A direct construction of a full family of Whitham solitary waves

Abstract

Starting with the periodic waves earlier constructed for the gravity Whitham equation, we parameterise the solution curves through relative wave height, and use a limiting argument to obtain a full family of solitary waves. The resulting branch starts from the zero solution, traverses unique points in the wave speed–wave height space, and reaches a singular highest wave at φ ( 0 ) = μ 2 \varphi (0) = \frac {\mu }{2} . The construction is based on uniform estimates improved from earlier work on periodic waves for the same equation, together with limiting arguments and a Galilean transform to exclude vanishing waves and waves levelling off at negative surface depth. In fact, the periodic waves can be proved to converge locally uniformly to a wave with negative tails, which is then transformed to the desired branch of solutions. The paper also contains some proof concerning uniqueness and continuity for signed solutions (improved touching lemma).

A Maximisation Technique for Solitary Waves: The Case of the Nonlocally Dispersive Whitham Equation

“…Different and modified perturbative proofs for small-amplitude solitary waves in (1.1) have since been suggested, for example by Stefanov-Wright [32], who used an implicit-function type theorem, and Hildrum [19], who also included more irregular nonlinearities. But it was first with Truong-Wahlén-Wheeler [35] and Ehrnström-Nik-Walker [12] that large solitary waves for the Whitham equation were found. The regularity theory for highest waves, including solitary such, had been established by Ehrnström-Wahlén in [13], and so had symmetry and decay by Bruell-Ehrnström-Pei in [8], but there were no proofs of large-amplitude existence in the solitary case.…”

Analysis on continuity of the solution map for the Whitham equation in Besov spaces

“…It should be mentioned that for negative order dispersive equations, Maehlen and Xue [14] considered the existence of the entropy solutions with initial datum in $$L^{2}\cap L^{\infty }(\mathbb {R})$$. Furthermore, the Whitham‐type equations have attracted a lot of interest for various reasons [1, 4, 9, 16].…”

Remarks on solitary waves in equations with nonlocal cubic terms

“…where u is a real-valued function of time and space, L is a Fourier multiplier and n is a local, nonlinear function. This class of equations includes many well-known models, such as the Korteweg-de Vries and Benjamin-Ono equations when the operator L is a positive-order operator, and Whitham-type equations when L has negative order [10,16]. We restrict our attention here to operators L of positive order, which physically corresponds to situations with surface tension and stronger dispersion [15].…”