The de Broglie soliton as a localized excitation of the action function

Guided by the formal analogy between the classical relativistic HamiltonJacobi equation and the dynamic equation for the premixed gas flame, a new class of time-dependent solutions for the relativistic quantum HamiltonJacobi equation, (1c2)(∂S∂t)2-(∇S)2=iLatin small letter h with stroke□S+m2c2, is revealed. The equation is shown to permit solutions in the form of breathers (nondispersive oscillating/spinning solitons) displaying simultaneous particle-like and wave-like behavior adaptable to the properties of the de Broglie clock. Within this formalism the de Broglie wave acquires the meaning of a localized excitation of the action function, a complex-valued potential in configuration space. For a free non-spinning particle in the rest system the breathing action function reads, S=-mc2t-iLatin small letter h with strokeln1+αexp[-i(mc2Latin small letter h with stroke)t]j0(kr), where j0(kr)=sin(kr)kr, k=3(mcLatin small letter h with stroke), r=x2+y2+z2, and |α| is a parameter controlling the breather's intensity. The problem of quantization in terms of the breathing action function and the double-slit experiment are discussed.