Considered herein is the stability problem of solitary wave solutions of a generalized Ostrovsky equation, which is a modification of the Korteweg-de Vries equation widely used to describe the effect of rotation on surface and internal solitary waves or capillary waves.
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.
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
We consider the stability of solitary waves of a class of 5th order KdV equations. It is known that their stability is determined by the second derivative of a function of the wave speed d(c). We perform a detailed investigation of the properties of this function, both analytically and numerically. For a class of homogeneous nonlinearities, we precisely determine the regions of wave speeds for which the solitary waves are stable or unstable.
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