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.

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In this paper, we shall obtain the symmetries of the mathematical model describing spontaneous relaxation of eastward jets into a meandering state and use these symmetries for constructing the conservation laws. The basic eastward jet is a spectral parameter of the model, which is in geostrophic equilibrium with the basic density structure and which guarantees the existence of nontrivial conservation laws.

Abstract. We derive the conserved vectors for the nonlinear two-dimensional Euler equations describing nonviscous incompressible fluid flows on a three-dimensional rotating spherical surface superimposed by a particular stationary latitude dependent flow. Under the assumption of no friction and a distribution of temperature dependent only upon latitude, the equations in question can be used to model zonal west-to-east flows in the upper atmosphere between the Ferrel and Polar cells. As a particualr example, the conserved densities are analyzed by visualizing the exact invariant solutions associated with the given model for the particular form of finite disturbances for which the invariant solutions are also exact solutions of Navier-Stokes equations.