Kinematic Condition For Soliton Motions of an $n$-dimensional Continuum in $R^{n+m}$

TL;DR

This paper introduces a new kinematic condition for soliton motions in n-dimensional continua within R^{n+m}, demonstrating its implications through various examples including strings, membranes, and knots, independent of physical specifics.

Contribution

It establishes a novel, physics-independent kinematic condition for soliton motions in higher-dimensional continua and explores its compatibility with equations of motion.

Findings

Kinematic condition proven for n-dimensional continua

Examples include rocking strings, rotating membranes, and classical wave knots

Conditions for all motions to be solitons are identified

Abstract

A new kinematic condition for soliton motions of an $n$-dimensional continuum in $R^{n+m}$, independent of the underlying physics, is proven. The condition and its consequences for different cases are demonstrated. A soliton in a 1D string that rocks back and forth, a rotating soliton in a 2D membrane, and various other cases are presented as examples. It is shown that traveling knots based on classical wave equation are plausible. Cases in which all the motions are solitons are also presented. Compatibility of equations of motion with the kinematic constraint is explored and demonstrated.