is sufficient because of the Hamiltonian structure (Clarke et al., 2021).

At larger amplitude, periodic Whitham waves exhibit superharmonic instability. Large-amplitude periodic traveling-wave solutions to both the Whitham equation and the cubic Whitham equation are unstable with respect to perturbations of the same period, and the threshold occurs at the maxima of the Hamiltonian and $$\widehat{\mathcal{K}}(k)=\sqrt{\frac{\tanh(k)}{k}}.$$7-norm (Carter et al., 2023). For $$\widehat{\mathcal{K}}(k)=\sqrt{\frac{\tanh(k)}{k}}.$$8, the Whitham thresholds reported in the comparison with finite-depth Euler are

$$\widehat{\mathcal{K}}(k)=\sqrt{\frac{\tanh(k)}{k}}.$$9

for modulational instability and

$u_t + 2uu_x + Lu_x = 0,\qquad Lf=K*f,$0

for superharmonic instability (Carter et al., 2023). A subsequent numerical study of near-extreme $u_t + 2uu_x + Lu_x = 0,\qquad Lf=K*f,$1-periodic solutions found that the Hamiltonian oscillates as a function of wave steepness and that a superharmonic instability is created at each extremum of the Hamiltonian; the first extremum occurs at

$u_t + 2uu_x + Lu_x = 0,\qquad Lf=K*f,$2

with further bifurcation at

$u_t + 2uu_x + Lu_x = 0,\qquad Lf=K*f,$3

(Carter, 2023).

6. Extensions, variants, and Whitham modulation theory beyond the classical equation

The Whitham framework has been extended in several directions while preserving the central idea of combining nonlocal dispersion with weakly nonlinear closure. The cubic-vortical Whitham equation incorporates constant vorticity and a cubic nonlinear term,

$u_t + 2uu_x + Lu_x = 0,\qquad Lf=K*f,$4

with vorticity-dependent symbol

$u_t + 2uu_x + Lu_x = 0,\qquad Lf=K*f,$5

It reduces to the classical Whitham equation when $u_t + 2uu_x + Lu_x = 0,\qquad Lf=K*f,$6, $u_t + 2uu_x + Lu_x = 0,\qquad Lf=K*f,$7, and $u_t + 2uu_x + Lu_x = 0,\qquad Lf=K*f,$8 (Carter et al., 2021). Numerically, all moderate- and large-amplitude periodic traveling waves are unstable, regardless of wavelength (Carter et al., 2021).

Surface tension produces the capillary or capillary-gravity Whitham equations. Their periodic bifurcation diagrams contain subharmonic bifurcations, branch crossings, self-crossings, connecting branches, and, at special resonant values of the Bond number, two-dimensional bifurcation sheets associated with double eigenvalues (Remonato et al., 2016, Ehrnström et al., 2019). In the capillary-gravity global numerics, the $u_t + 2uu_x + Lu_x = 0,\qquad Lf=K*f,$9 fundamental branch is orbitally stable for all amplitudes and all $$K(x)=\frac{1}{2\pi}\int_{-\infty}^{\infty}m(\xi)e^{ix\xi}\,d\xi,\qquad m(\xi)=\sqrt{\frac{\tanh \xi}{\xi}},$$0, whereas higher-harmonic branches can be unstable for sufficiently large amplitude when $$K(x)=\frac{1}{2\pi}\int_{-\infty}^{\infty}m(\xi)e^{ix\xi}\,d\xi,\qquad m(\xi)=\sqrt{\frac{\tanh \xi}{\xi}},$$1 (Charalampidis et al., 2021).

Bidirectional generalizations replace the one-way scalar equation by two-way systems for surface elevation and velocity-type variables. Examples include the ASMP, Hur–Pandey, Matsuno, Hamiltonian Hur–Pandey, and right-left systems, all of which retain exact Euler linear dispersion while differing in nonlinear structure, Hamiltonian form, and well-posedness properties (Dinvay et al., 2019). The sources state that the new Hamiltonian, semi-linear system introduced there performs well both with regard to approximating the full Euler system and with regard to well posedness properties (Dinvay et al., 2019).

“Whitham equation” also refers, in a broader modulation-theoretic sense, to averaged systems governing slow variation of periodic waves. For generalized Whitham scalar equations, the modulation system consists of three conservation laws for the periodic wave’s wavenumber, amplitude, and mean (Binswanger et al., 2020). In multidimensional nonlinear Schrödinger theory, the phrase “Whitham system” denotes hydrodynamic-type modulation equations for slowly varying periodic wavetrains. For the $$K(x)=\frac{1}{2\pi}\int_{-\infty}^{\infty}m(\xi)e^{ix\xi}\,d\xi,\qquad m(\xi)=\sqrt{\frac{\tanh \xi}{\xi}},$$2-dimensional NLS equation the complete Whitham system consists of six dynamical equations in evolutionary form and two constraints (Ablowitz et al., 2021). For the defocusing NLS equation in two and three spatial dimensions, the modulation equations are written in vector form and preserve rotational, scaling, and Galilean invariance (Abeya et al., 2023). For the radial NLS equation, the Whitham system in Riemann variables takes the form

$$K(x)=\frac{1}{2\pi}\int_{-\infty}^{\infty}m(\xi)e^{ix\xi}\,d\xi,\qquad m(\xi)=\sqrt{\frac{\tanh \xi}{\xi}},$$3

with explicit radial source terms absent in one dimension (Ablowitz et al., 2018).

A further recent variant is the spatial Whitham equation, designed for the spatial evolution of time series measured at fixed locations rather than temporal evolution of spatial profiles (Carter et al., 2024). The nondimensional spatial equation is

$$K(x)=\frac{1}{2\pi}\int_{-\infty}^{\infty}m(\xi)e^{ix\xi}\,d\xi,\qquad m(\xi)=\sqrt{\frac{\tanh \xi}{\xi}},$$4

For small amplitudes its traveling waves align with known results for the temporal Whitham equation, but at larger amplitudes the spatial equation does not appear to admit cusped solutions of maximal wave height and does not reproduce the modulational instability of deep-water wave packets (Carter et al., 2024).

7. Mathematical significance and open directions

The Whitham equation occupies a distinctive position in dispersive PDE because it is simultaneously close to water-wave physics and analytically nonlocal. The periodic and solitary highest-wave theorems confirm Whitham’s conjecture in both settings, with the extreme profile realized as a limit point of a global bifurcation curve and exhibiting optimal $$K(x)=\frac{1}{2\pi}\int_{-\infty}^{\infty}m(\xi)e^{ix\xi}\,d\xi,\qquad m(\xi)=\sqrt{\frac{\tanh \xi}{\xi}},$$5-regularity at the crest (Ehrnstrom et al., 2016, Truong et al., 2020). The wave-breaking theorem shows that exact water-wave dispersion does not prevent bounded solutions from developing unbounded slope in finite time (Hur, 2015). The modulation results show that formal Whitham averaging can be made rigorous and can be connected directly to spectral stability via Bloch perturbation theory (Clarke et al., 2021).

The comparison with classical water waves is especially sharp at the highest crest. In the periodic and solitary Whitham problems, the limiting wave has a cusp with square-root law, whereas the highest Stokes wave in classical water-wave theory has a corner (Ehrnstrom et al., 2016, Truong et al., 2020). This suggests a qualitative distinction between exact free-boundary hydrodynamics and the nonlocal scalar closure embodied by the Whitham equation.

At the same time, several sources point to a continuing research program rather than a closed theory. The capillary-gravity setting raises unresolved issues about nodal structure below the complete-monotonicity threshold $$K(x)=\frac{1}{2\pi}\int_{-\infty}^{\infty}m(\xi)e^{ix\xi}\,d\xi,\qquad m(\xi)=\sqrt{\frac{\tanh \xi}{\xi}},$$6 (Ehrnström et al., 2019). Numerical studies of near-extreme periodic waves reveal repeated Hamiltonian oscillations and repeated creation of superharmonic instabilities, indicating a more intricate large-amplitude spectral landscape than a single stability transition (Carter, 2023). The spatial Whitham equation reproduces some laboratory data well but appears structurally unable to capture modulational deep-water wave-packet dynamics (Carter et al., 2024). These developments suggest that the Whitham equation is best understood as both a specific nonlocal shallow-water model and a prototype for a wider class of fully dispersive nonlinear equations whose analytic behavior is governed by the interaction of singular kernels, weak nonlinearity, and global bifurcation geometry.