Single vortex pinning and shock waves in Burgers turbulence

The vortex line in a random potential is equivalent to the directed polymer problem that, through the Cole-Hopf transformation for the partition function, can be mapped to the Kardar-Parisi-Zhang equation and then to the viscous Burgers equation. The velocity in the Burgers equation then maps to the gradient of the free energy of the directed polymer (vortex line), the viscosity maps to the temperature T in the original directed polymer problem, and the pinning potential corresponds to the random pumping in the stochastic Burgers equation. We consider a model of a direct polymer subject to the pinning potential of well separated defects. At zero temperature the equilibrium shape of the polymer will be a set of straight line segments connecting defects via zigzag type way. This shape of the polymer translates to the characteristics of the Burgers equation. Branch crossing points for polymer correspond to shock waves. Changing the length of the polymer L and the positions of the pins changes the Labusch parameter κ. The appearance of the shock wave corresponds to the onset of strong pinning, κ(L) = 1. Analyzing the branch crossing for the directed polymer, with one defect one can easily get the breaking time of the shock and its subsequent evolution, x_s(t) and ?v(t). The Cole-Hopf transformation simplify the task: in the Burgers problem, we are dealing with a nonlinear partial differential equation, whereas for the directed polymer, we merely have to solve algebraic equations. For several defects, the branch crossing becoming more complicated. In the Burgers language the solution picks up many shock waves that can merge or get separated. At finite temperature thermal occupation of different branches of polymer gives the finite width of the shock wave.