Integrable Anisotropic Evolution Equations on a Sphere

Abstract

Abstract. V.V. Sokolov's modifying symmetry approach is applied to anisotropic evolution equations of the third order on the n-dimensional sphere. The main result is a complete classification of such equations. Auto-Bäcklund transformations are also found for all equations.

Superposition formulas for integrable vector evolutionary equations on a Sphere

“…The isotropic Schwartz-KdV equation on sphere S n was obtained in [1]. Also we can take the limit transition a → ∞ in (1.2) and get…”

“…Expression (2.11) coincides with the SF of the vector generalization of the LandauLifshitz equation [6]. It was proved in [4] …”

“…Integrable anisotropic evolutionary equations on the n-dimensional sphere were classified in [4]. However, superposition formulas were found only for a few equations, namely for the vector Schwarz-KdV [2], two generalizations of the mKdV [5], and the Landau-Lifshitz generalization [6].…”

On a classification of integrable vectorial evolutionary equations

“…5) where f k are scalar smooth functions depending on variables (1.3) and (1.4). In recent years the following third order equation…”

“…A componentless version of the symmetry approach has been developed in [5] for vector evolutionary equations of the following type u t = f n u n + f n−1 u n−1 + · · · + f 0 u 0 , (1. 5) where f k are scalar smooth functions depending on variables (1.3) and (1.4).…”