Nonlinear Dynamical Systems of Mathematical Physics

A vertex operator representation of solutions to the Gurevich-Zybin hydrodynamical equation

“…It should be noted that the effectiveness of our approach to studying the vertex operator representation of the Riemann type hierarchy owes much to the important exact representation (2.21) for the corresponding monodromy matrix, whose properties are described by means of applying the standard [16,17,19,21] Lie-algebraic techniques. As an indication of possible future research, it should also be mentioned that it would be interesting to generalize the vertex operator approach devised in this work to other linear spectral problems such as those related to dynamical systems with a parametrical spectral [12,18,19] dependence, spatially two-dimensional [11], Pavlov's and heavenly [13] dynamical systems.…”

“…The iso-spectrality condition imbedded in problem (2.2) gives rise [14,[16][17][18]21] naturally to a hierarchy of commuting to each other nonlinear bi-Hamiltonian dynamical systems on the functional manifold M in the general form…”

“…as λ → ∞ holds for all x ∈ R. The latter allows us to define the corresponding momentum mapping : M 2 →G − and to imbed the whole hierarchy of the Hamiltonian flows (2.3), related with the linear problem (1.1), into the standard [14,[16][17][18]21] Lie-algebraic scheme. HereG − ⊂G, whereG :=G + ⊕G − is the natural splitting into two affine subalgebras of positive and negative λ-expansions of the centrally extended [16,26] affine current sl(2)-algebraĜ :=G ⊕ C:…”

“…Moreover, this method was further effectively applied to studying solutions to a generalized [5] where N ∈ Z + , u ∈ M 1 ⊂ C ∞ (R/2πZ; R) is a smooth function on a periodic functional manifold M 1 and t ∈ R is the evolution parameter. Making use of novel methods, devised in [8,18,25] and based both on the spectral theory [10,17,19,20] and differential algebra techniques, the Lax type representations for the cases N = 1, 4 were constructed in explicit form.…”

Invariant measures for discrete dynamical systems and ergodic properties of generalized Boole-type transformations

“…It is well known that discrete dynamical systems on finite-dimensional manifolds play an important role [9,8,12,21] in describing evolution properties of many processes in the applied sciences. Of particular interest are discrete dynamical systems on manifolds with invariant measures, often possessing additional properties such as ergodicity or mixing, which allow to explain such phenomenon as chaotic behavior and instability of the physical objects being studied.…”