Example of a self-localizing nonlinear relativistic field

The real scalar relativistic field with self-interaction $(\lambda/4)\psi^4$ in the Lagrangian is studied. An exact solution is obtained for the one- and three-dimensional cases. It is shown that in the one-dimensional case the field is localized in a certain region whose size varies in accordance with a Lorentz transformation. The energy and momentum of the field are inversely proportional to the coupling constant, and they are related by the same relativistic equation as for a particle with a certain internal energy. In the three-dimensional case the field is localized only in one dimension.