We consider non-ultra local linear Poisson algebras on a continuous line. Suitable combinations of representations of these algebras yield representations of novel generalized linear Poisson algebras or “boundary” extensions. They are parametrized by a “boundary” scalar matrix and depend in addition on the choice of an antiautomorphism. The new algebras are the classical-linear counterparts of ...

Recently, the author and collaborators have developed a systematic program for proving the existence of homoclinic orbits in partial differential equations. Two typical forms of homoclinic orbits thus obtained are: (1) transversal homoclinic orbits, (2) Silnikov homoclinic orbits. Around the transversal homoclinic orbits in infinite-dimensional autonomous systems, the author was able to prove t...

A multilinear M -dimensional generalization of Lax pairs is introduced and its explicit form is given for the recently discovered class of time-harmonic, integrable, hypersurface motions in IR.

Noncommutative theories have been studied and probed from different viewpoints (see reviews [18, 34, 48]). For instance, a number of noncommutative generalizations of integrable systems were presented (see, e.g., [9, 16, 17, 24, 39]). Solutions were investigated using the dressing method and Riemann-Hilbert problems, formulations, and properties such as infinite sets of conserved quantities wer...

The paper continues nlin.SI/0212019 by giving three more examples of using cyclic bases of zero-curvature representations in studies of relation between strong Lax pairs and recursion operators.

The sets of the lattice equations, which generalize the Toda lattice equations, and their Lax pairs are presented. The multi-soliton solutions are constructed. PACS: 05.45.Yv; 05.50.+q