Strong coupling probe for the Kardar-Parisi-Zhang equation

We present an exact solution of the deterministic Kardar-Parisi-Zhang (KPZ) equation under the influence of a local driving force f. For substrate dimension d = 2 we recover the well-known result that for arbitrarily small f > 0, the interface develops a non-zero velocity v(f). Novel behaviour is found in the strong-coupling regime for d > 1, in which f must exceed a critical force f in order to drive the interface with constant velocity. We find v(f) ~ (f - f) for f ? f. In particular, the exponent a(d) = 2/(d-2) for 2 <d <4, but saturates at a(d) = 1 for d > 4, indicating that for this simple problem, there exists a finite upper critical dimension d = 4. For d > 2 the surface distortion caused by the applied force scales logarithmically with distance within a critical radius R ~ (f - f), where v(d) = a(d)/2. Connections between these results, and the critical properties of the weak/strong-coupling transition in the noisy KPZ equation are pursued.