The Whitham equation was proposed as a model for surface water waves that combines the quadratic flux nonlinearity
This content will become publicly available on August 31, 2025
 Award ID(s):
 2306319
 NSFPAR ID:
 10521657
 Publisher / Repository:
 SIAM Journal of Applied Mathematics
 Date Published:
 Journal Name:
 SIAM Journal on Applied Mathematics
 Volume:
 84
 Issue:
 4
 ISSN:
 00361399
 Page Range / eLocation ID:
 1337 to 1361
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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Abstract of the Korteweg–de Vries equation and the full linear dispersion relation of unidirectional gravity water waves in suitably scaled variables. This paper proposes and analyzes a generalization of Whitham's model to unidirectional nonlinear wave equations consisting of a general nonlinear flux function and a general linear dispersion relation . Assuming the existence of periodic traveling wave solutions to this generalized Whitham equation, their slow modulations are studied in the context of Whitham modulation theory. A multiple scales calculation yields the modulation equations, a system of three conservation laws that describe the slow evolution of the periodic traveling wave's wavenumber, amplitude, and mean. In the weakly nonlinear limit, explicit, simple criteria in terms of general and establishing the strict hyperbolicity and genuine nonlinearity of the modulation equations are determined. This result is interpreted as a generalized Lighthill–Whitham criterion for modulational instability. 
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Abstract Whitham modulation equations are derived for the nonlinear Schrödinger equation in the plane ((2+1)‐dimensional nonlinear Schrödinger [2d NLS]) with small dispersion. The modulation equations are obtained in terms of both physical and Riemann‐type variables; the latter yields equations of hydrodynamic type. The complete 2d NLS Whitham system consists of six dynamical equations in evolutionary form and two constraints. As an application, we determine the linear stability of one‐dimensional traveling waves. In both the elliptic and hyperbolic cases, the traveling waves are found to be unstable. This result is consistent with previous investigations of stability by other methods and is supported by direct numerical calculations.

A new type of wave–mean flow interaction is identified and studied in which a smallamplitude, linear, dispersive modulated wave propagates through an evolving, nonlinear, largescale fluid state such as an expansion (rarefaction) wave or a dispersive shock wave (undular bore). The Korteweg–de Vries (KdV) equation is considered as a prototypical example of dynamic wavepacket–mean flow interaction. Modulation equations are derived for the coupling between linear wave modulations and a nonlinear mean flow. These equations admit a particular class of solutions that describe the transmission or trapping of a linear wavepacket by an unsteady hydrodynamic state. Two adiabatic invariants of motion are identified that determine the transmission, trapping conditions and show that wavepackets incident upon smooth expansion waves or compressive, rapidly oscillating dispersive shock waves exhibit socalled hydrodynamic reciprocity recently described in Maiden et al. ( Phys. Rev. Lett. , vol. 120, 2018, 144101) in the context of hydrodynamic soliton tunnelling. The modulation theory results are in excellent agreement with direct numerical simulations of full KdV dynamics. The integrability of the KdV equation is not invoked so these results can be extended to other nonlinear dispersive fluid mechanic models.more » « less