Weierstrass's criterion and compact solitary waves

TL;DR

This paper extends Weierstrass's classical theory to identify differential equations that admit compact and semicompact solitary waves, linking mathematical analysis with continuum mechanics applications.

Contribution

It introduces a simple generalization of Weierstrass's criterion to find equations supporting compact solitary waves, bridging classical mechanics and continuum mechanics.

Findings

Identification of differential equations with compact solitary wave solutions

Connection between generalized Weierstrass's theory and shear waves in continuum mechanics

Potential for new wave solutions in specialized constitutive laws

Abstract

Weierstrass's theory is a standard qualitative tool for single degree of freedom equations, used in classical mechanics and in many textbooks. In this Brief Report we show how a simple generalization of this tool makes it possible to identify some differential equations for which compact and even semicompact traveling solitary waves exist. In the framework of continuum mechanics, these differential equations correspond to bulk shear waves for a special class of constitutive laws.