Necessary Condition For Integrability Research Articles

On Lr-norm-based derivatives and fuzzy Henstock-Kurzweil integrals with an application

The fundamental theorem of calculus not only plays an important role in the study of differential equation theory, but also in many practical problems solving. In order to enrich its content, in present paper, we extend fuzzy derivatives and fuzzy Henstock-Kurzweil integrals. Firstly, we discuss some properties of Lr-norm-based derivatives for fuzzy Henstock-Kurzweil integrable function (or briefly FLr-derivative). Then, we give some relationships with well-known fuzzy derivatives. After that, we also define a kind of fuzzy Henstock-Kurzweil integral based on FLr-derivative (or briefly Lr-FHK-integral). In addition, we give some important properties and convergence theorems for the Lr-FHK-integral. Meanwhile, we obtain a sufficient and necessary condition for integrability of this integral. As the application, we discuss the fuzzy Fourier series.

Discrete symmetries of equations of dynamics with polynomial integrals of higher degrees

We consider systems with toric configuration space and kinetic energy in the form of a "flat" Riemannian metric on the torus. The potential energy $V$ is a smooth function on the configuration torus. The dynamics of such systems is described by "natural" Hamiltonian systems of differential equations. If $V$ is replaced by $\varepsilon V$, where $\varepsilon$ is a small parameter, then the study of such Hamiltonian systems for small $\varepsilon$ is a part of the "main problem of dynamics" according to Poincaré. We discuss the well-known conjecture on the existence of single-valued momentum-polynomial integrals of motion equations: if there is a momentum-polynomial integral of degree $m$, then there exist a momentum-linear or momentum-quadratic integral. This conjecture was verified in full generality for $m=3$ and $m=4$. We study the cases of "higher" degrees $m=5$ and $m=6$. Similarly to the theory of perturbations of Hamiltonian systems, we introduce resonance lines on the momentum plane. If a system admits a polynomial integral, then the number of these lines is finite. The symmetries of the set of resonance lines are found, from which, in particular, necessary conditions for integrability are derived. Some new criteria for the existence of single-valued polynomial integrals are obtained.

Limit Cycles and Integrability of a Class of Quintic System

In this paper, a class of quintic systems is investigated. The first 13 focal values are computed with the aid of MATHEMATICA. Then the necessary conditions of integrability and linearizability are obtained and the sufficiency of every condition is proved. Meanwhile, bifurcation of limit cycles is discussed, 13 limit cycles can be bifurcated from the origin. As far as the number of limit cycles enclosing an isolated singular point is concerned, this is so far the best result for elementary singular points.

Comment on ,,On the integrability of 2D Hamiltonian systems with variable Gaussian curvature\u201d by A. A. Elmandouh

In the paper [1], the author formulates in Theorem 2 necessary conditions for integrability of a certain class of Hamiltonian systems with non-constant Gaussian curvature, which depends on local coordinates. We give a counterexample to show that this theorem is not correct in general. This contradiction is explained in some extent. However, the main result of this note is our theorem that gives new simple and easy to check necessary conditions to integrability of the system considered in [1]. We present several examples, which show that the obtained conditions are effective. Moreover, we justify that our criterion can be extended to wider class of systems, which are given by non-meromorphic Hamiltonian functions.

Multiple soliton solutions with chiral nonlinear Schrödinger’s equation in (2+1)-dimensions

This manuscript scrutinizes the (2+1)-dimensions chiral nonlinear Schrödinger’s equation with constant coefficients by utilizing two creative mix strategies. The creative mix strategies are namely the functional variable method and first integral method. Solution and singular soliton solutions are successfully recovered through integration techniques. The intermittent particular arrangements have been investigated for the side-effect of creative mix strategies. In order to have an assurance of these solutions, the various imperative relations jumped out to provide necessary conditions for integrability. Moreover, the dynamical attributes of the obtained results have been underlined and depicted in terms of 3D and 2D graphical illustrations.

Integrability Analysis of the Stretch\u2013Twist\u2013Fold Flow

We study the integrability of an eight-parameter family of three-dimensional spherically confined steady Stokes flows introduced by Bajer and Moffatt. This volume-preserving flow was constructed to model the stretch–twist–fold mechanism of the fast dynamo magnetohydrodynamical model. In particular we obtain a complete classification of cases when the system admits an additional Darboux polynomial of degree one. All but one such case are integrable, and first integrals are presented in the paper. The case when the system admits an additional Darboux polynomial of degree one but is not evidently integrable is investigated by methods of differential Galois theory. It is proved that the four-parameter family contained in this case is not integrable in the Jacobi sense, i.e. it does not admit a meromorphic first integral. Moreover, we investigate the integrability of other four-parameter {textit{STF}} systems using the same methods. We distinguish all the cases when the system satisfies necessary conditions for integrability obtained from an analysis of the differential Galois group of variational equations.

On Lie algebras responsible for integrability of (1+1)-dimensional scalar evolution PDEs

Zero-curvature representations (ZCRs) are one of the main tools in the theory of integrable PDEs. In [13], for any (1+1)-dimensional scalar evolution equation E, we defined a family of Lie algebras F(E) which are responsible for all ZCRs of E in the following sense. Representations of the algebras F(E) classify all ZCRs of the equation E up to local gauge transformations. In [12] we showed that, using these algebras, one obtains necessary conditions for existence of a Bäcklund transformation between two given equations.In this paper we show that, using the algebras F(E), one obtains some necessary conditions for integrability of (1+1)-dimensional scalar evolution PDEs, where integrability is understood in the sense of soliton theory. Using these conditions, we prove non-integrability for some scalar evolution PDEs of order 5. Also, we prove a result announced in [13] on the structure of the algebras F(E) for certain classes of equations of orders 3, 5, 7, which include KdV, mKdV, Kaup–Kupershmidt, Sawada–Kotera type equations. Among the obtained algebras for equations considered in this paper and in [12], one finds infinite-dimensional Lie algebras of certain polynomial matrix-valued functions on affine algebraic curves of genus 1 and 0.

Bi-homogeneity and integrability of rational potentials

In this paper we consider natural Hamiltonian systems with two degrees of freedom for which Hamiltonian function has the form H=12(p12+p22)+V(q1,q2) and potential V(q1,q2) is a rational function. Necessary conditions for the integrability of such systems are deduced from integrability of dominate term of the potential which usually is appropriately chosen homogeneous term of V. We show that introducing weights compatible with the canonical structure one can find new dominant terms which can give new necessary conditions for integrability. To deduce them we investigate integrability of a family of bi-homogeneous potentials which depend on two integer parameters. Unexpectedly systems with these potentials can be reduced to the Lotka–Volterra quadratic planar vector field. Then theorem of Jean Moulin Ollagnier, allows us to make complete classification of integrable cases. Moreover, the reduction is used for explicit integration of an exceptional case with a polynomial first integral of degree 4 in momenta.

Blow-up method to compute necessary conditions of integrability for planar differential systems

We provide a method to compute the necessary conditions of integrability for planar differential systems with a weak saddle at the origin. The method is used to characterize the local analytic integrability for resonant saddles of complex systems of the form x˙=px, and y˙=−y+f(y)x,0≠p∈N, with f analytic at the origin and with f(0)=0.

Lie algebras responsible for zero-curvature representations of scalar evolution equations

Zero-curvature representations (ZCRs) are one of the main tools in the theory of integrable PDEs. In particular, Lax pairs for (1+1)-dimensional PDEs can be interpreted as ZCRs.For any (1+1)-dimensional scalar evolution equation E, we define a family of Lie algebras F(E) which are responsible for all ZCRs of E in the following sense. Representations of the algebras F(E) classify all ZCRs of the equation E up to local gauge transformations. To achieve this, we find a normal form for ZCRs with respect to the action of the group of local gauge transformations.As we show in other publications, using these algebras, one obtains some necessary conditions for integrability of the considered PDEs (where integrability is understood in the sense of soliton theory) and necessary conditions for existence of a Bäcklund transformation between two given equations. Examples of proving non-integrability and applications to obtaining non-existence results for Bäcklund transformations are presented in other publications as well.In our approach, ZCRs may depend on partial derivatives of arbitrary order, which may be higher than the order of the equation E. The algebras F(E) generalize Wahlquist–Estabrook prolongation algebras, which are responsible for a much smaller class of ZCRs.In this paper we describe general properties of F(E) and present generators and relations for these algebras. In other publications we study the structure of F(E) for equations of KdV, Krichever–Novikov, Kaup–Kupershmidt, Sawada–Kotera types. Among the obtained algebras, one finds infinite-dimensional Lie algebras of certain matrix-valued functions on rational and elliptic algebraic curves.

Integrability of generalised type II defects in affine Toda field theory

The Liouville integrability of the generalised type II defects is investigated. Full integrability is not considered, only the existence of an infinite number of conserved quantities associated with a system containing a defect. For defects in affine Toda field theories (ATFTs) it is shown that momentum conservation is very likely to be a necessary condition for integrability. The defect Lax matrices which guarantee zero curvature, and so an infinite number of conserved quantities, are calculated for the momentum conserving Tzitzéica defect and the momentum conserving D4 ATFT defect. Some additional calculations pertaining to the D4 defect are also carried out to find a more complete set of defect potentials than has appeared previously.

Integrability of Lotka–Volterra Planar Complex Cubic Systems

In this paper, we study the Lotka–Volterra complex cubic systems. We obtain necessary conditions of integrability for these systems with some restriction on the parameters. The sufficiency is proved for all conditions, except one which remains open, using different methods.

Local Integrability and Linearizability of a $$(1:-1:-1)$$ ( 1 : - 1 : - 1 ) Resonant Quadratic System

In this paper we study the local integrability and linearizability of quadratic three dimensional systems of the form Open image in new window First, we obtain necessary and sufficient conditions for the complete integrability and linearizability of this system. Then, we discuss the problem of existence of one first integral of the form \(\psi ^{(1)}(x,y,z)=xy+O(|x,y,z|^3)\). Computation of resonant focus quantities and the decomposition of the variety of the ideal that they generate in the ring of polynomials of parameters \(a_{ij},b_{ij},c_{ij}\) of the system were used to obtain necessary conditions of integrability and linearizability. The theory of Darboux integrability and some other methods are used to show the sufficiency. In the investigation of the conditions for the existence of one first integral the decomposition of the variety mentioned above was performed using modular computations, its consequences are discussed.

The center problem for a [formula omitted] resonant quadratic system

The main objective of this paper is to find necessary and sufficient conditions for a 1:−4 resonant system of the formx˙=x−a10x2−a01xy−a12y2,y˙=−4y+b2,−1x2+b10xy+b01y2 to have a center at the origin. Since applying a linear change of variables any system of this form can be transformed either to system with a10=1 or a10=0, these are the two cases considered here. When a10=1 there appear 46 resonant center conditions and for a10=0 there are 9 center conditions. To obtain necessary conditions for integrability the computation of the resonant saddle quantities (focus quantities) and the decomposition of the variety of the ideal generated by an initial string of them were used. The theory of Darboux first integrals and some other methods, as the monodromy arguments for instance, are used to show the sufficiency. Since decomposition of the variety mentioned above was performed using modular computations the obtained conditions of integrability represent the complete list of the integrability conditions only with very high probability and there remains an open problem to verify this statement.

Integrability and Linearizability of Three Dimensional Vector Fields

This paper deals with the integrability and linearizability problem of three dimensional systems $$\begin{aligned} \dot{x}&= x(1 +ax+by+cz),\\ \dot{y}&= -y+ dx^2 +exy + fxz+gyz+hy^2+kz^2,\\ \dot{z}&= z(1 + \ell x + my +pz). \end{aligned}$$ More precisely, we give a complete set of necessary conditions for integrability and linearizability and then prove their sufficiency using Darboux method and Darboux inverse Jacobi multiplier, power series argument and a solution of a Riccati equation.

(Non)-integrability of geodesics in D-brane backgrounds

Motivated by the search for new backgrounds with integrable string theories, we classify the D-brane geometries leading to integrable geodesics. Our analysis demonstrates that the Hamilton-Jacobi equation for massless geodesics can only separate in elliptic or spherical coordinates, and all known integrable backgrounds are covered by this separation. In particular, we identify the standard parameterization of AdSp × Sq with elliptic coordinates on a flat base. We also find new geometries admitting separation of the Hamilton-Jacobi equation in the elliptic coordinates. Since separability of this equation is a necessary condition for integrability of strings, our analysis gives severe restrictions on the potential candidates for integrable string theories.

Darboux points and integrability analysis of Hamiltonian systems with homogeneous rational potentials

We study integrability in the Liouville sense for natural Hamiltonian systems with a homogeneous rational potential V(q). Strong necessary conditions for the integrability of such systems were obtained by analysis of differential Galois groups of variational equations along certain particular solutions. These conditions have the form of arithmetic restrictions on eigenvalues of the Hessians V″(d) calculated at non-zero solutions d of the equation gradV(d)=d. Such solutions are called proper Darboux points.It was recently proved that for generic polynomial homogeneous potentials, there exist universal relations between eigenvalues of Hessians of the potential taken at all proper Darboux points. The existence of such relations for rational potentials seems to be a difficult question. One of the reasons is the presence of points of indeterminacy of the potential and its gradient. Nevertheless, for two degrees of freedom we prove that such a relation exists. This result is important because it allows us to show that the set of admissible values for Hessian eigenvalues at a proper Darboux point for potentials satisfying the necessary conditions for integrability is finite. In turn, this provides a tool for classification of integrable rational potentials.It was also recently shown that for polynomial homogeneous potentials, additional necessary conditions for integrability can be deduced from the existence of improper Darboux points, that is, points d that are non-zero solutions of gradV(d)=0. These new conditions also take the form of arithmetic restrictions imposed on eigenvalues of V″(d). In this paper we prove that for rational potentials, improper Darboux points give the same necessary conditions for integrability.

Non-integrability by discrete quadratures

Abstract. We give a necessary condition for integrability by discrete quadratures of systems of difference equations: the discrete variational equations along algebraic solutions must have virtually solvable Galois groups. This necessary condition à la Morales and Ramis is used in order to prove that q-analogues of Painlevé I and Painlevé III equations are not integrable by discrete quadratures.

Constructing towers with skeletons from open Lie algebras and integrability

We provide a given algebraic structure with the structure of an infinitesimal algebraic skeleton. The necessary conditions for integrability of the absolute parallelism of a tower with such a skeleton are dispersive nonlinear models and related conservation laws given in the form of associated linear spectral problems.

Nonexistence of horizontal Sobolev surfaces in the Heisenberg group

Involutivity is a well known necessary condition for integrability of smooth tangent distributions. We show that this condition is still necessary for integrability with Sobolev surfaces. We specialize our study to the left invariant horizontal distribution of the first Heisenberg group ℍ 1 . Here we answer a question raised in a paper by Z.M. Balogh, R. Hoefer-Isenegger, and J.T. Tyson.