Nonlinear Evolution Equations of the Soliton Type: Old and New Results

An overview on the study of nonlinear evolution equations of soliton type is provided. In addition, 5th-order nonlinear evolution equations are shown to be connected to the Caudrey–Dodd–Gibbon–Sawada–Kotera (CDGSK) equation via Bäcklund transformations. The links are depicted in a wide net of links which we term a Bäcklund Chart. The links obtained previously by Rogers and Carillo and by Carillo and Fuchssteiner are revisited, and new results are obtained. A 5th-order nonlinear evolution equation, which does not seem to appear in any list of integrable equations, is provided. All the connected equations exhibit a very interesting symmetry structure enjoyed by the corresponding full hierarchies. Indeed, they all admit a hereditary recursion operator. Hence, each one of the mentioned equations represents the base member of a corresponding hierarchy of equations. These hierarchies are constructed via the recursive application of the respective recursion operators. The symmetry properties of such equations are recalled. Finally, we compare the net of links, derived via Bäcklund transformations, in the case of the fifth-order nonlinear evolution equations with an analog net of links connecting third-order Korteweg-de Vries (KdV) and modified Korteweg-de Vries (mKdV) equations. Analogies and discrepancies between the connections established in the case of fifth-order equations with respect to those established in the case of third-order equations are analyzed. This study aims to open the way for the construction of corresponding non-Abelian equations of the fifth order.