Cylindrically Symmetric Solitons with Nonlinear Self-gravitating Scalar Fields

Static, cylindrically symmetric solutions to nonlinear scalar-Einstein equations are considered. Regularity conditions on the symmetry axis and flat or string asymptotic conditions are formulated in order to select soliton-like solutions. Some non-existence theorems are proved, in particular, theorems asserting (i) the absence of black-hole and wormholelike cylindrically symmetric solutions for any static scalar fields minimally coupled to gravity and (ii) the absence of solutions with a regular axis for scalar fields with the Lagrangian L = F (I) , I = φαφα , for any function F (I) possessing a correct weak field limit. Exact solutions for scalar fields with an arbitrary potential function V (φ) are obtained by quadratures and are expressed in a parametric form in a few ways, where the parameter may be either the coordinate x , or the φ field itself, or one of the metric coefficients. It is shown that soliton-like solutions exist only with V (φ) having a variable sign. Some explicit examples of the solutions (including a soliton-like one) and their flat-space limit are discussed.