On the Bifurcation Diagram of the Capillary–Gravity Whitham Equation

Abstract

We study the bifurcation of periodic travelling waves of the capillary-gravity Whitham equation. This is a nonlinear pseudodifferential equation that combines the canonical shallow water nonlinearity with the exact (unidirectional) dispersion for finite-depth capillarygravity waves. Starting from the line of zero solutions, we give a complete description of all small periodic solutions, unimodal as well bimodal, using simple and double bifurcation via Lyapunov-Schmidt reductions. Included in this study is the resonant case when one wavenumber divides another. Some bifurcation formulas are studied, enabling us, in almost all cases, to continue the unimodal bifurcation curves into global curves. By characterizing the range of the surface tension parameter for which the integral kernel corresponding to the linear dispersion operator is completely monotone (and therefore positive and convex; the threshold value for this to happen turns out to be T = 4 π 2 , not the critical Bond number 1 3 ), we are able to say something about the nodal properties of solutions, even in the presence of surface tension. Finally, we present a few general results for the equation and discuss, in detail, the complete bifurcation diagram as far as it is known from analytical and numerical evidence. Interestingly, we find, analytically, secondary bifurcation curves connecting different branches of solutions; and, numerically, that all supercritical waves preserve their basic nodal structure, converging asymptotically in L 2 (S) (but not in L ∞ ) towards one of the two constant solution curves. 4 π 2 , we are able in Theorem 2.6 to show that the kernel is completely monotone, a delicate structural property shared by the kernel for the linear dispersion in the pure gravity case (not its inverse). Moreover, we

Solitary waves for weakly dispersive equations with inhomogeneous nonlinearities

“…which corresponds to s = 1 2 . Modelled on the water wave problem with surface tension, the capillary-gravity Whitham equation is known to admit generalized solitary waves in the case T < 1 3 (weak surface tension) [11], and decaying solitary waves for T > 0 (both weak and strong surface tension) [3], as well as periodic steady waves, including rippled solutions in the case of weak surface tension [7]. In the case T < 1 3 the solitary waves have wave speeds ν smaller than m(0) (called subcritical), whereas the generalized waves exhibit supercritical wave speeds ν > m(0); for strong surface tension we are only aware of sub-critical solutions.…”

Solitary waves in a Whitham equation with small surface tension

“…Indeed, one of the hypotheses of Faye & Scheel's result is that both of the functions 5 F −1 (m −1 τ ) and ∂ x F −1 (m −1 τ ) are integrable and exhibit exponential decay, which are highly non-trivial properties. While the exponential decay and integrability of F −1 (m −1 τ ) was recently established in [11], this reference also unfortunately shows that ∂ x F −1 (m −1 τ ) is not an integrable function. By carefully considering the methodologies used in [14,15], Truong, Wahlén and Wheeler were recently able to circumvent this difficulty in [30], where they present a refinement of the result in [14,15] which does not rely on the integrability 6 of F −1 (m −1 τ ) .…”

“…The profile equation ( 4) has received several treatments in recent years and theoretical frameworks for studying them are expanding. Existence results for (4) include periodic waves by Hur & Johnson [20] in 2015 and Ehrnström, Johnson, Maehlen & Remonato [11] in 2019, solitary (e.g. integrable) waves for both strong and weak surface tension by Arnesen [3] in 2016, solitary waves of depression for strong surface tension τ > 1/3 and subcritical wave speed c < 1 by Johnson & Wright [23] in 2018, as well as generalized solitary waves for weak surface tension τ ∈ (0, 1/3) and supercritical wave speed c > 1 also by [23].…”

Solitary waves in a Whitham equation with small surface tension

“…The profile equation ( 6) has received several treatments in recent years and theoretical frameworks for studying them are expanding. Existence results for (6) include periodic waves by Hur and Johnson 12 in 2015 and Ehrnström et al 11 in 2019, solitary (e.g., integrable) waves for both strong and weak surface tension by Arnesen 27 in 2016, solitary waves of depression for strong surface tension 𝜏 > 1∕3 and subcritical wave speed 𝑐 < 1 by Johnson and Wright 13 in 2018, as well as generalized solitary waves for weak surface tension 𝜏 ∈ (0, 1∕3) and supercritical wave speed 𝑐 > 1 also by Ref. 13.…”