The observation that the soliton-like solutions of a given second-order nonlinear differential equation define the separatrix of the equivalent autonomous system is used to obtain the one-soliton solutions for theφ⁴ theories (the usual and the one with the wrong sign of the mass term), theφ⁶, theφ⁸, the sine-Gordon theories and the KdV equation. Transformations are given which transform the sine-Gordon equation into an equation belonging to theφ²ⁿ class of theories. A procedure is evolved for obtaining the two-soliton solutions for the sine-Gordon theory without the use of Backlund transformations; it is suggested that this procedure may be useful for investigating the existence of similar solutions for theories of the polynomial type.

A virial theorem for solitons derived by Friedberg, Lee and Sirlin is used to reduce a system of second order equations to an equivalent first order set. It is shown that this theorem, when used in conjunction with our earlier observation that soliton-like solutions lie on the separatrix, helps in obtaining soliton-like solutions of theories involving coupled fields. The method is applied to a system of equations studied extensively by Rajaraman. The ’t-Hooft-Polyakov monopole equations are then studied and we obtain the well-known monopole solutions in the Prasad-Sommerfeld limit (λ=0); for the case λ≠0, we succeed in obtaining a non-trivial algebraic constraint between the fields of the theory.