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The general concept of nonlinear self-adjointness of differential equations is introduced. It includes the linear self-adjointness as a particular case. Moreover, it embraces the previous notions of self-adjoint and quasi self-adjoint…
Lie symmetry analysis is applied to study the nonlinear rotating shallow water equations. The 9-dimensional Lie algebra of point symmetries admitted by the model is found. It is shown that the rotating shallow water equations are related…
In this paper we characterize the definiteness of the discrete symplectic system, study a nonhomogeneous discrete symplectic system, and introduce the minimal and maximal linear relations associated with these systems. Fundamental…
We consider a net of *-algebras, locally around any point of observation, equipped with a natural partial order related to the isotony property. Assuming the underlying manifold of the net to be a differentiable, this net shall be…
We give a pedagogical introduction to the Generalized Hydrodynamic approach to inhomogeneous quenches in integrable many-body quantum systems. We review recent applications of the theory, focusing in particular on two classes of problems:…
This paper discusses the interplay of symmetries and stability in the analysis and control of nonlinear dynamical systems and networks. Specifically, it combines standard results on symmetries and equivariance with recent convergence…
Conformal nets are a classical topic in quantum field theory: they assign operator algebras to one-dimensional manifolds, and have close connections with one-dimensional topological field theories. It seems to be well-known that the usual…
Systems of partial differential equations which appear in classical field theories can be studied geometrically using different geometrical structures, for example, k-symplectic geometry, k-cosymplectic geometry, multisymplectic geometry,…
We consider higher symmetries and operator symmetries of linear partial differential equations. The higher symmetries form a Lie algebra, and operator ones form an associative algebra. The relationship between these symmetries is…
In this investigation, symmetry properties of the nonlinear heat conductivity equations of general form $u_t = [E(x, u)u_x]_x + H(x, u)$ are studied. The point symmetry analysis of these equations is considered as well as an equivalence…
Reversible part of evolution equations of physical systems is often generated by a Poisson bracket. We discuss geometric means of construction of Poisson brackets and their mutual coupling (direct, semidirect and matched-pair products) as…
Heteroclinic connections are trajectories that link invariant sets for an autonomous dynamical flow: these connections can robustly form networks between equilibria, for systems with flow-invariant spaces. In this paper we examine the…
This paper is devoted to the study of symplectic manifolds and their connection with Hamiltonian dynamical systems. We review some properties and operations on these manifolds and see how they intervene when studying the complete…
This is a survey article on ordinary differential equations over nonarchimedean fields based on the author's lecture at the 2015 Simons Symposium on nonarchimedean and tropical geometry. Topics include: the convergence polygon associated to…
In this paper, we investigate multidimensional first-order quasi-linear systems and find necessary conditions for them to admit Hamiltonian formulation. The insufficiency of the conditions is related to the Poisson cohomology of the…
Within the framework of weighted integrable Hamiltonian systems, we study the long-time behavior of the associated statistical ensembles. We construct an action-dependent angular conjugacy that rectifies the nonuniform angular flow into a…
This paper is devoted to a description of integrable Hamiltonian hydrodynamic chains associated with Dorfman Poisson brackets. Three main classes of these hydrodynamic chains are selected. Generating functions of conservation laws and…
In this paper we study oriented quantum coalgebras which are structures closely related to oriented quantum algebras. We study the relationship between oriented quantum coalgebras and oriented quantum algebras and the relationship between…
This paper describes the notion of \sigma -symmetry, which extends the one of \lambda-symmetry, and its application to reduction procedures of systems of ordinary differential equations and of dynamical systems as well. We also consider…
This note is concerned in so called harmonic complex structures introduced by the author previously. I will recall some previous results and emphasize the motivation: Provide an attempt to a fundamental problem in geometry--determining the…