Related papers: Integrability and dynamics of the Rajeev-Ranken mo…
We study the classical Rajeev-Ranken model, a Hamiltonian system with three degrees of freedom describing nonlinear continuous waves in a 1+1-dimensional nilpotent scalar field theory pseudodual to the SU(2) principal chiral model. While it…
The Rajeev-Ranken (RR) model is a Hamiltonian system describing screw-type nonlinear waves of wavenumber $k$ in a scalar field theory pseudodual to the 1+1D SU(2) principal chiral model. Classically, the RR model is Liouville integrable.…
The integrable 1+1-dimensional SU(2) principal chiral model (PCM) serves as a toy-model for 3+1-dimensional Yang-Mills theory as it is asymptotically free and displays a mass gap. Interestingly, the PCM is 'pseudodual' to a scalar field…
We review some essential aspects of classically integrable systems. The detailed outline of the lectures consists of: 1. Introduction and motivation, with historical remarks; 2. Liouville theorem and action-angle variables, with examples…
We study the integrability of two-dimensional theories that are obtained by a dimensional reduction of certain four-dimensional gravitational theories describing the coupling of Maxwell fields and neutral scalar fields to gravity in the…
We describe classical top-like integrable systems arising from the quantum exchange relations and corresponding Sklyanin algebras. The Lax operator is expressed in terms of the quantum non-dynamical $R$-matrix even at the classical level,…
We generalize to some PDEs a theorem by Nekhoroshev on the persistence of invariant tori in Hamiltonian systems with $r$ integrals of motion and $n$ degrees of freedom, $r\leq n$. The result we get ensures the persistence of an…
The notion of integrability is discussed for classical nonautonomous systems with one degree of freedom. The analysis is focused on models which are linearly spanned by finite Lie algebras. By constructing the autonomous extension of the…
In our recent paper we suggested a natural construction of the classical relativistic integrable tops in terms of the quantum $R$-matrices. Here we study the simplest case -- the 11-vertex $R$-matrix and related ${\rm gl}_2$ rational…
We give new estimates for a critical elliptic system introduced by Rivi\`ere-Struwe in \cite{riviere_struwe} (see also the work of Rupflin \cite{rupflin} and Schikorra \cite{schikorra_frames}), which generalises PDE solved by harmonic (and…
A procedure allowing for the construction of Lorentz invariant integrable models living in d+1 dimensional space-time and with an n dimensional target space is provided. Here, integrability is understood as the existence of the generalized…
The R-matrix formalism for the construction of integrable systems with infinitely many degrees of freedom is reviewed. Its application to Poisson, noncommutative and loop algebras as well as central extension procedure are presented. The…
We study the integrability of a two-dimensional Hamiltonian system with a gyroscopic term and a non-homogeneous potential composed of two homogeneous components of different degrees. The model describes the motion of a particle in a plane…
We consider the issue of correspondence between symmetries and conserved quantities in the class of linear relativistic higher-derivative theories of derived type. In this class of models the wave operator is a polynomial in another…
A model for planar phenomena introduced by Jackiw and Pi and described by a Lagrangian including a Chern-Simons term is considered. The associated equations of motion, among which a 2+1 gauged nonlinear Schr\"odinger equation, are rewritten…
The purpose of this article is to develop an algebraic approach to the problem of integrable classification of differential-difference equations with one continuous and two discrete variables. As a classification criterion, we put forward…
For dynamical systems of dimension three or more the question of integrability or nonintegrability is extended by the possibility of chaotic behaviour in the general solution. We determine the integrability of isotropic cosmological models…
We analyze the integrability properties of models defined on the symmetric space SU(2)/U(1) in 3+1 dimensions, using a recently proposed approach for integrable theories in any dimension. We point out the key ingredients for a theory to…
In the paper, we study the dynamics of planar $n$-gons, which can be considered as discrete models of threads. The main result of the paper is that, under some weak assumptions, these systems are not integrable in the sense of Liouville.…
We present a theoretical framework for analyzing longitudinal coupled-bunch instabilities in double-rf systems with even filling patterns, accounting for potential-well distortion and multiple azimuthal modes. The linearized Vlasov equation…