Dispersive Decay for the 1D Klein-Gordon Equation with Variable Coefficient Nonlinearities

TL;DR

This paper investigates the dispersive decay of solutions to the 1D Klein-Gordon equation with variable coefficient nonlinearities, revealing novel normal-form transformations that facilitate decay and smoothness analysis.

Contribution

It introduces new quadratic and cubic normal-form transformations for variable coefficient nonlinearities, enabling decay and regularity results for the 1D Klein-Gordon equation.

Findings

Proved dispersive decay for the 1D Klein-Gordon with variable cubic coefficients.

Established smoothness of solutions in weighted spaces.

Developed novel normal-form transformations for cubic interactions.

Abstract

We study the 1D Klein-Gordon equation with variable coefficient nonlinearity. This problem exhibits an interesting resonant interaction between the spatial frequencies of the nonlinear coefficients and the temporal oscillations of the solutions. In the case when only the cubic coefficients are variable we prove dispersive decay and smoothness of the solution in weighted spaces with the help of quadratic and cubic normal-forms transformations. In the case of cubic interactions these normal forms appear to be novel.