Stability and weak rotation limit of solitary waves of the Ostrovsky equation

Abstract

In this paper we study several aspects of solitary wave solutions of the Ostrovsky equation. Using variational methods, we show that as the rotation parameter goes to zero, ground state solitary waves of the Ostrovsky equation converge to solitary waves of the Korteweg-deVries equation. We also investigate the properties of the function d(c) which determines the stability of the ground states. Using an important scaling identity, together with numerical approximations of the solitary waves, we are able to numerically approximate d(c). These calculations suggest that d is convex everywhere, and therefore all ground state solitary waves of the Ostrovsky equation are stable.

On the ground states of the Ostrovskyi equation and their stability

“…Similar result holds for limit waves of (9). For 1 < < 5 and > 0, let be a constrained minimizer for (12). Then, again are weakly nondegenerate and under (17), they are spectrally stable as well.…”

“…These are fourth-order nonlinear ordinary differential equations (ODEs), for which there is not very well-developed theory. In particular, for noninteger values of , existence has been proved by variational methods, [11][12][13] so that (8) is an Euler-Lagrange equation for these constrained minimizers. Regarding uniqueness, which is well known to be a hard issue, is only known in the case = 2.…”

“…Our approach to (8) is variational, but rather different from the other works. [11][12][13] More precisely, Levandosky and Liu construct their waves as minimizers of energy, subject to a fixed +1 norm. This method allows for a construction of waves for any power of > 1.…”

“…• For 1 < < 5, the minimization problems (11) and (12) have constrained minimizers, ∈ 4 (ℝ), ∈ 2 ∩̇− 2 (ℝ) ∶ ′ = , which satisfy the…”

On a model for internal waves in rotating fluids

“…• 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.…”