  In this monograph, two sets of parabolic differential equations are studied,
each with nonlinear medium response. The equations are generally referred to as
"Newell-Whitehead-Segel equation," which model a wide variety of nonlinear
physical, mechanical and biological systems. Nonlinear medium response can be
viewed in many perspectives, such as, memory response from the medium, whereby,
the medium "remembers" earlier influences; reactive responses, whereby, the
medium is actively responsive to input, such as, chemical reactivity,
turbulence and many other circumstances; these equations arise often in the
biological sciences when modeling population dynamics, whether the population
be genomic, such as, alleles, or animal species in the environment; finally,
these sets of equations are often employed to model neurological responses from
excitable cellular media. The solutions provided are of a very general nature,
indeed, whereby, a canonical set of solutions are given for a class of
nonlinear parabolic partial differential equations with nonlinear medium
response expressed as either a p-times iterative convolution or p-times
multiplicative response. The advantage of canonical solution sets are these
solutions involve classic representative forms, such as, Green's function or
Green's heat kernel and aid researchers in further complication, analysis and
understanding of the systemic behavior of these types of nonlinear systems.
