Integrable evolution equations with constant separant

Abstract

The survey provides classification results for integrable one-field evolution equations of orders 2, 3 and 5 with the constant separant. The classification is based on necessary integrability conditions following from the existence of the formal recursion operator for integrable equations. Recurrent formulas for the whole infinite sequence of necessary conditions are presented for the first time. The most of the classification statements can be found in papers by S.I. Svinilupov and V.V. Sokolov but the proofs have never been published before. The result concerning the fifth order equations is stronger than obtained before.

51 pages

References in corpus (2)

• Two-Field Integrable Evolutionary Systems of the Third Order and Their Differential Substitutions
• Nonlocal symmetries of integrable two-field divergent evolutionary systems

Cited by in corpus (13)

• Lie algebras responsible for zero-curvature representations of scalar evolution equations
• Necessary integrability conditions for evolutionary lattice equations
• Generalized symmetries and integrability conditions for hyperbolic type semi-discrete equations
• KdV-type equations linked via Baecklund transformations: remarks and perspectives
• Integrability test for evolutionary lattice equations of higher order
• On the integrability of a discrete analogue of the Kaup-Kupershmidt equation
• On Lie algebras responsible for integrability of (1+1)-dimensional scalar evolution PDEs
• Higher jet prolongation Lie algebras and Backlund transformations for (1+1)-dimensional PDEs
• Integrable evolution Hamiltonian equations of the third order with the Hamiltonian operator ${\mathbf{D}}_x$
• On Lie algebras responsible for zero-curvature representations and Backlund transformations of (1+1)-dimensional scalar evolution PDEs
• Bäcklund transformations: a tool to study Abelian and non-Abelian nonlinear evolution equations
• On the integrability of a lattice equation with two continuum limits
• Symmetry approach to integrability and non-associative algebraic structures