More Basic Solutions

Abstract

The purpose of this chapter is to extend the results of Chapters 2–4 in three ways. This will be carried out in Sections 5.1–5.3 that follow. In Section 5.1, we briefly indicate how to modify Theorems 3.2 and 4.40 to obtain more complex heteroclinic solutions of (PDE). These new solutions, U_k, $3 \leq k \leq n, \$ are higher-dimensional analogues of U₁ of Theorem 3.2 and U₂ of Theorem 4.40. By higher-dimensional we mean that U_k is periodic in x_k+1,...,x_n, lies between U_k-1 and $\tau_{-1}^{k-1} U_k-1,\$ and is heteroclinic in x_k from U_k-1 to $\tau_{-1}^{k-1} U_{k-1}.$ All of these new solutions have rotation vector α = 0.