Stability of Solitary Waves of a Generalized Ostrovsky Equation

Abstract

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.

Spectral Stability for Classical Periodic Waves of the Ostrovsky and Short Pulse Models

“…Indeed, Hence, we establish that the 2L periodization of the function z constructed here is not even continuous at ±L. On the other hand, z ∈ C ∞ (−L, L) and it satisfies the eigenvalue equation (12) for ξ ∈ (−L, L) by tracing back the changes of variables.…”

Hamiltonian formulation, nonintegrability and local bifurcations for the Ostrovsky equation

“…Surprisingly, even if the mass of the solitary wave of the KdV equation is not zero, it is shown [21] that the limit of the solitary waves of the Ostrovsky equation tends to the solitary wave of the KdV equation as the rotation parameter γ tends to zero. For β < 0, solitary waves in the form of stationary localized pulses cannot exist at all [8,20,31].…”

Sharp well-posedness of the Cauchy problem for a generalized Ostrovsky equation with positive dispersion

“…The parameter β determines the type of dispersion, more precisely, β < 0 (negative dispersion) for surface and internal waves in the ocean or surface waves in a shallow channel with an uneven bottom and β > 0 (positive dispersion) for capillary waves on the surface of liquid or for oblique magneto-acoustic waves in plasma [1,[13][14][15]. Some authors have investigated the stability of the solitary waves or soliton solutions of (1.2); for instance, see [16][17][18]. Many people have studied the Cauchy problem for (1.2), for instance, see [17,[19][20][21][22][23][24][25][26][27][28][29][30].…”

“…Levandosky [18] studied the stability of ground state solitary waves of (1.4) with homogeneous nonlinearities of the form f (u) = c 1 |u|…”

“…Recently, Li et al [34] proved that the Cauchy problem for the Ostrovsky equation with negative dispersion is locally well-posed in H -3 4 (R). Levandosky and Liu [16] studied the stability of solitary waves of the generalized Ostrovsky equation,…”