A numerical study of the Whitham equation as a model for steady surface water waves

Modulational Instability in the Whitham Equation for Water Waves

“…The underlying periodic traveling wave is then modulationally unstable if the characteristic polynomial P(−iτ · ; τ, a) admits a pair of complex roots, i.e., τ,a := 18d 3…”

“…Whitham therefore concocted (1) as an alternative to the KdV equation, incorporating the full range of the dispersion of surface water waves (rather than a second-order approximation) and the nonlinearity that compels blowup in the shallow water equations. As a matter of fact, (1) better approximates short-wavelength, periodic traveling waves in water than (4) does; see [3], for instance. Moreover Whitham advocated that (1) would explain "breaking" and "peaking."…”

On Whitham and Related Equations

“…We will focus here on nonperiodic solitary wave solutions of the Whitham equation (1), which are solutions of the form u(x − ct). We refer to [44], [5], [45], [6], and [8] for interesting (theoretical and numerical) studies on periodic traveling waves. In particular, the existence of a global bifurcation branch of 2π -periodic smooth traveling-wave solutions is established in [45].…”

Solitary waves in dispersive evolution equations of Whitham type with nonlinearities of mild regularity

“…Significant breakthrough in the last decade, however, has put the original Whitham equation, and also other full-dispersion models, in the spotlight, beginning with the existence of periodic traveling waves by Ehrnström and Kalisch [9] in 2009 and solitary-wave solutions by Ehrnström, Groves and Wahlén [8] in 2012; see also [30]. Research has furthermore confirmed Whitham's conjectures for qualitative wave breaking (bounded wave profile with unbounded slope) in finite time [16] and the existence of highest, cusp-like solutions [10, 12]-now known to also have a convex profile between the stagnation points [13].Additional analytical and numerical results for the Whitham equation include modulational instability of periodic waves [17,29], local well-posedness in Sobolev spaces H s , s > 3 2 , for both solitary and periodic initial data [7,11,19], non-uniform continuity of the data-to-solution map [1], symmetry and decay of traveling waves [3], analysis of modeling properties, dynamics and identification of scaling regimes [19], and wave-channel experiments and other numerical studies [2,5,18,32].In total, these investigations have demonstrated the potential usefulness of full-dispersion versions of traditional shallow-water models.1.2 Assumptions and main results. In this paper we contribute to the longstanding mathematical program of fully understanding the interplay between dispersive and nonlinear effects for the formation of traveling waves.…”

“…Under Assumptions A 1 and A 2 , we study (2) in the Sobolev space H s on the real line and in the corresponding P-periodic analogue H s P in the periodic setting (see section 2.1 for definitions) for s > 0 satisfying…”