A class of integrable evolutionary vector equations

Abstract

We present the results of classifying integrable evolutionary N -component vector equations and construct Bäcklund transformations for each equation as proof of the exact integrability.

Integrable Anisotropic Evolution Equations on a Sphere

“…and (3) are called the integrability conditions. The modified symmetry approach has been recently applied to the equations on S n [8] and on R n [9,10], where the third order equations in the form…”

On a classification of integrable vectorial evolutionary equations

“…Equation (3.11) has the auto-Bäcklund transformation of the following form: Remark 7. The auto-Bäcklund transformations (4.9), (4.10), (4.18) and (4.19) have been found in [7]. The auto-Bäcklund transformations for equations (3.3)-(3.7) have been found in [6].…”

“…Equations (3.1) and (3.2) differ from (1.1) by a dilatation of u; equations (3.3) -(3.7) have been presented in [6]. Equations (3.14) and (3.15) have been presented in [7] in a different form.…”

“…A complete classification of the integrable divergent equations (1.6) has been presented in [6]. A complete list of the isotropic integrable equations (1.6) with f 2 = 0 has been presented in [7]. The anisotropic integrable equations (1.6) on a sphere have been completely classified in [8].…”

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

“…Moreover, about 50 two-field integrable systems of the form (1.1) with a = 1 can be extracted from papers [15,16,17,18,19] that deal with vector evolutionary equations.…”