Dissipative Soliton - Particle Properties and Universality

Particle Properties and Universality

DSs in many different systems show universal particle-like properties. To understand and describe the latter, one may try to derive "particle equations" for slowly varying order parameters like position, velocity or amplitude of the DSs by adiabatically eliminating all fast variables in the field description. This technique is known from linear systems, however mathematical problems arise from the nonlinear models due to a coupling of fast and slow modes.

Similar to low-dimensional dynamic systems, for supercritical bifurcations of stationary DSs one finds characteristic normal forms essentially depending on the symmetries of the system. E.g., for a transition from a symmetric stationary to an intrinsically propagating DS one finds the Pitchfork normal form

for the velocity v of the DS, here σ represents the bifurcation parameter and σ₀ the bifurcation point. For a bifurcation to a "breathing" DS, one finds the Hopf normal form

for the amplitude A of the oscillation. It is also possible to treat "weak interaction" as long as the overlap of the DSs is not too large. In this way, a comparison between experiment and theory is facilitated., Note that the above problems do not arise for classical solitons as inverse scattering theory yields complete analytical solutions.