Exact solitary wave solutions of nonlinear evolution and wave equations using a new direct algebraic method

W. Hereman, P. P. Banerjee, A. Korpel, G. Assanto, and A. Van Immerzeele

J. Phys. A: Math. Gen. 19, 607-628 (1986)

Abstract:

We present a systematic and formal approach toward finding solitary wave solutions of non-linear evolution and wave equations from the real exponential solutions of the underlying linear equations. The physical concept is one of the mixing of these elementary solutions through the non-linearities in the system. In the present paper the emphasis is, however, on the mathematical aspects, i.e. the formal procedure necessary to find single solitary wave solutions. By means of examples we show how various cases of pulse-type and kink-type solutions are to be obtained by this method. An exhaustive list of equations so treated is presented in tabular form, together with the particular intermediate relations necessary for deriving their solutions. We also outline the extension of our technique to construct N-soliton solutions and indicate connections with other existing methods.