Slow passage through parametric resonance for a weakly nonlinear dispersive wave

TL;DR

This paper analyzes how a weakly nonlinear dispersive wave described by the Klein-Gordon equation responds to a slowly varying parametric resonance, deriving formulas that connect the wave's behavior before and after resonance.

Contribution

It provides a novel connection formula for the asymptotic solution of the nonlinear Klein-Gordon equation during parametric resonance crossing.

Findings

Derived a connection formula for the wave solution across resonance

Identified the impact of slow frequency variation on wave dynamics

Enhanced understanding of parametric resonance in nonlinear dispersive waves

Abstract

A solution of the nonlinear Klein-Gordon equation perturbed by a parametric driver is studied. The frequency of the parametric perturbation varies slowly and passes through a resonant value. It yields a change in a solution. We obtain a connection formula for the asymptotic solution before and after the resonance.