The Completely Integrable Differential Systems are Essentially Linear Differential Systems

El títol de la versió pre-print de l'article és: The Completely Integrable Differential Systems are Essentially LinearAgraïments: Grant UNAB13-4E-1604, and from the recruitment program of high-end foreign experts of China. The second author is supported by Portuguese national funds through FCT-Fundação para a Ciência e a Tecnologia: Project PEst-OE/EEI/LA0009/2013 (CAMGSD). The third author is partially supported by NNSF of China Grant Number 11271252, by RFDP of Higher Education of China Grant Number 20110073110054 and by innovation program of Shanghai municipal education commission Grant 15ZZ012.Let ˙x = f(x) be a C k autonomous differential system with k ∈ N ∪ {∞, ω} defined in an open subset Ω of R n. Assume that the system ˙x = f(x) is C r completely integrable, i.e. there exist n−1 functionally independent first integrals of class C r with 2 ≤ r ≤ k. If the divergence of system ˙x = f(x) is non-identically zero, then any Jacobian multiplier is functionally independent of the n − 1 first integrals. Moreover the system ˙x = f(x) is C r−1 orbitally equivalent to the linear differential system ˙y = y in a full Lebesgue measure subset of Ω. For Darboux and polynomial integrable polynomial differential systems we characterize their type of Jacobian multipliers

Nonintegrability, Chaos, and Complexity

Two-dimensional driven dissipative flows are generally integrable via a conservation law that is singular at equilibria. Nonintegrable dynamical systems are confined to n*3 dimensions. Even driven-dissipative deterministic dynamical systems that are critical, chaotic or complex have n-1 local time-independent conservation laws that can be used to simplify the geometric picture of the flow over as many consecutive time intervals as one likes. Those conserevation laws generally have either branch cuts, phase singularities, or both. The consequence of the existence of singular conservation laws for experimental data analysis, and also for the search for scale-invariant critical states via uncontrolled approximations in deterministic dynamical systems, is discussed. Finally, the expectation of ubiquity of scaling laws and universality classes in dynamics is contrasted with the possibility that the most interesting dynamics in nature may be nonscaling, nonuniversal, and to some degree computationally complex

On a unified formulation of completely integrable systems

The purpose of this article is to show that a $\mathcal{C}^1$ differential system on $\R^n$ which admits a set of $n-1$ independent $\mathcal{C}^2$ conservation laws defined on an open subset $\Omega\subseteq \R^n$, is essentially $\mathcal{C}^1$ equivalent on an open and dense subset of $\Omega$, with the linear differential system $u^\prime_1=u_1, \ u^\prime_2=u_2,..., \ u^\prime_n=u_n$. The main results are illustrated in the case of two concrete dynamical systems, namely the three dimensional Lotka-Volterra system, and respectively the Euler equations from the free rigid body dynamics.Comment: 11 page

Galoisian obstructions to non-Hamiltonian integrability

We show that the main theorem of Morales--Ramis--Simo about Galoisian obstructions to meromorphic integrability of Hamiltonian systems can be naturally extended to the non-Hamiltonian case. Namely, if a dynamical system is meromorphically integrable in the non-Hamiltonian sense, then the differential Galois groups of the variational equations (of any order) along its solutions must be virtually AbelianComment: 5 page

Noncommutative Burgers Equation

We present a noncommutative version of the Burgers equation which possesses the Lax representation and discuss the integrability in detail. We find a noncommutative version of the Cole-Hopf transformation and succeed in the linearization of it. The linearized equation is the (noncommutative) diffusion equation and exactly solved. We also discuss the properties of some exact solutions. The result shows that the noncommutative Burgers equation is completely integrable even though it contains infinite number of time derivatives. Furthermore, we derive the noncommutative Burgers equation from the noncommutative (anti-)self-dual Yang-Mills equation by reduction, which is an evidence for the noncommutative Ward conjecture. Finally, we present a noncommutative version of the Burgers hierarchy by both the Lax-pair generating technique and the Sato's approach.Comment: 24 pages, LaTeX, 1 figure; v2: discussions on Ward conjecture, Sato theory and the integrability added, references added, version to appear in J. Phys.

Non-perturbative Quantum Theories and Integrable Equations

I review the appearance of classical integrable systems as an effective tool for the description of non-perturbative exact results in quantum string and gauge theories. Various aspects of this relation: spectral curves, action-angle variables, Whitham deformations and associativity equations are considered separately demonstrating hidden parallels between topological 2d string theories and naively non-topological 4d theories. The proofs are supplemented by explicit illustrative examples.Comment: 35 pages, LaTeX, to appear in Int.J.Mod.Phys.

New Fundamental Symmetries of Integrable Systems and Partial Bethe Ansatz

We introduce a new concept of quasi-Yang-Baxter algebras. The quantum quasi-Yang-Baxter algebras being simple but non-trivial deformations of ordinary algebras of monodromy matrices realize a new type of quantum dynamical symmetries and find an unexpected and remarkable applications in quantum inverse scattering method (QISM). We show that applying to quasi-Yang-Baxter algebras the standard procedure of QISM one obtains new wide classes of quantum models which, being integrable (i.e. having enough number of commuting integrals of motion) are only quasi-exactly solvable (i.e. admit an algebraic Bethe ansatz solution for arbitrarily large but limited parts of the spectrum). These quasi-exactly solvable models naturally arise as deformations of known exactly solvable ones. A general theory of such deformations is proposed. The correspondence ``Yangian --- quasi-Yangian'' and ``$XXX$ spin models --- quasi-$XXX$ spin models'' is discussed in detail. We also construct the classical conterparts of quasi-Yang-Baxter algebras and show that they naturally lead to new classes of classical integrable models. We conjecture that these models are quasi-exactly solvable in the sense of classical inverse scattering method, i.e. admit only partial construction of action-angle variables.Comment: 49 pages, LaTe

Open problems, questions, and challenges in finite-dimensional integrable systems

The paper surveys open problems and questions related to different aspects of integrable systems with finitely many degrees of freedom. Many of the open problems were suggested by the participants of the conference “Finite-dimensional Integrable Systems, FDIS 2017” held at CRM, Barcelona in July 2017.Postprint (updated version