The Construction of Implicit and Explicit Solitary Wave Solutions of Nonlinear Partial Differential Equations.

By means of easy examples, such as the Korteweg-de Vries, the Harry Dym, the sine-Gordon equations, and the Hirota coupled system, it is shown how nonlinear partial differential equations can be exactly solved by a direct algebraic method. The physical concept, on which the method relies, is one of generation and mixing of the real exponential solutions of the underlying linear equations. This approach leads in a straightforward way to single solitary waves of pulse, kink and cusp shape. The extension of the method towards the construction of multi-soliton solutions and the connections with other direct methods are outlined.