The Existence of Soliton-Like Solutions

Abstract

In this chapter by way of an example of the φ⁶-model we shall consider some problems concerning the existence of localized soliton-like solutions. The choice of the model is determined by its importance in applications (see Chapter 1 and 7) and by the rather rich set of solution-like solutions in vacuum and condensate. Firstly, we seek virial relations which exist for Lagrangian systems. From these relations the necessary conditions for the existence of soliton-like and particle-like solutions will follow^*). We then formulate the concept of a mechanical analogy, whereby the conditions for the existence of soliton-like and particle-like solutions may be visualized and, in principle, can be developed for non-Lagrangian systems. Here we consider D-dimensional models.