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We discuss algebraic and analytic structure of rational Lax operators. With algebraic reductions of Lax equations we associate a reduction group-a group of twisted automor-phisms of the corresponding infinite dimensional Lie algebra. We present a complete study of dihedral reductions for sl(2, C) Lax operators with simple poles and corresponding integrableâ€¦ (More)

- Alexander V. Mikhailov, Vladimir S. Novikovand, Jing Ping Wang
- 2008

We study partial differential equations of second order (in time) that possess a hierarchy of infinitely many higher symmetries. The famous Boussinesq equation is a member of this class after the extension of the differential polynomial ring. We develop the perturbative symmetry approach in symbolic representation. Applying it, we classify the integrableâ€¦ (More)

In this paper we give definitions of basic concepts such as symmetries, first integrals, Hamilto-nian and recursion operators suitable for ordinary differential equations on associative algebras, and in particular for matrix differential equations. We choose existence of hierarchies of first integrals and/or symmetries as a criterion for integrability andâ€¦ (More)

Integrable discretisations for a class of coupled (super) nonlinear SchrÃ¶dinger (NLS) type of equations are presented. The class corresponds to a Lax operator with entries in a Grassmann algebra. Elementary Darboux transformations are constructed. As a result, Grassmann generalisations of the Toda lattice and the NLS dressing chain are obtained. Theâ€¦ (More)

A two-population firing-rate model describing the dynamics of excitatory and inhibitory neural activity in one spatial dimension is investigated with respect to formation of patterns, in particular stationary periodic patterns and spatiotemporal oscillations. Conditions for existence of spatially homogeneous equilibrium states are first determined, and theâ€¦ (More)

We present a solitary solution of the three-wave nonlinear partial differential equation ~PDE! modelâ€” governing resonant space-time stimulated Brillouin or Raman backscatteringâ€”in the presence of a cw pump and dissipative material and Stokes waves. The study is motivated by pulse formation in optical fiber experiments. As a result of the instability anyâ€¦ (More)

In this paper we discuss the concept of cosymmetries and coâ€“recursion operators for difference equations and present a coâ€“recursion operator for the Viallet equation. We also discover a new type of factorisation for the recursion operators of difference equations. This factorisation enables us to give an elegant proof that the recursion operator given inâ€¦ (More)

- V. S. Gerdjikov, Georgi G. Grahovski, Alexander V. Mikhailov, Tihomir I. VALCHEV
- 2011

A special class of integrable nonlinear differential equations related to A.III-type symmetric spaces and having additional reductions are analyzed via the inverse scattering method (ISM). Using the dressing method we construct two classes of soliton solutions associated with the Lax operator. Next, by using the Wronskian relations, the mapping between theâ€¦ (More)

In this paper we present a family of second order in time nonlinear partial differential equations, which have only one higher symmetry. These equations are not integrable, but have a solution depending on one arbitrary function.

Integrable generalisations of the Benjaminâ€“Ono equation are constructed. The integrable equations of this type are classified by using the perturbative symmetry approach. 2000 Math. Subj. Class. 35Q58.