Narrow Results

In this paper, the categorization of nonlocal symmetries of the (1+1)-dimensional nonlinear Vakhnenko equation is presented. The governing equation is turned into a set of invertibly equivalent partial differential equations (PDE) via the invertible transformation of the canonical coordinates. The nonlocal symmetries of the aforementioned PDE were derived by establishing an inverse potential system and a locally related subsystem of the system of invertibly equivalent PDEs. In addition, using similarity variables and group invariant solutions connected to nonlocal symmetry, the exact solution of the nonlinear Vakhnenkov equation is achieved. Finally, the conservation laws for the nonlinear Vakhnenko equation were developed using the multiplier method.

In this paper, by introduction of pseudopotentials, the nonlocal symmetry is obtained for the Ablowitz–Kaup–Newell–Segur system, which is used to describe many physical phenomena in different applications. Together with some auxiliary variables, this kind of nonlocal symmetry can be localized to Lie point symmetry and the corresponding once finite symmetry transformation is calculated for both the original system and the prolonged system. Furthermore, the nth finite symmetry transformation represented in terms of determinant and exact solutions are derived.

The fifth order Kaup–Kupershmidt equation

We review the theory of nonlocal symmetries of nonlinear partial differential equations and, as examples, we present infinite-dimensional Lie algebras of nonlocal symmetries of the Fokas–Qiao and Kaup–Kupershmidt equations. Then, we consider nonlocal symmetries of a family which contains the Korteweg–de Vries (KdV) and (a subclass of) the Rosenau–Hyman compacton-bearing K(m, n) equations. We find that the only member of the family which possesses nonlocal symmetries (of a kind specified in Sec. 3 below) is precisely the KdV equation. We take this fact as an indication that the K(m, n) equations are not integrable in general, and we use the formal symmetry approach of Shabat to check this claim: we prove that the only integrable cases of the full K(m, n) family are the KdV and modified KdV equations.

An application of solvable structures to the reduction of ODEs with a lack of local symmetries is given. Solvable structures considered here are all defined in a nonlocal extension, or covering space, of a given ODE. Examples of the reduction procedure are provided.

We discuss nonlocal symmetries and nonlocal conservation laws that follow from the systematic potentialisation of evolution equations. Those are the Lie point symmetries of the auxiliary systems, also known as potential symmetries. We define higher-degree potential symmetries which then lead to nonlocal conservation laws and nonlocal transformations for the equations. We demonstrate our approach and derive second degree potential symmetries for the Burgers' hierarchy and the Calogero–Degasperis–Ibragimov–Shabat hierarchy.