Solitary waves of the rotation-generalized Benjamin-Ono equation

Abstract

This work studies the rotation-generalized Benjamin-Ono equation which is derived from the theory of weakly nonlinear long surface and internal waves in deep water under the presence of rotation. It is shown that the solitary-wave solutions are orbitally stable for certain wave speeds.Mathematical subject classification: 35Q35, 76B55, 76U05, 76B25, 35B35

On a model for internal waves in rotating fluids

“…The main theoretical results below will be established for (8), although the particular case of (1) (for which f (u) = u 2 /2 and p = 2)) may be of more interest.…”

“…A third mathematical property of (8), under study in the present paper, is concerned with the existence of solitary wave solutions. These are solutions of permanent form ζ = ϕ(x − c s t) that travel with constant speed of propagation c s = 0 and decay, along with its derivatives, to zero as X = x − c s t → ±∞.…”

“…These are solutions of permanent form ζ = ϕ(x − c s t) that travel with constant speed of propagation c s = 0 and decay, along with its derivatives, to zero as X = x − c s t → ±∞. Substituting into (8) and integrating once, the profile ϕ = ϕ(X) must satisfy…”

“…The purpose in this section is to discuss the existence of solitary-wave solutions of (8) in terms of the parameters α, β, γ and δ. The discussion is based on the corresponding studies in the literature for the Ostrovsky equation, [23,18], and generalized versions, [19], as well as the RMBenjamin-Ono equation, [8]. In all the cases, some results of existence are obtained by applying the Concentration-Compactness theory, [22].…”

“…• Sufficient conditions on the parameters of the model are given in order to obtain existence and uniqueness of solutions of the associated linear problem. The result makes use of the theory on oscillatory integrals and regularity of dispersive equations developed in [14] (see also [15,16,8,19,23]). • The equation is shown to admit three conserved quantities by decaying to zero at infinity and smooth enough solutions.…”