Formation of envelop solitary waves from the localised pulses within the Ostrovsky equation

Abstract: We study the formation of envelope solitons from the Korteweg–de Vries (KdV) solitons in the long term evolution within the framework of the Ostrovsky equation. This equation was derived by L.A. Ostrovsky in 1978 for the description of weakly nonlinear oceanic waves affected by the Earth' rotation. Subsequently, it became clear that this equation is rather universal; currently, it is widely used for the description of nonlinear waves in various media. This equation is, apparently, non-integrable and even does not possess steady solitary wave solutions in application to media with negative small-scale dispersion. As has been shown by Grimshaw and Helfrich (2008), long-term evolution of initial pulses in the form of small-amplitude KdV soliton results in the emergence of envelope solitons which can be described by the nonlinear Schrodinger (NLS) equation. However, the generalised NLS equation derived by Grimshaw and Helfrich (2008) provides the results which are in contradiction with the numerical simulations. The problem was later revisited by Grimshaw and Stepanyants (2020) and was shown that the wave packet asymptotically appearing after a long-term evolution of a KdV soliton can be described by the conventional NLS equation. The solution obtained for an envelope soliton agrees well with the results of numerical simulations.

mathematical physicsanalysis of PDEsclassical analysis and ODEsdynamical systemsnumerical analysisexactly solvable and integrable systemsfluid dynamics

Audience: researchers in the topic

Mathematical models and integration methods