Dark Sharma–Tasso–Olver Equations and Their Recursion Operators

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

“…tively, and which prove to be completely integrable bi-Hamiltonian systems, actively studied [5,[17][18][19][20][21][22] during past decades. Some other more nontrivial examples can be taken from the work [4], devoted to description of the integrable hierarchies of modified Burgers type nonlinear dynamical systems.…”

“…Some twenty years ago, a new class of nonlinear dynamical systems, called 'dark equations' was introduced by Boris Kupershmidt [1,2], and shown to possess unusual properties that were not particularly well-understood at that time. 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].…”