Solitary waves in a Whitham equation with small surface tension

Abstract

Using a nonlocal version of the center-manifold theorem and a normal form reduction, we prove the existence of small-amplitude generalized solitary-wave solutions and modulated solitary-wave solutions to the steady gravity-capillary Whitham equation with weak surface tension. Through the application of the center-manifold theorem, the nonlocal equation for the solitary wave profiles is reduced to a four-dimensional system of ODEs inheriting reversibility. Along particular parameter curves, relating directly to the classical gravitycapillary water wave problem, the associated linear operator is seen to undergo either a reversible 0 2+ (i𝑘 0 ) bifurcation or a reversible (i𝑠) 2 bifurcation. Through a normal form transformation, the reduced system of ODEs along each relevant parameter curve is seen to be well approximated by a truncated system retaining only second-order or third-order terms. These truncated systems relate directly to systems obtained in the study of the full gravity-capillary water wave equation and, as such, the existence of generalized and modulated solitary waves for the truncated systems is guaranteed by classical works, and they are readily seen to persist as solutions of the gravity-capillary Whitham equation dueThis is an open access article under the terms of the Creative Commons Attribution-NonCommercial License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited and is not used for commercial purposes.

Mass and spring dimer Fermi–Pasta–Ulam–Tsingou nanopterons with exponentially small, nonvanishing ripples

“…The exponential localization of the Friesecke-Pego monatomic solitary waves[8, Proposition 5.5] provides sufficient decay for this series to converge. However, if 𝑟 𝑘 is a nanopteron, then the periodic ripple will contribute to the series (24); for example, if relative displacement is given by ( 18), then one must contend with the series…”

“…We are also curious if our spatial dynamics method can provide another perspective on the Whitham problem for which long wave nanopterons and micropterons exist. 23,24 A challenge here is that in these Whitham papers, the derivative is integrated out of the traveling wave equation, leaving a strictly nonlocal equation that does not appear to have the form of our general problem (65). Somewhat contemporaneously with our research, Hilder, de Rijk, and Schneider 49 used a lucid modernization of these spatial dynamics techniques to construct supersonic fronts in a monatomic lattice with nearest+next-to-nearest-neighbor interactions.…”

“…This small core naturally arises from the long wave scaling, which is, of course, not present in the material limits. It is quite interesting to note that Johnson, Truong, and Wheeler 24 also studied the gravity‐capillary Whitham equation using a nonlocal center manifold reduction and found, effectively, micropterons; they obtained an $$\mathcal {O}(\epsilon ^2)$$ core but an explicit nonzero $$\mathcal {O}(\epsilon )$$ leading order term for their periodic ripples.…”