9 research outputs found
Hierarchies of Manakov-Santini Type by Means of Rota-Baxter and Other Identities
The Lax-Sato approach to the hierarchies of Manakov-Santini type is
formalized in order to extend it to a more general class of integrable systems.
For this purpose some linear operators are introduced, which must satisfy some
integrability conditions, one of them is the Rota-Baxter identity. The theory
is illustrated by means of the algebra of Laurent series, the related
hierarchies are classified and examples, also new, of Manakov-Santini type
systems are constructed, including those that are related to the dispersionless
modified Kadomtsev-Petviashvili equation and so called dispersionless r-th
systems
On deformations of standard R-matrices for integrable infinite-dimensional systems
Simple deformations, with a parameter , of classical -matrices
which follow from decomposition of appropriate Lie algebras, are considered. As
a result nonstandard Lax representations for some well known integrable systems
are presented as well as new integrable evolution equations are constructed.Comment: 14 page
Central extensions of cotangent universal hierarchy: (2+1)-dimensional bi-Hamiltonian systems
We introduce the cotangent universal hierarchy that extends the so-called
universal hierarchy (as for the latter, see e.g. arXiv:nlin/0202008,
arXiv:nlin/0312043 and arXiv:nlin/0310036). Then we construct a
(2+1)-dimensional double central extension of the cotangent universal hierarchy
and show that this extension is bi-Hamiltonian. This yields, as a byproduct,
the central extension of the original universal hierarchy.Comment: 12 pages, LaTeX 2e, minor changes (typos fixed, English improved,
etc.
Novikov algebras and a classification of multicomponent Camassa-Holm equations
A class of multi-component integrable systems associated to Novikov algebras,
which interpolate between KdV and Camassa-Holm type equations, is obtained. The
construction is based on the classification of low-dimensional Novikov algebras
by Bai and Meng. These multi-component bi-Hamiltonian systems obtained by this
construction may be interpreted as Euler equations on the centrally extended
Lie algebras associated to the Novikov algebras. The related bilinear forms
generating cocycles of first, second and third order are classified. Several
examples, including known integrable equations, are presented.Comment: V2: some comments and references are adde
Construction and separability of nonlinear soliton integrable couplings
A very natural construction of integrable extensions of soliton systems is
presented. The extension is made on the level of evolution equations by a
modification of the algebra of dynamical fields. The paper is motivated by
recent works of Wen-Xiu Ma et al. (Comp. Math. Appl. 60 (2010) 2601, Appl.
Math. Comp. 217 (2011) 7238), where new class of soliton systems, being
nonlinear integrable couplings, was introduced. The general form of solutions
of the considered class of coupled systems is described. Moreover, the
decoupling procedure is derived, which is also applicable to several other
coupling systems from the literature.Comment: letter, 10 page