Can one hear the full nonlinearity of a PDE from its small excitations?

TL;DR

This paper demonstrates how to reconstruct a nonlinear PDE of sine-Gordon type from linearized data around a stationary kink, revealing the full nonlinearity from small excitations.

Contribution

It introduces a method to recover the entire nonlinear PDE from linearization around a stationary solution, linking the ground state to the Goldstone mode.

Findings

Successfully reconstructs the sine-Gordon type PDE from linearized data

Establishes the ground state as the Goldstone mode of the nonlinear PDE

Provides a framework for understanding nonlinearity from small excitations

Abstract

In this article, we show how one can restore an unknown nonlinear partial differential equation of a sine-Gordon type from its linearization around an unknown stationary kink. The key idea is to regard the ground state of the linear problem as the translation-related Goldstone mode of the nonlinear PDE sought after.