Conoidal waves in the $\varphi^4$ model with self-interacting currents

A new form of solutions for relativistic wave equations in the form of rotating “conoidal” waves is found and studied. These waves are more general solutions of periodic type. These essentially nonlinear waves, expressed in terms of elliptic functions, include as special cases nonperiodic solutions of soliton type. In a limiting case, the nonlinear waves can go over into linear oscillations, forming a vacuum condensate.