A discrete spectrum of solutions of the wave equation with strong cubic nonlinearity

Abstract

The nonlinear wave equation, φ _tt −Δφ+φ³=0, has many solutions that are periodic in time and localized in space, all with infinte energies. The search for spherically symmetric solutions that are well represented by the simple approximation, φ(r, t)≡A(r) sin ωt, leads to a discrete spectrum of solutions{φ _N (r, t; ω)}. The solutions are nonlinear wavepackets, and they can be regarded as particles. The asymptotic theory (ω→∞) of the motion of the guiding center of theNth wavepacket, in the presence of a specified potential, is characterized by an infinite mechanical mass and an infinte interaction mass, and they are compatible. The rest mass in the classical relativistic mechanics of guiding centers ism ₀ c ²=ħ _N ω; i.e. the spectrum {φ _N } determines a spectrum of Planck's constants.