One-dimensional field theories with odd-power self-interactions

Classical solutions to nonlinear field theories are considered as model particles. Two fields are examined here, the $\lambda {\phi }^3$ field and a generalization of the sine-Gordon system. Each of these fields is in one space dimension and quantization is accomplished using the WKB method. Static solutions to the $\lambda {\phi }^3$ field are shown to represent objects with an internal structure resembling a dumbbell. The quantum mass of these objects is computed in the weak-coupling limit and an approximate expression for the classical force between two of these objects is obtained. This force seems to be attractive and constant at large separations. In the case of the generalized sine-Gordon field it is shown that classical solutions to the field equation may be obtained by a transformation from known solutions to the sine-Gordon equation. The behavior of this field is therefore similar to that of the sine-Gordon field.