Theoretical continuous equation derived from the microscopic dynamics for growing interfaces in quenched media

We present an analytical continuous equation for the Tang and Leschhorn model [Phys. Rev. A 45, R8309 (1992)] derived from their microscopic rules using a regularization procedure. As well in this approach, the nonlinear term (nablah)(2) arises naturally from the microscopic dynamics even if the continuous equation is not the Kardar-Parisi-Zhang equation [Phys. Rev. Lett. 56, 889 (1986)] with quenched noise (QKPZ). Our equation is similar to a QKPZ equation but with multiplicative quenched and thermal noise. The numerical integration of our equation reproduces all the scaling exponents of the directed percolation depinning model.