Epilogue

The essence of this book is the description of the method of inverse scattering. The book is too short to do more than outline the chief properties of solitons and indicate some lesser properties by a few remarks and problems. For that reason the chief properties stand out more clearly.

Research into the physical, earth and life sciences has led to the study of hundreds of nonlinear equations. We have mentioned only a little of this. Indeed, it is too wide to be described in any single volume. Each reader of this introduction to solitons may, however, go on to study the derivation of the equations used in his own specialist field. On having obtained an appropriate nonlinear system, it is natural to seek the waves of permanent form and to test whether they are stable and whether they are solitons. The existence of soliton interactions is the exception rather than the rule. Yet scores of nonlinear systems are already known to have soliton solutions. Even now it is not known how to ascertain definitely whether any given nonlinear system has solitons which may preserve their identities after interacting with one another. The preceding chapters, nonetheless, outline many ideas which may determine, or at least suggest, that a given nonlinear system has, or has not, soliton solutions and which indicate how the properties of the solitons may be found. The ideas are summarized in the following points.