Whitham–Boussinesq Systems

• The Whitham–Boussinesq system is a family of bidirectional water-wave models that couples Boussinesq-type quadratic nonlinearities with a nonlocal dispersion operator matching finite-depth water-wave dispersion.
• It is designed to accurately reproduce water-wave physics, and its formulations extend to include effects like surface tension, variable bathymetry, and modulation in stationary cnoidal waves.
• Rigorous analyses using energy methods, variational techniques, and dispersive estimates have established well-posedness, Hamiltonian structures, and solitary wave characteristics for these models.

The Whitham–Boussinesq system is a family of bidirectional, fully dispersive, weakly nonlinear models for free-surface water waves in which Boussinesq-type quadratic coupling is combined with a Whitham-type nonlocal dispersion operator chosen to reproduce the finite-depth linear water-wave dispersion relation. In the contemporary literature, the term most often refers to two-component evolution systems for the surface elevation and a surface velocity variable, but it is also used in a distinct modulation-theoretic sense for a four-component Whitham system governing stationary cnoidal waves of the Boussinesq equation (Dinvay, 2018, Tesfahun, 2022, Biondini et al., 2023).

1. Terminological scope and defining characteristics

In the water-wave modeling literature, the defining feature of a Whitham–Boussinesq system is the coexistence of a Boussinesq-type bidirectional structure with a nonlocal dispersion operator whose Fourier symbol matches, or closely matches, the full linear dispersion of finite-depth gravity waves. In this sense, “Whitham-type” refers to the use of exact or full dispersion rather than a low-order Taylor truncation, while “Boussinesq-type” refers to the coupling of a free-surface variable and a horizontal velocity variable through quadratic shallow-water nonlinearities (Dinvay, 2018, Dinvay et al., 2019).

A representative one-dimensional model is

$$\begin{aligned} \eta_t &= - v_x - i \tanh D (\eta v),\ v_t &= - i \tanh D \eta - i \tanh D \frac{v^2}{2}, \end{aligned}$$

with $D=-i\partial_x$, $\eta(x,t)$ the dimensionless free-surface elevation, and $v(x,t)$ a dual surface variable related to the horizontal fluid velocity at the surface (Dinvay, 2018). The linearization around $(\eta,v)=(0,0)$ yields

$\eta_t=-v_x,\qquad v_t=-i\tanh D\,\eta,$

so the linear dispersion relation is

$\omega(\xi)^2=\xi\,\tanh\xi,$

which is the standard nondimensional finite-depth gravity-wave dispersion relation. This is the basic reason the system is both fully dispersive and bidirectional (Dinvay, 2018).

The term also has a second established meaning in modulation theory. In the stationary reduction of the KP–Whitham system, the Boussinesq equation gives rise to a four-component first-order quasilinear system for $(r_1,r_2,r_3,p)$, obtained by imposing conservation of waves and setting the frequency $\omega=0$. In that setting, the “Whitham–Boussinesq system” is not a direct evolution system for $(\eta,v)$, but a genus‑1 modulation system for stationary cnoidal waves (Biondini et al., 2023).

2. Canonical formulations and dispersive operators

Several non-equivalent formulations appear in the literature, all sharing the same organizing principle. A standard one-dimensional formulation can be written as

$D=-i\partial_x$0

so that $D=-i\partial_x$1 (Dinvay et al., 2019). In two spatial dimensions, one encounters

$D=-i\partial_x$2

with $D=-i\partial_x$3 curl-free and $D=-i\partial_x$4 the shallowness and nonlinearity parameters (Tesfahun, 2022).

These formulations recover classical Boussinesq models only after a long-wave expansion. For $D=-i\partial_x$5,

$D=-i\partial_x$6

and dropping $D=-i\partial_x$7 terms yields a classical Boussinesq system. The fully dispersive model is therefore a regularization of the truncated polynomial dispersion by the exact symbol $D=-i\partial_x$8 (Tesfahun, 2022).

Variants with additional physics are also standard. With surface tension, the one-dimensional system becomes

$D=-i\partial_x$9

where $\eta(x,t)$0 is the capillarity parameter (Dinvay, 2019). For variable bathymetry, fully dispersive Whitham–Boussinesq systems incorporate the bottom operator $\eta(x,t)$1 and the Fourier multiplier

$\eta(x,t)$2

leading to uneven-bottom ASMP, Hur–Pandey, and regularised systems (Carter et al., 2020).

The same terminology is also used for the general class

$\eta(x,t)$3

with Fourier multiplier $\eta(x,t)$4 defined by a real, even symbol $\eta(x,t)$5; the bi-directional Whitham system corresponds to $\eta(x,t)$6 (Nilsson et al., 2018). This formulation emphasizes that many analytical arguments depend more on symbol properties than on a single canonical model.

3. Hamiltonian structure and well-posedness theory

A recurrent feature of the water-wave Whitham–Boussinesq systems is Hamiltonian structure. For the one-dimensional fully dispersive model

$\eta(x,t)$7

the Hamiltonian is

$\eta(x,t)$8

with skew-adjoint structure operator

$\eta(x,t)$9

and there is also a momentum invariant

$v(x,t)$0

(Dinvay et al., 2019). The surface-tension model has an analogous Hamiltonian,

$v(x,t)$1

which is conserved along the flow (Dinvay, 2019).

Analytically, the first local well-posedness result for the capillarity-free Dinvay system established existence, uniqueness, and continuous dependence for data

$v(x,t)$2

with no smallness assumption, by an energy method and a compactness argument (Dinvay, 2018). The energy used there is equivalent to the phase-space norm,

$v(x,t)$3

and satisfies the a priori differential inequality

$v(x,t)$4

for sufficiently smooth solutions (Dinvay, 2018).

Subsequent work lowered the regularity threshold by exploiting dispersion more directly. In one dimension, local well-posedness was proved for

$v(x,t)$5

and in two dimensions for

$v(x,t)$6

using dyadic dispersive estimates, localized Strichartz estimates, Bourgain spaces, and bilinear estimates (Dinvay et al., 2019). In the same work, one-dimensional small-data global well-posedness was obtained from Hamiltonian conservation at the energy level $v(x,t)$7 (Dinvay et al., 2019).

For the two-dimensional $v(x,t)$8-dependent system, a different mechanism yields long-time control. If $v(x,t)$9, then

$(\eta,v)=(0,0)$0

exists on a time interval

$(\eta,v)=(0,0)$1

and in the KdV regime $(\eta,v)=(0,0)$2 this becomes

$(\eta,v)=(0,0)$3

(Tesfahun, 2022). The proof uses frequency-localized dispersive estimates, localized Strichartz estimates, and bilinear estimates in Strichartz norms (Tesfahun, 2022).

With surface tension, the nonlinearity is not symmetric enough for the standard Sobolev energy to close. The remedy is a modified energy

$(\eta,v)=(0,0)$4

whose cubic correction cancels the problematic term. This yields local well-posedness for $(\eta,v)=(0,0)$5 under noncavitation and global well-posedness for small one-dimensional data for every $(\eta,v)=(0,0)$6 (Dinvay, 2019).

4. Derivation from water waves and bathymetric generalizations

The family is motivated by the free-surface Euler equations in the shallow-water regime. One rigorous route starts from the one-dimensional water-wave system in Zakharov variables $(\eta,v)=(0,0)$7 and introduces

$(\eta,v)=(0,0)$8

The corresponding Whitham–Boussinesq system is

$(\eta,v)=(0,0)$9

and the paper derives the Whitham equation from water waves either by approximate Riemann invariants, adapted to unidirectional propagation, or by a generalized Birkhoff normal form algorithm for almost smooth Hamiltonians, adapted to bidirectional propagation (Emerald, 2021).

A second rigorous route treats a general class

$\eta_t=-v_x,\qquad v_t=-i\tanh D\,\eta,$0

with admissible Fourier multipliers $\eta_t=-v_x,\qquad v_t=-i\tanh D\,\eta,$1. For the choice

$\eta_t=-v_x,\qquad v_t=-i\tanh D\,\eta,$2

the resulting Whitham–Boussinesq systems are proved locally well-posed on a time interval of size $\eta_t=-v_x,\qquad v_t=-i\tanh D\,\eta,$3, uniformly in $\eta_t=-v_x,\qquad v_t=-i\tanh D\,\eta,$4, and solutions of the full water-wave equations satisfy

$\eta_t=-v_x,\qquad v_t=-i\tanh D\,\eta,$5

for $\eta_t=-v_x,\qquad v_t=-i\tanh D\,\eta,$6 (Emerald, 2022). This gives a rigorous justification of the model as an approximation to water waves on the Boussinesq time scale.

Bathymetric generalizations preserve the fully dispersive philosophy while changing the operator structure. Three weakly nonlinear but fully dispersive systems for uneven bathymetry were compared against Dingemans’s laboratory measurements: the ASMP system, the Hur–Pandey system, and a regularised system (Carter et al., 2020). All three use

$\eta_t=-v_x,\qquad v_t=-i\tanh D\,\eta,$7

and a bottom operator $\eta_t=-v_x,\qquad v_t=-i\tanh D\,\eta,$8, but the accuracy differs markedly. Over flat bottom the predictions are essentially identical, whereas over the seamount the ASMP system performs best by far, the regularised system is intermediate, and the Hur–Pandey system is less accurate (Carter et al., 2020). This suggests that the precise placement of the nonlocal operators relative to the nonlinear terms matters decisively once bathymetric forcing is present.

5. Solitary waves, travelling waves, and numerical interactions

Travelling waves are central to the subject. For the capillarity-free Dinvay system, a travelling ansatz $\eta_t=-v_x,\qquad v_t=-i\tanh D\,\eta,$9, $\omega(\xi)^2=\xi\,\tanh\xi,$0 yields

$\omega(\xi)^2=\xi\,\tanh\xi,$1

and numerical Petviashvili iteration shows solitary waves for all $\omega(\xi)^2=\xi\,\tanh\xi,$2. Their amplitude–speed curves match Euler solitary waves very closely for moderate amplitudes, the relative discrepancy tends to $\omega(\xi)^2=\xi\,\tanh\xi,$3 as the amplitude tends to $\omega(\xi)^2=\xi\,\tanh\xi,$4, and for amplitudes around $\omega(\xi)^2=\xi\,\tanh\xi,$5 the relative error is below $\omega(\xi)^2=\xi\,\tanh\xi,$6 (Dinvay, 2018).

Existence theory for solitary waves was later developed variationally. For the specific Whitham–Boussinesq system

$\omega(\xi)^2=\xi\,\tanh\xi,$7

with $\omega(\xi)^2=\xi\,\tanh\xi,$8, the travelling-wave problem reduces to a scalar Euler–Lagrange equation with a non-local operator of positive order in both the linear and nonlinear parts. Existence is obtained by constrained minimization and concentration–compactness, and the small-amplitude branch admits a KdV asymptotic description (Dinvay et al., 2019). For the more general class

$\omega(\xi)^2=\xi\,\tanh\xi,$9

the travelling-wave system reduces to a single nonlocal profile equation, and solitary waves exist under general symbol assumptions by a similar constrained minimization strategy (Nilsson et al., 2018).

The small-amplitude asymptotics are explicitly KdV-like. After the long-wave scaling $(r_1,r_2,r_3,p)$0, the Whitham–Boussinesq solitary waves converge to the KdV solitary-wave profile $(r_1,r_2,r_3,p)$1 in $(r_1,r_2,r_3,p)$2, and the associated $(r_1,r_2,r_3,p)$3 and $(r_1,r_2,r_3,p)$4 converge, after the same scaling, in $(r_1,r_2,r_3,p)$5 and $(r_1,r_2,r_3,p)$6, respectively (Dinvay et al., 2019). This places the fully dispersive solitary waves in the same asymptotic universality class as the classical KdV soliton, while retaining exact finite-depth linear dispersion.

Large-amplitude numerics reveal phenomena absent from the small-amplitude theory. For the HP and ASMP bidirectional Whitham–Boussinesq systems,

$(r_1,r_2,r_3,p)$7

and

$(r_1,r_2,r_3,p)$8

solitary waves can be continued numerically to speeds close to a singular limit. The branches approach peakon-type profiles, and overtaking collisions exhibit the geometric Lax categorization for both models. For the HP system, the numerics also indicate an algebraic categorization: $(r_1,r_2,r_3,p)$9 whereas no such categorization is possible for the ASMP system (Flamarion et al., 2022).

6. Modulation systems, spectral instabilities, and open directions

A different but important use of the term occurs in Whitham modulation theory. Starting from the stationary reduction of the KP–Whitham system and imposing conservation of waves, one obtains a four-component system

$\omega=0$0

with $\omega=0$1, where $\omega=0$2 are KdV–Whitham parameters and $\omega=0$3 is an auxiliary mean component (Biondini et al., 2023). In this stationary Boussinesq setting, the Whitham–Boussinesq system is integrable in the hydrodynamic sense: the Haantjes tensor vanishes, and the system exhibits hyperbolic or elliptic regimes depending on the sign structure of the parameters (Biondini et al., 2023).

Stability theory for periodic travelling waves is less settled than the solitary-wave theory. For the Hur–Pandey–Tao Boussinesq–Whitham system, small-amplitude periodic travelling waves were shown numerically to exhibit high-frequency instabilities, and a perturbative analysis identifies isolated unstable spectral loops generated by nonzero eigenvalue collisions. The first such instability, corresponding to $\omega=0$4, is numerically present for all $\omega=0$5, while the $\omega=0$6 instability disappears at the special aspect ratio $\omega=0$7 (Creedon et al., 2021). This places the spectral theory of Whitham–Boussinesq systems closer to that of full water waves than to that of classical polynomial-dispersion Boussinesq equations.

Several open problems are explicit in the literature. For the capillarity-free Dinvay system, the authors point to rigorous consistency with Euler, modulational instability and wave breaking, global well-posedness, asymptotic stability of solitary waves, and higher-dimensional extensions as natural next steps (Dinvay, 2018). For the surface-tension system, large-data global theory in one dimension and any global theory in two dimensions remain open (Dinvay, 2019). For the collision theory, rigorous existence, uniqueness, and stability of the peakon-limit waves, as well as a mathematical explanation of the HP collision thresholds, are not yet available (Flamarion et al., 2022).

Taken together, these works present the Whitham–Boussinesq system not as a single equation but as a mathematically diverse class of nonlocal bidirectional models. Their common core is the replacement of polynomial long-wave dispersion by the exact or near-exact finite-depth water-wave symbol, while preserving a Boussinesq-type two-field nonlinear structure. The resulting theory connects local and nonlocal PDE, Hamiltonian structure, low-regularity well-posedness, rigorous asymptotics from Euler, solitary-wave variational theory, and Whitham modulation in both dynamical and stationary settings (Dinvay, 2018, Emerald, 2022, Biondini et al., 2023).