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 {bf 45}, R8309 (1992)] derived from his microscopic rules using a regularization procedure. As well in this approach the nonlinear term $( abla h)^2$ arises naturally from the microscopic dynamics even if the continuous equation is not the Kardar-Parisi-Zhang equation [Phys. Rev. Lett. {bf 56}, 889 (1986)] with quenched noise (QKPZ). Our equation looks like a QKPZ but with multiplicative quenched and thermal noise. The numerical integration of our equation reproduce the scaling exponents of the roughness of this directed percolation depinning model.