We use methods from dynamical systems to study the fourth Painleve equation PIV. Our starting point is the symmetric form of PIV, to which the Poincare compactification is applied. The motion on the sphere at infinity can be completely characterized. There are fourteen fixed points, which are classified into three different types. Generic orbits of the full system are curves from one of four asymptotically unstable points to one of four asymptotically stable points, with the set of allowed transitions depending on the values of the parameters. This allows us to give a qualitative description of a generic real solution of PIV.
We study a Hamiltonian system without the Painlevé property and show that it admits a kind of regularisation on a bundle of rational surfaces with certain divisors removed, generalising Okamoto's spaces of initial conditions for the Painlevé differential equations. The system in question was obtained by Takasaki as part of the Painlevé-Calogero correspondence and possesses the algebro-Painlevé property, being related by an algebraic transformation to the fourth Painlevé equation. We provide an atlas for the bundle of surfaces in which the system has a global Hamiltonian structure, with all Ham…
Painleve transcendents are usually considered as complex functions of a complex variable, but in applications it is often the real cases that are of interest. Under a reasonable assumption (concerning the behavior of a dynamical system associated with Painleve IV, as discussed in a recent paper), we give a number of results towards a classification of the real solutions of Painleve IV (or, more precisely, symmetric Painleve IV), according to their asymptotic behavior and singularities. We show the existence of globally nonsingular real solutions of symmetric Painleve IV for arbitrary nonzero v…
In this paper we study the integrability of the Hamiltonian system associated to the fourth Painlevé equation. We prove that one two parametric family of this Hamiltonian system is not integrable in the sense of the Liouville-Arnold theorem. Computing explicitly the Stokes matrices and the formal invariants of the second variational equations we deduce that the connected component of the unit element of the corresponding Galois grou is not Abelian. Thus the Morales-Ramis-Simó theory leads to a non-integrable result. Moreover, combining the new result with our previous one we establish that for…
The last decades saw growing interest across multiple disciplines in nonlinear phenomena described by partial differential equations (PDE). Integrability of such equations is tightly related with the Painleve property - solutions being free from moveable critical singularities. The problem of Painleve classification of ordinary and partial nonlinear differential equations lasting since the end of XIX century saw significant advances for the equation of lower (mainly up to fourth with rare exceptions) order, however not that much for the equations of higher orders. Recent works of the author ha…
Classical Symmetries and Painleve Analysis for the (1+3) Kudryashov-Sinelshchikov Equation
Woxsen University（沃克森大学）；Pondicherry University（本地治里大学）；Durban University of Technology（德班理工大学）；Stellenbosch University（斯泰伦博斯大学）；Universidad Católica del Norte（北天主教大学）；University of KwaZulu-Natal（夸祖鲁-纳塔尔大学）
In this paper, we show how to relate $n$-dimensional cubes on which ABS equations hold to the symmetry groups of discrete Painlevé equations. We here focus on the reduction from the 4-dimensional cube to the $q$-discrete third Painlevé equation, which is a dynamical system on a rational surface of type $A_5^{(1)}$ with the extended affine Weyl group $\widetilde{\mathcal W}\bigl((A_2+A_1)^{(1)}\bigr)$. We provide general theorems to show that this reduction also extends to other discrete Painlevé equations at least of type A.
We propose multidimensional versions of the Painlevé VI equation and its degenerations. These field theories are related to the isomonodromy problems of flat holomorphic infinite rank bundles over elliptic curves and take the form of non-autonomous Hamiltonian equations. The modular parameter of curves plays the role of "time". Reduction of the field equations to the zero modes leads to ${\rm SL}(N, {\mathbb C})$ monodromy preserving equations. The latter coincide with the Painlevé VI equation for $N\!=\!2$. We consider two types of the bundles. In the first one the group of automorphisms is t…
This manuscript develops a novel understanding of non-polar solutions of the discrete Painlevé I equation (dP1). As the non-autonomous counterpart of an analytically completely integrable difference equation, this system is endowed with a rich dynamical structure. In addition, its non-polar solutions, which grow without bounds as the iteration index $n$ increases, are of particular relevance to other areas of mathematics. We combine theory and asymptotics with high-precision numerical simulations to arrive at the following picture: when extended to include backward iterates, known non-polar so…
We present a complete dynamical description of the higher-order modulation instability for a fourth-order nonlinear Schrödinger equation. For two-breather solutions of this equation, we have identified the locus in a geometrical space where the growth rates for the breathers are equal in parameter space. We show that a circle bounds the entire parameter space for the nonlinear Schrödinger equation. In contrast, it is bound by an intersecting circle and an ellipse for the fourth-order equation. We show that, for all the higher-order equations in the nonlinear Schrödinger equation hierarchy, the…
A geometric study for an integrable 4-dimensional dynamical system so called the Fuji-Suzuki-Tsuda system is given. By the resolution of indeterminacy, the group of its Bäklund transformations is lifted to a group of pseudo-isomorphisms between rational varieties obtained from $(P^1)^4$ by blowing-up along eight 2-dimensional subvarieties and four 1-dimensional subvarieties. The root basis is realised in the Néron-Severi bilattices. A discrete Painlevé system with quadratic degree growth is also realised as its translational element.
