Structures on manifolds and algebraic integrability of dynamical systems

Quasi-linearization and stability analysis of some self-dual, dark equations and a new dynamical system †

“…To proceed with the problem of classifying integrable dark dynamical systems on the functional manifold M, we need from the very beginning to analyze the conditions under which the reduced quasi-linearized system (2.3) possesses an infinite hierarchy of suitably ordered conservation laws. To do this, we will make use of the geometrically motivated gradient-holonomic integrability scheme devised in [16] and further developed in [12][13][14], to first transform the vector fields (2.3) to their following equivalent form on the functional manifold M:…”

“…with a Frechét smooth [13,14,[23][24][25][26] vector field P: M u → T(M u ) on the manifold M u . It is a strongly nonlinear dispersive evolution flow on M u , whose special perturbations can be eventually used for modeling from a physical point of view [27,28] interaction of atmospheric magneto-sonic Alfvén plasma waves.…”

“…on the manifold M u for  Î + n . Since the dynamical system (3.19) seems to be integrable only when n = 0, as was argued in the work [15], we will consider this case, for which we shall pose the direct integrability problem and successively state that the flow (3.17) is actually self-dual with respect to the change of variables u := w ε ä M u for an arbitrary  e Î ⧹{ } 0 and show the existence of an infinite hierarchy of conservation laws for the dynamical system (3.17) using methods developed in [13,14,26], thus presenting another dark nonlinear dynamical system.…”

“…Later, in related developments, some Burgers-type [3][4][5] and also Korteweg-de Vries type [6,7] dynamical systems were studied in detail, and it was proved that they have a finite number of conservation laws, a linearization and degenerate Lax representations, among other properties. In what follows, we provide a description of a class of self-dual dark-type (or just, dark, for short) nonlinear dynamical systems, which a priori allows their quasi-linearization, whose integrability can be effectively studied by means of a geometrically motivated [8,[9][10][11] gradient-holonomic approach [12][13][14]. Moreover, we study a slightly modified form of a self-dual nonlinear dark dynamical system on a functional manifold, whose integrability was recently analyzed in [15].…”

“…Not only did this dynamical system appear to be Lax integrable, it also seemed to have a rich mathematical architecture including compatible Poisson structures and an infinite hierarchy of nontrivial mutually commuting conservation laws. In the sequel, we shall prove these properties using the gradient-holonomic integrability scheme devised in our prior work with several collaborators [12][13][14].…”