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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.

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