Emergence of Envelope Solitary Waves from Initial Localized Pulses within the Ostrovsky Equation

Abstract

We study the emergence of steady wave packets in the form of envelope solitary waves (envelope solitons) which evolve from localized pulse-type initial conditions in the long-term evolution within the framework of the Ostrovsky equation. Ostrovsky equation derived in 1978 describes weakly nonlinear oceanic waves affected by the Earth’s rotation. Subsequently it became clear that this equation is universal, and is widely used for the description of waves in various settings. It is well-known that in applications to media with a negative small-scale dispersion this equation does not possess steady solitary wave solutions. In particular, the evolution of an initial Korteweg-de Vries (KdV) soliton results in the emergence of envelope solitons which can be described by the nonlinear Schr¨odinger equation (NLS) when the amplitude of the initial KdV soliton is sufficiently small. However, the derivation of the corresponding NLS still has not been sufficiently comprehensive, which resulted in contradiction with some numerical simulations. Here, we revisit this problem and suggest a new approach to the construction of the emerging envelope solitons.