High order amplitude equation for steps on creep curve

We consider a model proposed by one of the authors for a type of plastic instability found in creep experiments which reproduces a number of experimentally observed features. The model consists of three coupled non-linear differential equations describing the evolution of three types of dislocations. The transition to the instability has been shown to be via Hopf bifurcation leading to limit cycle solutions with respect to physically relevant drive parameters. Here we use reductive perturbative method to extract an amplitude equation of up to seventh order to obtain an approximate analytic expression for the order parameter. The analysis also enables us to obtain the bifurcation (phase) diagram of the instability. We find that while supercritical bifurcation dominates the major part of the instability region, subcritical bifurcation gradually takes over at one end of the region. These results are compared with the known experimental results. Approximate analytic expressions for the limit cycles for different types of bifurcations are shown to agree with their corresponding numerical solutions of the equations describing the model. The analysis also shows that high order nonlinearities are important in the problem. This approach further allows us to map the theoretical parameters to the experimentally observed macroscopic quantities.Comment: LaTex file and eps figures; Communicated to Phys. Rev.

Condensation of bosons in kinetic regime

We study the kinetic regime of the Bose-condensation of scalar particles with weak $\lambda \phi^4$ self-interaction. The Boltzmann equation is solved numerically. We consider two kinetic stages. At the first stage the condensate is still absent but there is a nonzero inflow of particles towards ${\bf p} = {\bf 0}$ and the distribution function at ${\bf p} ={\bf 0}$ grows from finite values to infinity in a finite time. We observe a profound similarity between Bose-condensation and Kolmogorov turbulence. At the second stage there are two components, the condensate and particles, reaching their equilibrium values. We show that the evolution in both stages proceeds in a self-similar way and find the time needed for condensation. We do not consider a phase transition from the first stage to the second. Condensation of self-interacting bosons is compared to the condensation driven by interaction with a cold gas of fermions; the latter turns out to be self-similar too. Exploiting the self-similarity we obtain a number of analytical results in all cases.Comment: 23 pages plus 11 uuencoded figures, LaTeX, REVTEX 3.0 versio