--- title: 'Whitham Equation: Nonlocal Water-Wave Model' url: https://www.emergentmind.com/topics/whitham-equation type: topic --- # Whitham Equation: Nonlocal Water-Wave Model 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 [1410.8299] [2009.04713]. 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 [1602.05384] [1506.04075]. ## 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 [1410.8299]. 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 \[ -\mu\varphi+L\varphi+\varphi^2=0 \] depending on normalization [1602.05384] [2009.04713]. 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 [2009.02350]. In the derivation from Hamiltonian theory, the unidirectional Whitham equation takes the form \[ \eta_t+\frac{3}{2}\frac{c_0}{h_0}\eta\eta_x+K_{h_0}*\eta_x=0, \] with Fourier symbol \[ \mathcal{F}K_{h_0}(\xi)=\sqrt{\frac{g\tanh(h_0\xi)}{\xi}}, \] so the exact linear dispersion relation of the surface-water problem is retained while the nonlinearity remains quadratic [1410.8299]. This nonlocality is analytically decisive. The kernel is singular near the origin, with \[ K(x)\sim \frac{1}{\sqrt{2\pi|x|}} \quad \text{as } |x|\to 0, \] and in the periodic highest-wave analysis the integral kernel corresponding to the symbol \(m(\xi)\) is completely monotone on \((0,\infty)\) [1602.05384] [1506.04075]. 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 [1410.8299]. 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 [1410.8299]. 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 [1410.8299]. 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 [1410.8299]. For capillary-gravity waves the same modeling principle leads to the capillary-gravity Whitham equation \[ u_t+c_{\mathrm{ww}}(|\partial_x|;T)u_x+(u^2)_x=0, \] with multiplier \[ \widehat{c_{\mathrm{ww}}(|\partial_x|;T)f}(k) = \sqrt{(1+Tk^2)\frac{\tanh(k)}{k}}\;\widehat{f}(k), \] or, equivalently in another normalization, \[ u_t+2uu_x+K_T*u_x=0,\qquad \widehat{K_T}(\xi)=\sqrt{\frac{(1+T\xi^2)\tanh(\xi)}{\xi}}. \] This preserves the exact capillary-gravity dispersion relation and produces a much richer periodic bifurcation diagram than in the pure-gravity case [2102.01905] [1604.08324]. 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 [1410.8299] [2002.09946]. ## 3. Traveling waves, bifurcation, and highest waves Traveling waves are among the central objects in Whitham theory. For the \(2\pi\)-periodic problem, the normalized profile equation is \[ -\mu\varphi+\varphi^2+K*\varphi=0, \] and global bifurcation for smooth periodic traveling waves was established along a branch emerging from the trivial state [1303.6721]. For each integer \(k\ge 1\), the local bifurcation point is \[ \mu_k=\left(\frac{\tanh(k)}{k}\right)^{1/2}, \] and there exists an analytic local curve of nontrivial \(2\pi/k\)-periodic even solutions with leading term \(\cos(kx)\) [1303.6721]. 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 \(C^\alpha\), \(\alpha<\frac12\) [1303.6721]. Numerical results along the principal branch exhibit a turning point and suggest convergence to a highest cusped wave [1303.6721]. 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 \(P\)-periodic solutions and has optimal \(C^{1/2}\)-regularity at its crests [1602.05384]. Near a crest, \[ \varphi(x)\sim \frac{\mu}{2}-C|x|^{1/2}, \] and more precisely \[ c_1|x|^{1/2}\le \frac{\mu}{2}-\varphi(x)\le c_2|x|^{1/2} \] for small \(|x|\) [1602.05384]. The solitary-wave problem is subtler because the small-amplitude limit is singular and standard Crandall–Rabinowitz theory does not apply [2009.04713]. In that setting the traveling-wave equation is \[ c\varphi-K*\varphi-\varphi^2=0, \] for decaying profiles \(\varphi:\mathbb{R}\to\mathbb{R}\) [2009.04713]. 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 \((\varphi,c)=(0,1)\) [2009.04713]. Near small amplitude, the reduced equation is a perturbed KdV-type ODE, \[ \varphi''=-6\varphi+\frac{19}{5}(\varphi')^2+6(c-1)\varphi+\text{higher order terms}, \] which admits a unique even positive exponentially decaying solution for each small \(c-1>0\) [2009.04713]. Global analytic bifurcation then continues this local solitary-wave branch into a connected global curve \(C\) [2009.04713]. 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 \[ \int_{\mathbb{R}}\varphi(x)(\varphi(x)-(c-1))\,dx=0. \] The only remaining global alternative is that \[ \sup_{x\in\mathbb{R}}\varphi(x)\nearrow \frac12(1+c) \] and simultaneously \(\|\varphi\|_{H^3}\to\infty\) [2009.04713]. The limiting profile is an extreme solitary wave that is even, bounded, continuous, exponentially decaying, smooth except at the crest, and satisfies \[ \varphi(0)=\frac12(1+c), \qquad C_1|x|^{1/2}\le \frac12(1+c)-\varphi(x)\le C_2|x|^{1/2}. \] 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 [2009.04713]. ## 4. Kernel analysis and regularity mechanisms The kernel corresponding to the symbol \(m(\xi)=\sqrt{\tfrac{\tanh \xi}{\xi}}\) plays a central role in the regularity theory [1602.05384]. One of the key periodic highest-wave results is that this kernel is completely monotone on \((0,\infty)\), meaning \[ (-1)^nK^{(n)}(x)>0,\qquad x>0,\ n\ge 0. \] The paper also provides an explicit representation formula, \[ K(x)=\frac{1}{\pi}\sum_{n=1}^{\infty}\int_{\frac{(2n-1)\pi}{2}}^{n\pi} e^{-s|x|}\sqrt{\frac{|\tan s|}{s}}\,ds, \] and shows exponential decay as \(|x|\to\infty\) [1602.05384]. Near zero the asymptotic behavior \[ K(x)\sim \frac{1}{\sqrt{2\pi|x|}} \] is precisely what drives the \(C^{1/2}\)-regularity threshold at highest crests [1602.05384]. In the solitary-wave theory the principal linear operator is \[ T:=\varphi-K*\varphi:H^j_{-\eta}\to H^j_{-\eta}, \] and its Fredholm properties are established using pseudodifferential operator theory [2009.04713]. 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 [2009.04713]. 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 [1602.05384]. 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 \[ T\ge \frac{4}{\pi^2}, \] which is distinct from the classical Bond number threshold \(T=\frac13\) [1901.03534]. 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 [1602.05384] [2009.04713]. ## 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 \[ \inf_x \partial_x u(x,t)\to -\infty \quad \text{as } t\to T^-, \] for some finite \(T\) [1506.04075]. The proof follows characteristics \[ \frac{dX}{dt}(t;x)=u(X(t;x),t),\qquad X(0;x)=x, \] and derives an ODE for the slope along characteristics, \[ \frac{dv_1}{dt}+v_1^2+\phi_1(t;x)=0, \] with careful control of the nonlocal forcing term \(\phi_1\) using the kernel singularity [1506.04075]. The result resolves a Whitham conjecture and extends to KdV-type equations with fractional dispersion for \(0<\alpha<2/3\) [1506.04075]. 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 \(kh>1.146\) [2110.02072]. In the cubic-vortical Whitham equation, the zero-vorticity small-amplitude cutoff is \[ kh>1.252, \] which is closer to the full water-wave threshold \(kh>1.363\) than the classical Whitham value [2110.02072]. A general modulational framework is available for generalized Whitham equations \[ u_t+f(u)_x+\mathcal{K}*u_x=0, \] with general nonlinear flux \(f(u)\) and general dispersion relation \(\Omega(k)\) [2009.02350]. 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 \[ n(u,k)\,\Omega''(k)>0, \] where \[ n(u,k)= k\left[ \frac{[f''(u)]^2}{\Omega'(k)} + \frac{k[f''(u)]^2}{2\Omega(k)-\Omega(2k)} +\frac12 f'''(u) \right]. \] The modulational instability criterion is the sign reversal \[ n(u,k)\,\Omega''(k)<0 \] [2009.02350]. 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 [2112.00049]. 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 \(\mathcal{L}_2\)-norm [2306.11889]. For \(L=2\pi\), the Whitham thresholds reported in the comparison with finite-depth Euler are \[ s^\dagger=0.062,\quad c^\dagger=0.925 \] for modulational instability and \[ s^*=0.103,\quad c^*=0.974 \] for superharmonic instability [2306.11889]. A subsequent numerical study of near-extreme \(2\pi\)-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 \[ s=0.1045, \] with further bifurcation at \[ s=0.1216 \] [2308.06583]. ## 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, \[ u_t+\mathcal{K}*u_x+\alpha uu_x+\beta u^2u_x=0, \] with vorticity-dependent symbol \[ \mathcal{K}(k)= -\frac{\Omega\tanh(k)}{2k} + \sqrt{ \frac{\tanh(k)}{k} + \frac{\Omega^2\tanh^2(k)}{4k^2} }. \] It reduces to the classical Whitham equation when \(\Omega=0\), \(\alpha\to \tfrac32\), and \(\beta\to 0\) [2110.02072]. Numerically, all moderate- and large-amplitude periodic traveling waves are unstable, regardless of wavelength [2110.02072]. 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 [1604.08324] [1901.03534]. In the capillary-gravity global numerics, the \(k=1\) fundamental branch is orbitally stable for all amplitudes and all \(T>0\), whereas higher-harmonic branches can be unstable for sufficiently large amplitude when \(0