Related papers: The Canonical Symmetry for Integrable Systems
The properties of the canonical symmetry of the nonlinear Schr\"odinger equation are investigated. The densities of the local conservation laws for this system are shown to change under the action of the canonical symmetry by total space…
We show that nonlocal reductions of systems of integrable nonlinear partial differential equations are the special discrete symmetry transformations.
In a recent paper [TMP, 200:1 (2019), 966--984] by the authors, a series of integrable discrete autonomous equations on a square lattice with a non-standard structure of generalized symmetries is constructed. We build modified series by…
Sequences of canonical conservation laws and generalized symmetries for the lattice Boussinesq and the lattice modified Boussinesq systems are successively derived. The interpretation of these symmetries as differential-difference equations…
We investigate various types of symmetries and their mutual relationships in Hamiltonian systems defined on manifolds with different geometric structures: symplectic, cosymplectic, contact and cocontact. In each case we pay special…
Quantum canonical transformations are defined in analogy to classical canonical transformations as changes of the phase space variables which preserve the Dirac bracket structure. In themselves, they are neither unitary nor non-unitary. A…
This work builds on an existing model of discrete canonical evolution and applies it to the general case of a linear dynamical system, i.e., a finite-dimensional system with configuration space isomorphic to $ \mathbb{R}^{q} $ and linear…
For classical discrete system under constant composition, typically reffered to as substitutional alloys, correspondence between interatomic many-body interactions and structure in thermodynamic equilibrium exhibit profound, complicated…
A certain notion of canonical equivalence in quantum mechanics is proposed. It is used to relate quantal systems with discrete ones. Discrete systems canonically equivalent to the celebrated harmonic oscillator as well as the quartic and…
The conservation laws for a class of nonlinear equations with variable coefficients on discrete and noncommutative spaces are derived. For discrete models the conserved charges are constructed explicitly. The applications of the general…
For classical discrete systems under constant composition, canonical average provides equilibrium configuration from a set of many-body interactions, which typically acts as nonlinear map. The nonlinearity has recently been investigated in…
Discrete canonical evolution is a key tool for understanding the dynamics in discrete models of spacetime, in particular those represented by a triangular Regge lattice. We consider a finite-dimensional system whose evolution is realized by…
In this survey paper we review classical results and recent progress about a certain topic in the spectral theory of two-dimensional canonical systems. Namely, we consider the questions whether the spectrum $\sigma$ is discrete, and if it…
Symmetry properties of conservation laws of partial differential equations are developed by using the general method of conservation law multipliers. As main results, simple conditions are given for characterizing when a conservation law…
The one-dimensional modified shallow water equations in Lagrangian coordinates are considered. It is shown the relationship between symmetries and conservation laws in Lagrangian coordinates, in mass Lagrangian variables, and Eulerian…
A class of generalized nonlinear p-Laplacian evolution equations is studied. These equations model radial diffusion-reaction processes in $n\geq 1$ dimensions, where the diffusivity depends on the gradient of the flow. For this class, all…
Symmetry groups allow to transform solutions of differential equations continuously into other solutions. This property can be used for the observability analysis of infinite-dimensional systems with input and output. In this contribution,…
It is demonstrated that the canonical distribution for a subsystem of a closed system follows directly from the solution of the time-reversible Newtonian equation of motion in which the total energy is strictly conserved. It is shown that…
Variational and divergence symmetries are studied in this paper for linear equations of maximal symmetry in canonical form, and the associated first integrals are given in explicit form. All the main results obtained are formulated as…
To study discrete dynamical systems of different types --- deterministic, statistical and quantum --- we develop various approaches. We introduce the concept of a system of discrete relations on an abstract simplicial complex and develop…