Dynamical scaling laws in two types of extended Hamiltonian systems at dissipation onset

The transition to dissipation in one-dimensional extended Hamiltonian systems with saddle-node bifurcations of stationary solutions is characterized. Three different systems are studied: (i) nonlinear Schrِdinger flow past a localized obstacle; (ii) sine-Gordon pendulum chains forced by a local torque; (iii) electrically charged nonlinear Schrِdinger flows. In case (i), no frequency gap is present in the dispersion relation. In contrast, in cases (ii) and (iii) a minimum frequency for propagating waves exists. In the gapless case, the growth rates of the unstable modes and the frequency of supercritical soliton emission are found to scale as the square root of the bifurcation parameter. No subcriticality is observed. In contrast, when a frequency gap is present, subcritical soliton emission takes place. Logarithmic and one-fourth power scaling laws are found, respectively, at the bottom and top of the subcriticality window.