Classical solutions by inverse scattering transformation in any number of dimensions. I. The gap equation and the effective action

A method to find space-dependent extrema (soliton or instanton) of one-loop effective actions (local terms plus a logarithm of a functional determinant) is given. This method is based on a suitable inverse scattering transformation and can be used in any number of space dimensions, provided the field configurations depend on only one variable. The effective action of ${\left({\overrightarrow{\phi }}^2\right)}^2$ theory for the $\frac{1}{N}$ series in one, two, three, and four dimensions is worked out in detail. Explicit expressions for the effective action in terms of scattering data are derived. It is found that the gap equation for massless ${\left({\overrightarrow{\phi }}^2\right)}^2$ theory (in four dimensions) is analytically solvable for spherically symmetric fields.