On the class of nonlinear PDEs that can be treated by the modified method of simplest equation. Application to generalized Degasperis-Processi equation and \(b\)-equation
The method of simplest equation relies on exact solutions of some simple equations such as the equations of Bernoulli and Riccati, which are embedded into much more complicated equations such as the Degasperi-Processi equation. This method was proposed by \textit{N.A. Kudryashov} in [Phys. Lett., A 342, No.~1--2, 99--106 (2005; Zbl 1222.35054)]. The authors propose a modification of this method, where the procedure analogous to the search for a Painlevé property is developed to identify the balance equations and the underlying ``simplest'' equation. The authors apply this modification to the construction of exact traveling wave solutions to two complicated equations, the Degasperis-Processi equation and the \(b\)-equation. The solutions are constructed analytically and represented graphically. The authors also compare their method with the known exp-function method.
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