On symmetry classification of third-order evolutionary systems of divergent type

Vector hyperbolic equations with higher symmetries

“…There also exist Bäcklund transformations relating solutions of two different equations. In many cases, Bäcklund transformations are first-order equations (see, e.g., [10]), but examples of higher-order Bäcklund transformations are known (see, e.g., [11]). Bäcklund transformations depending on nonlocal variables are also known [12].…”

Two-Field Integrable Evolutionary Systems of the Third Order and Their Differential Substitutions

“…It was shown in [21] that the first sequence of the canonical densities ρ n for system (1.1) obtained from (2.11) is equivalent to the first sequence of the canonical densities τ n obtained from (2.12) and the second sequence of the canonical densitiesρ n is equivalent to the second sequence of the canonical densitiesτ n . Hence,…”

“…It is shown in [21] that systems (3.10) and (3.12) are unique divergent systems of the form (2.21) that satisfy the integrability conditions.…”

“…Partial solutions of the classification problem for a = 0 and ord G 1 have been obtained in [20], and in [21] for divergent systems with a = 1. A complete list of integrable systems of the form (1.1) does not exist today because the problem is too cumbersome and the set of integrable systems is very large.…”

“…Review paper [27] is devoted to higher symmetries, exact integrability and related problems. Peculiarities of systems (1.1) have been discussed in [21]. For the sake of completeness, the main points of the symmetry method and some results necessary for understanding of this paper are considered in the Sections 2-4.…”