Symmetry of solitary waves

Particle trajectories in solitary water waves

“…The parameters c > 0 and h > 0 cannot be arbitrarily chosen: given the wave speed c > 0, the inequality (7) c > gh must hold for nontrivial solutions (see [1]). Moreover, all solitary waves are a priori of positive elevation above their asymptotic limit h, symmetric about a single crest and with a strictly monotone wave profile on either side of this crest, as shown by Craig and Sternberg [8].…”

“…In view of Lemma 1, we deduce that x(t) is strictly increasing for t > 0. On the other hand, (8) and (12) …”

“…On the other hand, the rigorous theory of solitary water waves started with the work of Friedrichs and Hyers [11] and was subsequently readdressed by Beale [4]. Further substantial developments of the analytical theory are mainly due to the work 424 ADRIAN CONSTANTIN AND JOACHIM ESCHER of Toland and collaborators (see [1,2,3,12]) and to the work of Craig and collaborators (see [6,7,8], unveiling to a large extent the structure of these waves. The aim of this contribution is to describe the particle trajectories in the fluid as the solitary wave propagates on the water's free surface.…”

“…The Hamiltonian function for (13) is ψ(X, Y ) in view of (8). Notice that X(t) describes precisely the position of the particle with respect to the wave crest at time t, assuming that initially (at time t = 0) the wave crest is located at x = 0: (X(t), Y (t)) ∈ Ω.…”

Steady Water Waves

“…In this fashion they established the existence of supercritical (c 2 > gh) solitary waves on water of finite depth which are symmetric, positive (η > 0) and monotonically decaying on either side of the crest. These properties are not restrictions: Craig & Sternberg [21] used the method of moving planes to show that any supercritical solitary wave is symmetric, positive and monotonically decaying on either side of its crest. Furthermore, McLeod [47] proved by elementary means that any symmetric, positive solitary wave which decays monotonocally on either side of its crest is supercritical, while Craig [19] has recently demonstrated that there are no solitary waves on water of infinite depth that are single signed (η > 0 or η < 0).…”