Solution spectrum of nonlinear diffusion equations

The stationary version of the nonlinear diffusion equation–c/t +D c = ₁c – ₂c² can be solved with the ansatz c = _p ^{= 1/} A_p(coshkx) ^–p , inducing a band structure with regard to the ratio ₁/₂. The resulting solution manifold can be related to an equilibrium of fluxes of nonequilibrium thermodynamics. The modification of this ansatz yielding the expansion c = skp = 1/ A _pq (coshkx)^–p (cosht) ^–q–1 sinht + b(cosht)^–q] represents a solution spectrum of the time-dependent nonlinear equation, and the stationary version can be found from the asymptotic behavior of the expansion. The solutions can be associated with reactive processes propagating along molecular chains, and their applicability to biophysical processes such as active transport phenomena and control circuit problems is discussed. There are also applications to cellular kinetics of clonogenic cell assays and spheroids.