Soliton Coupling Driven by Phase Fluctuations in Auto-Parametric Resonance

Introduction. For nonlinear field theory models in 1+1–dimensional space–time the equations of motion admit finite energy and finite width solutions called solitons [1]. The interest in such low-dimensional sine– Gordon (SG) models as an universal concept in nonlinear science arises from their integrability, duality properties, non-perturbative aspects, and electric-magnetic duality in gauge theories. The solitons retain their identity after collisions, can annihilate with anti–solitons, many– soliton solutions obey Pauli’s exclusion principle. As pointed out by Skyrme [2] this can be interpreted as a fermion–like behavior. Vertex solitons are localized and non-singular solutions of a non-linear field theory, whereas the energy-momentum transfer between solitons can be assigned to linear bosons. In this paper we will try to determine the coupling strength of vertex soliton and linear waves. With the focus on low-dimensional autonomous behavior, the interaction can be approached by the equation of Lord Rayleigh with coupling and dissipation driven by phase fluctuations. The second order equation includes two terms that balance wave dissipation and regeneration as a special case of self– excited auto–parametric resonance [3]. This system can i.e. model simplified music instruments (in the original work of Rayleigh a clarinet reed).