Whitham Modulation Theory

• Whitham modulation theory is a framework that replaces fast oscillations in nonlinear dispersive PDEs with a quasilinear system for slow wave parameters.
• It employs an averaging technique over local conservation laws to derive modulation equations that capture large-scale dynamics, shock formation, and modulational instability.
• The approach uses analytic and Gevrey space methods to ensure well-posedness and quantify error bounds for multiphase and nonlinear dispersive systems.

Whitham modulation theory is a framework for describing the slow evolution of parameters associated with nonlinear periodic or multiphase waves in weakly dispersive media. The theory replaces rapidly oscillatory solutions of nonlinear dispersive PDEs by a closed quasilinear system for the local wave parameters, whose solution captures large-scale features, dispersive shock formation, and modulational instability phenomena that are inaccessible to perturbative or integrable methods.

1. Theoretical Foundations and Derivation

Whitham's approach begins with a multiscale ansatz for the solution of a nonlinear dispersive PDE, such as the coupled nonlinear Schrödinger system (CNLS), in which fast oscillatory phases are modulated by slow variables $(X,T)=\epsilon(x,t)$:

$$Y_j(x,t) = v_j(\omega, k) \, e^{i \theta_j}, \qquad \theta_j = k_j x - \omega_j t + \theta_{j0}$$

The modulation variables $\omega_j, k_j$ are promoted to slowly varying functions: $\omega_j = \omega^0_j + \epsilon \Omega_j(X,T)$, $k_j = k^0_j + \epsilon K_j(X,T)$.

The core analytical step is averaging either the local conservation laws or the Lagrangian density over the fast phases, resulting in the averaged Lagrangian $L_{\mathrm{avg}}(\omega, k)$:

$$L_{\mathrm{avg}}(\omega, k) = \frac{1}{(2\pi)^2} \int_0^{2\pi}\int_0^{2\pi} L\big(Y(\theta; \omega, k)\big) \; d\theta_1 d\theta_2$$

The Whitham equations arise by imposing the stationarity of the action for the averaged Lagrangian under slow variations, yielding the modulation equations for multiphase systems:

$$\boxed{ \begin{aligned} &\partial_T \left( \frac{\partial L_{\mathrm{avg}}}{\partial \omega_j} \right) + \partial_X \left( \frac{\partial L_{\mathrm{avg}}}{\partial k_j} \right) = 0 \ &\partial_T K_j = \partial_X \Omega_j, \qquad j=1,2 \end{aligned} }$$

These equations are first-order quasilinear PDEs for the modulation variables $I_j = \partial_{\omega_j} L_{\mathrm{avg}}$, $B_j = \partial_{k_j} L_{\mathrm{avg}}$.

2. Linear and Nonlinear Modulation Regimes

The Whitham system's character—hyperbolic, elliptic, or mixed—depends crucially on the eigenvalue structure of the characteristic polynomial:

$\Delta = \det D^2_{(\omega, k)} L_{\mathrm{avg}}$

• Hyperbolic regime: All characteristic speeds $\lambda$ real ($\Delta>0$), signifying long-wavelength modulational stability.
• Elliptic regime: Some $\lambda$ complex or occur in conjugate pairs ($\Delta<0$), leading to modulational instability.
• Mixed type: The sign of $\Delta$ varies, and the system changes type along the solution trajectory.

Change of hyperbolicity in multiphase systems is linked to the modulational stability or instability of the underlying nonlinear waves (Bridges et al., 2020). Notably, the type can change dynamically due to dependence of $D^2 L_{\mathrm{avg}}$ on $(I, B)$.

3. Gevrey Spaces and Well-Posedness

Standard Sobolev spaces do not guarantee well-posedness of Whitham modulation systems in elliptic or mixed regimes; ill-posedness manifests in loss of control over solutions due to exponential growth of high-frequency perturbations. Analyticity or Gevrey regularity is required, ensuring the solution is analytic in a strip $S_\sigma = \{X + iY: |Y|<\sigma\}$.

The Gevrey norm is defined by:

$$\|u\|_{G^s_\sigma}^2 = \int_{\mathbb{R}} e^{2\sigma(1+|\xi|^2)^{1/(2s)}} (1+|\xi|^2)^s |\hat{u}(\xi)|^2 d\xi$$

Key properties include:

• Algebra property: $$\|uv\|_{G^s_\sigma} \leq C \|u\|_{G^s_\sigma} \|v\|_{G^s_\sigma}$$ for $s>1/2$.
• Shrinking strip control: For $0<\sigma'<\sigma$, $$\|u\|_{G^s_{\sigma'}} \leq C(p, \sigma', \sigma) \|u\|_{G^s_\sigma}$$.
• Composition: For entire $P(z)$, $$\|P(u)\|_{G^s_\sigma} \leq \Phi(\|u\|_{G^s_\sigma})$$.

4. Well-Posedness, Validity, and Error Estimates

A Cauchy–Kowalevskaya-type theorem can be established for Whitham systems in Gevrey spaces. Given initial data $u_0 \in G^s_\sigma$, the solution $u(X,T)$ to

$\partial_T u = M(u) \partial_X u$

remains analytic for $T \in [0, \sigma_0/\gamma)$, with existence, uniqueness, and a uniform bound in $\|u\|_{G^s_{\sigma(T)}}$.

For the original perturbed system $\partial_T u = M(u)\partial_X u + \epsilon F(Du)$, one constructs higher-order Whitham expansions

$$u \sim u^0 + \epsilon u^1 + \cdots + \epsilon^N u^N$$

with each $u^k$ solving a linear “equation of variation” about $u^0$. Energy estimates ensure validity and analyticity of each corrector on the strip $S_{\sigma - kT}$, and the residual satisfies

$\|Res^N\|_{G^s_{\sigma'}} \leq C \epsilon^{N+1}$

Rigorous comparison yields:

$$\|u(x, t) - u_{\mathrm{Whitham}}(x, t)\|_{H^s_x} \leq C \epsilon^N, \qquad t = O(1/\epsilon)$$

for $t$ up to the natural modulation time scale (Bridges et al., 2020).

5. Application to Two-Phase Systems and Coupled NLS

The theory in (Bridges et al., 2020) is applied to CNLS of the form, in real-vector notation,

$$\partial_t u = M(u) \partial_x u + \epsilon^2 F(Du)$$

with explicit modulation equations for two-phase wavetrains. The averaged Lagrangian is used to derive action–angle variables, and the Whitham system tracks the dynamics of mass and momentum densities via:

$$\partial_T (\partial_{\omega_j} L_{\mathrm{avg}}) + \partial_X (\partial_{k_j} L_{\mathrm{avg}}) = 0, \qquad j=1,2$$

The validity theory ensures the Whitham system remains a good approximation in both elliptic/hyperbolic/mixed cases for the class of analytic initial data on $t=O(1/\epsilon)$. The framework extends to an arbitrary number of phases, provided appropriate toral symmetry is present.

6. Higher-Order Expansions and Multiphase Bifurcations

Near criticality, when two characteristics coalesce (e.g., at the Benjamin–Feir threshold), the modulation equations can morph into dispersive PDEs of Boussinesq type via a normal form reduction. This captures the emergence of complex wave structures and dispersive instabilities at the transition between hyperbolic and elliptic regimes (Bridges et al., 2020).

These transitions encode the precise bifurcation mechanism underlying the onset of modulational instability and generate phenomena such as DSWs, resonant radiation, and soliton trains in weakly nonlinear dispersive media.

7. Key Formulas and Summary Table

Mathematical Entity	Formula/Definition	Role
Lagrangian density	$L(Y, Y^*, Y_x, Y_x^*)$ (see above)	Governing variational principle
Averaged Lagrangian	$L_{\mathrm{avg}}(\omega, k)$	Modulation variable generator
Whitham PDEs	$\partial_T(\frac{\partial L_{\mathrm{avg}}{\partial\omega_j}) + \partial_X(\frac{\partial L_{\mathrm{avg}}{\partial k_j}) = 0$	Conservation laws for slow modulation
Gevrey norm	$$\|u\|_{G^s_\sigma}^2 = \int e^{2\sigma(1+|\xi|^2)^{1/(2s)}}(1+|\xi|^2)^s|\hat{u}(\xi)|^2d\xi$$	Analyticity control for PDE solutions
Error estimate	$$\|u_{\mathrm{exact}} - u_{\mathrm{Whitham}}\|_{H^s} \leq C \epsilon^N,\, t=O(1/\epsilon)$$	Rigorous validity on modulation time scale

Rigorous, uniform-in-time error bounds for Whitham approximations have been established in analytic/Gevrey regularity classes, and these bounds hold for nonuniformly hyperbolic, elliptic, or mixed-type systems. The analysis confirms the predictive accuracy of Whitham modulation for the slow, large-scale evolution of nonlinear wavetrains in dispersive systems with small parameters, especially in the presence of transitions between hyperbolic and elliptic regimes. The methodology is extensible to arbitrary numbers of phases, higher-order corrections, and general Hamiltonian dispersive systems.