Approximate particle-like solutions for a nonlinear relativistic scalar complex field model in 3+1 dimensions

Abstract

A weakly nonlinear Lorentz invariant complex field model in 3+1 dimensions is studied by an asymptotic perturbation method, based on Fourier expansion and spatio-temporal rescaling. It is shown that a nonlinear system of partial differential equations describes oscillation amplitudes of Fourier modes. This system is C-integrable, i.e., can be linearized through a suitable transformation of the dependent and independent variables. We resolve the Cauchy problem and demonstrate that localized nondispersive waves (envelope solitons) with finite energy exist under appropriate initial conditions. These particle-like solutions propagate with the group velocity of their carrier wave. During a collision solitons maintain their shape, because the only change is a phase shift. Energy E and momentum p of solitons are identical to those of a relativistic particle. If the Planck constant is connected to the spatial dimension of the envelope soliton, then we obtain at the lowest order of approximation the quantum relations E=ℏ ω, λ= h/ p, where λ and ω are wavelength and frequency of the carrier wave. This work represents a possible way to achieve the Einstein–de Broglie soliton–particle concept.