The diffusion–reaction (D–R) Hamiltonian and the solutions of certain types of linear and nonlinear D–R equations in one dimension

Abstract

With a view to having further insight into the mathematical content of the non-Hermitian Hamiltonian associated 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 case (a) we use the standard techniques to handle the quadratic and cubic nonlinearities in the D–R equation whereas for 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.