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 less than or equal to 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 > 2, in which f must exceed a critical force f(c) in order to drive the interface with constant velocity. We find v(f) similar to (f - f(c))(alpha(d)) for f SE arrow f(c). In particular, the exponent alpha(d) = 2/(d-2) for 2 < d < 4, but saturates at alpha(d) = 1 for d > 4, indicating that for this simple problem, there exists a finite upper critical dimension d(u) = 4. For d > 2 the surface distortion caused by the applied force scales logarithmically with distance within a critical radius R(c) similar to (f - f(c))(-v(d)) = where v(d) = alpha(d)/2. Connections between these results, and the critical properties of the weak/strong-coupling transition in the noisy KPZ equation are pursued.