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

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.