Elastic rods, Weierstrass’ theory and special travelling waves solutions with compact support

By an extension of the usual Weierstrass discussion for first-order differential equations, we determine a simple criterion which allows us to determine when compact structures are possible in second-order wave equations. For higher-order wave equations, such as the modified improved Boussinesq equation, this criterion may still be used, but the appearance of an overdetermined system of differential equations which is not trivially satisfied, unlike in the second-order case, renders the possibility of compact structures a happenstance.