  With a view to having further insight into the mathematical content of the
non-Hermitian Hamiltonian associaterd with the diffusion-reaction (D-R)
equation in one dimension, we investigate (a) the solitary wave solutions of
certain types of its nonlinear versions, and (b) the problem of real eigenvalue
spectrum associated with its linear version or with this class of non-Hermitian
Hamiltonians. For the case (a) we use the standard techniques to handle the
quadratic and cubic nonlinearities in the D-R equation whereas for the case (b)
a newly proposed method, based on an extended complex phase space, is employed.
For a particular class of solutions, an Ermakov system of equations is also
found for the linear case. Further, corresponding to the 'classical' version of
the above one-dimensional complex Hamiltonian, an equivalent integrable system
of two, two-dimensional real Hamiltonians is suggested.
