Shock waves in the dissipative Toda lattice

We consider the propagation of a shock wave (SW) in the damped Toda lattice. The SW is a moving boundary between two semi-infinite lattice domains with different densities. A steadily moving SW may exist if the damping in the lattice is represented by an 'inner' friction, which is a discrete analogue of the second viscosity in hydrodynamics. The problem can be considered analytically in the continuum approximation, and the analysis produces an explicit relation between the SW's velocity and the densities of the two phases. Numerical simulations of the lattice equations of motion demonstrate that a stable SW establishes if the initial velocity is directed towards the less dense phase; in the opposite case, the wave gradually spreads out. The numerically found equilibrium velocity of the SW turns out to be in very good agreement with the analytical formula even in a strongly discrete case. If the initial velocity is essentially different from the one determined by the densities (but has the correct sign), the velocity does not alter significantly, but instead the SW adjusts itself to the given velocity by sending another SW in the opposite direction.