  We turn high energy elastic scattering of hadrons into an initial value
problem using an evolution equation based on the Regge Field Theory, which has
a form of the complex nonlinear reaction-diffusion equation, with time being
played by the logarithm of energy. The initial conditions are provided by the
data-driven models for the real and imaginary parts of the amplitude. Numerical
calculations of pp differential cross sections and forward quantities for LHC
energies agree very well with experimental data extending up to, and including
the diffractive cone. Furthermore, we show that at current accessible energies
the non-linear effects play an important role, as the impact parameter space
profiles approach the unitarity bound. The equation also predicts some other
effects discussed in the literature, like the hollowness of nuclear matter or
existence of stationary points in momentum transfer t.
