A direct construction of a full family of Whitham solitary waves

On the precise cusped behaviour of extreme solutions to Whitham-type equations

“…All assumptions required for Proposition 5.5 and Proposition 5.7 are therefore satisfied, and we may settle a conjecture posed in the aforementioned paper. Our result also applies equally well to the highest solitary waves recently found in [34] and [13]. As shown in [7,8], such solitary waves are necessarily even, and smooth and decreasing on R + .…”

“…but did not establish the full limit described in (1.4). More recently, the existence of full global curves of solitary waves up to a highest wave has also been proved [13,34]. The same asymptotic estimates (1.5) from [15] apply equally well for these.…”

“…where φ represents the surface profile and KW (ξ) = ˆR K W (x)e −ixξ dx := tanh(ξ) ξ , is a fully dispersive variant of the classical Korteweg-de Vries (KdV) equation, originally proposed in [35]. It features some properties that the KdV equation lacks, such as wave breaking [23,32], highest waves [13,15,34], and better high-frequency modelling [17]. While Whitham added the dispersion in an ad hoc manner, the model has since been both justified experimentally and derived from the full water-wave problem in several ways.…”

Highest Cusped Waves for the Burgers-Hilbert equation

“…This conjecture was proved by Enciso, Gómez-Serrano and Vergara [20]. See also [17] for some recent remarks. For the fractional KdV equations the traveling waves assumption allows us to write the equation as…”