Darboux Integrability Research Articles

Darboux integrability for diagonal systems of hydrodynamic type

We prove that (1) diagonal systems of hydrodynamic type are Darboux integrable if and only if the corresponding systems for commuting flows are Darboux integrable, (2) systems for commuting flows are Darboux integrable if and only if the Laplace transformation sequences terminate, (3) Darboux integrable systems are necessarily semihamiltonian. We give geometric interpretation for Darboux integrability of such systems in terms of congruences of lines and in terms of solution orbits with respect to symmetry subalgebras, discuss known and new examples.

Integral preserving discretization of 2D Toda lattices

There are different methods of discretizing integrable systems. We consider semi-discrete analog of two-dimensional Toda lattices associated to the Cartan matrices of simple Lie algebras that was proposed by Habibullin in 2011. This discretization is based on the notion of Darboux integrability. Generalized Toda lattices are known to be Darboux integrable in the continuous case (that is, they admit complete families of characteristic integrals in both directions). We prove that semi-discrete analogs of Toda lattices associated to the Cartan matrices of all simple Lie algebras are Darboux integrable. By examining the properties of Habibullin’s discretization we show that if a function is a characteristic integral for a generalized Toda lattice in the continuous case, then the same function is a characteristic integral in the semi-discrete case as well. We consider characteristic algebras of such integral-preserving discretizations of Toda lattices to prove the existence of complete families of characteristic integrals in the second direction.

Painlevé equations, integrable systems and the stabilizer set of Virasoro orbit

We study a geometrical formulation of the nonlinear second-order Riccati equation (SORE) in terms of the projective vector field equation on [Formula: see text], which in turn is related to the stability algebra of Virasoro orbit. Using Darboux integrability method, we obtain the first integral of the SORE and the results are applied to the study of its Lagrangian and Hamiltonian descriptions. Using these results, we show the existence of a Lagrangian description for SORE, and the Painlevé II equation is analyzed.

Darboux and analytic integrability of five‐dimensional semiclassical Jaynes–Cummings system

We study the dynamics of a version of the Jaynes–Cummings system without the rotating wave approximation, which essentially consists of the interaction of two systems, two‐level atoms and a single system of electromagnetic field. In particular, we investigate some types of first integrals such as Darboux, analytic, and time‐dependent first integrals of this system. We prove that only two Darboux first integrals and two analytic first integrals exist for this system. Moreover, we determine all exponential factors and invariant algebraic hypersurfaces of this system.

The Darboux Polynomials and Integrability of Polynomial Levinson–Smith Differential Equations

We provide the necessary and sufficient conditions of Liouvillian integrability for nondegenerate near infinity polynomial Levinson–Smith differential equations. These equations generalize Liénard equations and are used to describe self-sustained oscillations. Our results are valid for arbitrary degrees of the polynomials arising in the equations. We find a number of novel Liouvillian integrable subfamilies. We derive an upper bound with respect to one of the variables on the degrees of irreducible Darboux polynomials in the case of nondegenerate or algebraically degenerate near infinity polynomial Levinson–Smith equations. We perform the complete classification of Liouvillian first integrals for the nondegenerate or algebraically degenerate near infinity Rayleigh–Duffing–van der Pol equation that is a cubic Levinson–Smith equation.

An algebraic criterion of the Darboux integrability of differential-difference equations and systems

The article investigates systems of differential-difference equations of hyperbolic type, integrable in sense of Darboux. The concept of a complete set of independent characteristic integrals underlying Darboux integrability is discussed. A close connection is found between integrals and characteristic Lie–Rinehart algebras of the system. It is proved that a system of equations is Darboux integrable if and only if its characteristic algebras in both directions are finite-dimensional.

Darboux integrability of a cubic differential system with two parallel invariant straight lines

In this paper we prove the Darboux integrability of a cubic differential system with a singular point of a center typer having at least two parallel invariant straight lines.

Characteristic Lie algebras of integrable differential-difference equations in 3D

The purpose of this article is to discuss an algebraic method for studying integrable differential-difference lattices with two discrete and one continuous independent variables. The main idea is based on the hypothesis that any integrable equation of this type admits an infinite sequence of reductions, which are Darboux integrable systems of differential-difference equations with two independent variables. To detect the Darboux integrability property of the reductions the authors use the characteristic Lie–Rinehart algebras. The hypothesis is confirmed by integrable three-dimensional lattices from the list given by Ferapontov et al (2015 Int. Math. Res. Not. 2015 4933). The required reductions are found for these lattices. Characteristic algebras corresponding to small-order reductions are described, integrals are calculated in both directions, which proves Darboux integrability. The approach might be useful for the integrable classification in 3D.

Darboux first integrals and linearizability of quadratic–quintic Duffing–van der Pol oscillators

Nonlinear oscillators described by polynomial Liénard differential equations arise in a variety of mathematical and physical applications. For a family of generalized Duffing–van der Pol oscillators we classify Darboux integrable cases and explicitly construct the corresponding generalized Darboux first integrals. We demonstrate that Darboux integrability is in strong correlation with the linearizability via the generalized Sundman transformations. We establish that the general solutions can be written in a parametric form. We prove that there are no limit cycles in integrable cases with autonomous Darboux first integrals.

Dynamics and Darboux Integrability of the D2 Polynomial Vector Fields of Degree 2 in $\mathbb {R}^{3}$

We characterize the Darboux integrability and the global dynamics of the 3-dimensional polynomial differential systems of degree 2 which are invariant under the D2 symmetric group, which have been partially studied previously by several authors.

Qualitative study of a model with Rastall gravity

We consider the Rastall theory for the flat Friedmann–Robertson–Walker Universe filled with a perfect fluid that satisfies a linear equation of state. The corresponding dynamical system is a two dimensional system of polynomial differential equations depending on four parameters. We show that this differential system is always Darboux integrable. In order to study the global dynamics of this family of differential systems we classify all their non-topological equivalent phase portraits in the Poincaré disc and we obtain 16 different dynamical situations for our spacetime.

On a class of 2D integrable lattice equations

We develop a new approach to the classification of integrable equations of the form uxy=f(u,ux,uy,△zu△z¯u,△zz¯u), where △z and △z¯ are the forward/backward discrete derivatives. The following two-step classification procedure is proposed: (1) First, we require that the dispersionless limit of the equation is integrable, that is, its characteristic variety defines a conformal structure, which is Einstein–Weyl, on every solution. (2) Second, to the candidate equations selected at the previous step, we apply the test of Darboux integrability of reductions obtained by imposing suitable cutoff conditions.

Integrability and zero-Hopf bifurcation in the Sprott A system

The first objective of this paper is to study the Darboux integrability of the polynomial differential systemx˙=y,y˙=−x−yz,z˙=y2−a and the second one is to show that for a>0 sufficiently small this model exhibits one small amplitude periodic solution that bifurcates from the origin of coordinates when a=0. This model was introduced by Hoover as the first example of a differential equation with a hidden attractor and it was used by Sprott to illustrate a differential equation having a chaotic behavior without equilibrium points, and now this system is known as the Sprott A system.

Darboux integrability and dynamics of the Basener–Ross population model

We deal with the Basener and Ross model for the evolution of human population in Easter island. We study the Darboux integrability of this model and characterize all its global dynamics in the Poincare disc, obtaining 15 different topological phase portraits.

Darboux integrability of the simple chaotic flow with a line equilibria differential system

Abstract In this paper, we study the first integrals of Darboux type of the differential system x ˙ = y , y ˙ = − x + y z , z ˙ = − x − a x y − b x z , which exhibits chaotic phenomena for suitable chosen values of the real parameters a and b. We show that the system has no polynomial, rational, or Darboux first integrals for any value of a and b. All the Darboux polynomials of the system are derived together with its exponential factors. This study effectively exploits the benefits inherent in weight-homogenous polynomials to solve linear partial differential equations.

Darboux Integrable System with a Triple Point and Pseudo-Abelian Integrals

In this paper we consider the degeneracies of the third type. More exact, the perturbations of the Darboux integrable foliation with a triple point, i.e. the case where three of the curves {P i = 0} meet at one point, are considered. Assuming that this is the only non-genericity, we prove that the number of zeros of the corresponding pseudo-abelian integrals is bounded uniformly for close Darboux integrable foliations. Let F denote the foliation with triple point (assume it to be at the origin), and let F λ = {M λ dH λ H λ = 0}, M λ is a integrating factor, be the close foliation. The main problem is that F λ can have a small nest of cycles which shrinks to the origin as λ → 0. A particular case of this situation, namely H λ = (x − λ) ǫ (y − x) ǫ + (y + x) ǫ − ∆ with ∆ non-vanishing at the origin (and generic in appropriate sense). Mathematics subject classification: 34C07, 34C08

On the integrability of some forced nonlinear oscillators

We consider integrability properties of a family of forced nonlinear oscillators, which generalizes the Liénard equation. We demonstrate that some forced oscillators with previously known first integrals can be linearized via certain nonlocal transformations. Furthermore, we show that the whole family of Liénard (n,n+1) equations with arbitrary external forcing admits a first integral. We study in detail the case of the Liénard (3,4) equation due to its value for applications. We prove that despite the fact that this equation possesses one Darboux first integral and can be linearized, it does not have an additional Darboux integral and, hence, is not Darboux integrable. Therefore, we demonstrate that certain nonlocal transformations do not preserve the property of Darboux integrability.

Integrability analysis of the Shimizu–Morioka system

The aim of this paper is to give some new insights into the Shimizu–Morioka systemx˙=y,y˙=x−λy−xz,z˙=−αz+x2,from the integrability point of view. Firstly, we propose a linear scaling in time and coordinates which converts the Shimizu–Morioka system into a special case of the Rucklidge system when α ≠ 0 and discuss the relationship between Shimizu–Morioka system and Rucklidge system. Based on this observation, Darboux integrability of the Shimizu–Morioka system with α ≠ 0 is trivially derived from the corresponding results on the Rucklidge system. When α=0, we investigate Darboux integrability of the Shimizu–Morioka system by the Gröbner basis in algebraic geometry. Secondly, we use the stability of the singular points and periodic orbits to study the nonexistence of global C1 first integrals of the Shimizu–Morioka system. Finally, in the case α ≠ 0, we prove it is not rationally integrable for almost all parameter values by an extended Morales-Ramis theory, and in the case α=0, we show that it is not algebraically integrable by quasi-homogeneous decompositions and Kowalevski exponents. Our results are in accord with the fact that this system admits chaotic behaviors for a large range of its parameters.

Isometric embedding and Darboux integrability

Given a smooth 2-dimensional Riemannian or pseudo-Riemannian manifold $$(M, \varvec{g})$$ and an ambient 3-dimensional Riemannian or pseudo-Riemannian manifold $$(N, \varvec{h})$$ , one can ask under what circumstances does the exterior differential system $$\mathcal {I}$$ for an isometric embedding $$M\hookrightarrow N$$ have particularly nice solvability properties. In this paper we give a classification of all 2-dimensional metrics $$\varvec{g}$$ whose isometric embedding system into flat Riemannian or pseudo-Riemannian 3-manifolds $$(N, \varvec{h})$$ is Darboux integrable. As an illustration of the motivation behind the classification, we examine in detail one of the classified metrics, $$\varvec{g}_0$$ , showing how to use its Darboux integrability in order to construct all its embeddings in finite terms of arbitrary functions. Additionally, the geometric Cauchy problem for the embedding of $$\varvec{g}_0$$ is shown to be reducible to a system of two first-order ODEs for two unknown functions—or equivalently, to a single second-order scalar ODE. For a large class of initial data, this reduction permits explicit solvability of the geometric Cauchy problem for $$\varvec{g}_0$$ up to quadrature. The results described for $$\varvec{g}_0$$ also hold for any classified metric whose embedding system is hyperbolic.

Liouvillian integrability of a general Rayleigh-Duffing oscillator

We give a complete description of the Darboux and Liouville integrability of a general Rayleigh-Duffing oscillator through the characterization of its polynomial first integrals, Darboux polynomials and exponential factors.