On the Oscillatory Tails with Arbitrary Phase Shift for Solutions of the Perturbed Korteweg--de Vries Equation

Abstract

A singularly perturbed Korteweg--de Vries (KdV) equation $ \e^2 \eta_{xxxx} +\eta_{xx}-\eta +\eta^2=0$ for $ -\infty < x < + \infty$ is studied, where $\e > 0$ is a small parameter. It is known that the equation has homoclinic solutions connected to small periodic orbits at infinity. The relationship between amplitude and phase shift of the periodic orbits was established for noncritical cases. To study the critical cases, an asymptotic method is introduced here to formally derive the homoclinic solutions to periodic orbits with arbitrary phase shift. The exact asymptotic formula relating the amplitude of oscillations to the phase shift is obtained. The amplitude of oscillations is determined by the phase shift, and the range of the amplitude can be from $o(\e^n)$ with any $n\ge 1 $ up to $O(\e^{-2} \exp (-\pi /\e ))$ for different phase shift of the oscillations.