Symmetries and conservation laws of the generalized Krichever–Novikov equation

Abstract

Abstract. A computational classification of contact symmetries and higher-order local symmetries that do not commute with t, x, as well as local conserved densities that are not invariant under t, x is carried out for a generalized version of the Krichever-Novikov equation. Several new results are obtained. First, the Krichever-Novikov equation is explicitly shown to have a local conserved density that contains t, x. Second, apart from the dilational point symmetries known for special cases of the Krichever-Novikov equation and its generalized version, no other local symmetries with low differential order are found to contain t, x. Third, the basic Hamiltonian structure of the Krichever-Novikov equation is used to map the local conserved density containing t, x into a nonlocal symmetry that contains t, x. Fourth, a recursion operator is applied to this nonlocal symmetry to produce a hierarchy of nonlocal symmetries that have explicit dependence on t, x. When the inverse of the Hamiltonian map is applied to this hierarchy, only trivial conserved densities are obtained.

A family of wave equations with some remarkable properties

“…On the other hand, the techniques [5,6,7] can equally be employed for the same purpose. In [3] the reader can find applications in this direction regarding Krichever-Novikov type equations. For further discussion, see [4].…”

“…In spite of being nonlinear, equation (1) has some intriguing and interesting properties regarding its solutions due to its homogeneity, that is, invariance under transformations (x, t, u) → (x, t, λu), λ = const. A very simple observation is the fact that if one assumes u(x, t) = e i(kx−ωt) , where i = √ −1 and ω = ω(k), then we conclude that (1) has plane waves 2 provided that k is a real number and ω(k) = ( − 2)ak 3 .…”

Equivalence transformations and conservation laws for a generalized variable-coefficient Gardner equation

“…In this section we determine the equivalence transformation of class (3). These transformations allow us to reduce class (3) to a subclass with simpler form, for instance, reducing the number of arbitrary elements.…”

“…This work is organised as follows. In Section 2, we obtain the continuous equivalence group of equation (3). Next, in Section 3 we obtain Lie symmetries of the reduced equation obtained by using equivalence transformations.…”

Generalized Camassa–Holm Equations: Symmetry, Conservation Laws and Regular Pulse and Front Solutions

“…Then, we will derive some conservation laws of Equation (2). A great many authors have used the conservation laws to study PDEs as they have a key role in the resolution of problems in which some physical properties do not change along the time [15][16][17][18]. We will use a general method to derive conservation laws given in [19][20][21][22].…”