Modified Korteweg-de Vries Equation as a System with Benign Ghosts

TL;DR

This paper investigates the modified Korteweg-de Vries equation's dynamics in space, demonstrating that higher derivatives, termed 'ghosts,' are benign and do not cause blow-ups, implying the quantum version may also be well-defined.

Contribution

It introduces the concept of benign ghosts in the modified Korteweg-de Vries equation and analyzes their impact on the system's classical and potential quantum behavior.

Findings

Higher derivatives lead to benign ghosts

Classical dynamics do not involve blow-up

Quantum problem likely well-defined

Abstract

We consider the modified Korteweg-de Vries equation, $ u_{xxx} + 6u^2 u_x + u_t \ =\ 0 $, and explore its dynamics in {\it spatial} direction. Higher $x$ derivatives bring about the {\it ghosts}. We argue that these ghosts are benign, i.e. the classical dynamics of this system does not involve a blow-up. This probably means that also the associated quantum problem is well defined.