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

Abstract

We describe a class of self-dual dark nonlinear dynamical systems \textit{a priori} allowing their quasi-linearization, whose integrability can be effectively studied by means of a geometrically motivated gradient-holonomic approach. Using this integrability scheme, we study a new self-dual, dark nonlinear dynamical system on a smooth functional manifold, which models the interaction of atmospheric magneto-sonic Alfv\'{e}n plasma waves. We prove that this dynamical system possesses a Lax representation that allows its full direct linearization and compatible Poisson structures. Moreover, we construct an infinite hierarchy of mutually commuting conservation laws for the dynamical system and prove its complete integrability.

Symplectic Geometry Aspects of the Parametrically-Dependent Kardar–Parisi–Zhang Equation of Spin Glasses Theory, Its Integrability and Related Thermodynamic Stability

“…for all (v, u) ∈ M v × M u jointly with the condition that dγ/dt = 0 for t ∈ [0, T]. The obtained functional relationship (18) reduces to the following generalized Noether-Lax condition…”

“…To solve this problem effectively, we made use of the gradient-holonomic algorithm, motivated by symplectic geometry techniques on functional manifolds [14,15], recently devised for studying the integrability properties of nonlinear dynamical systems with hidden symmetries and, in part, related with the optimal control theory [16,17] approach to parametrically dependent processes. Being applied to the parametrically-extended Kardar-Parisi-Zhang Equation ( 1) this algorithm allowed one to state that it belongs to a so called dark type class [18][19][20][21] of integrable Hamiltonian dynamical systems on functional manifolds with hidden symmetry. Namely, the parametrically-extended Kardar-Parisi-Zhang system of Equation ( 4) reduces to the evolution flow…”

On parametric generalizations of the Kardar-Parisi-Zhang equation and their integrability

“…To solve this problem effectively, we made use of the symplectic geometry based analytic gradient-holonomic algorithm [5,19], recently devised for studying integrability properties of nonlinear dynamical systems with hidden symmetries and partially motivated by the classical optimal control theory [3,17] approach. Being applied to the parametrically extended Kardar-Parisi-Zhang equation (2), this algorithm allowed us to solve the posed above problem and to to state that it belongs to the class of completely integrable dark type [5,6,4,12,15,16] Hamiltonian systems on functional manifolds with hidden symmetry. Namely, there was stated the following proposition.…”