Painleve Hierarchy Research Articles

Rational and Special Solutions for Some Painlevé Hierarchies

A self-similar reduction of the Korteweg–de Vries hierarchy is considered. A linear system of equations associated with this hierarchy is presented. This Lax pair can be used to solve the Cauchy problem for the studied hierarchy. It is shown that special solutions of the self-similar reduction of the KdV hierarchy are determined via the transcendents of the first Painleve hierarchy. Rational solutions are expressed by means of the Yablonskii–Vorob’ev polynomials. The map of the solutions for the second Painleve hierarchy into the solutions for the self-similar reduction of the KdV hierarchy is illustrated using the Miura transformation. Lax pairs for equations of the hierarchy for the Yablonskii–Vorob’ev polynomial are discussed. Special solutions to the hierarchy for the Yablonskii–Vorob’ev polynomials are given. Some other hierarchies with properties of the Painleve hierarchies are presented. The list of nonlinear differential equations whose general solutions are expressed in terms of nonclassical functions is extended.

Nonlinear differential equations associated with the first Painlevé hierarchy

The first Painlevé hierarchy with general solutions in the form of the Painlevé transcendents is considered. The linear system associated with this hierarchy is given. Some new hierarchies with properties similar to the Painlevé hierarchies are presented. It is shown that the solutions of these hierarchies are expressed via the transcendents of the first Painlevé hierarchy. Thus, the list of nonlinear differential equations whose solutions are expressed in terms of non-classical functions is extended.

Hankel Determinant Approach to Generalized Vorob’ev–Yablonski Polynomials and Their Roots

Generalized Vorob’ev–Yablonski polynomials have been introduced by Clarkson and Mansfield in their study of rational solutions of the second Painleve hierarchy. We present new Hankel determinant identities for the squares of these special polynomials in terms of Schur polynomials. As an application of the identities, we analyze the roots of generalized Vorob’ev–Yablonski polynomials and provide a partial characterization for the boundary curves of the highly regular patterns observed numerically in Clarkson and Mansfield (Nonlinearity 16(3):R1–R26, 2003).

Instanton-type solutions for the second and the fourth Painlevé hierarchies with a large parameter

A construction of general formal solutions for members $(P_{\mathrm{J}})_{m}$ $({\rm J = II,\,IV})$ of the second and the fourth Painleve hierarchies with a large parameter is discussed. We also investigate a relation between formal solutions of $(P_{\mathrm{J}})_{m}$ $({\rm J = II,\,IV})$.

Auxiliary differential polynomials for the first Painlevé hierarchy

This article concerns with the first Painleve hierarchy, i.e. the $2n$-th order analogues of the first Painleve equation. Several higher order analogues of the first Painleve equation are proposed by several authors however, we investigate one derived from the KdV hierarchy by studying singularity manifold. We give the auxiliary differential polynomials of the first Painleve hierarchy.

Higher Painlevé transcendents as special solutions of some nonlinear integrable hierarchies

It is well known that the self-similar solutions of the Korteweg-de Vries equation and the modified Korteweg-de Vries equation are expressed via the solutions of the first and second Painleve equations. In this paper we solve this problem for all equations from the Korteveg-de Vries, modified Korteweg-de Vries, Kaup-Kupershmidt, Caudrey-Dodd-Gibbon and Fordy-Gibbons hierarchies. We show that the self-similar solutions of equations corresponding to hierarchies mentioned above can be found by means of the general solutions of higher-order Painleve hierarchies introduced more than ten years ago.

Bäcklund Transformations for First and Second Painlevé Hierarchies

We give Backlund transformations for first and second Painleve hierarchies. These Backlund transformations are generalization of known Backlund transformations of the first and second Painleve equations and they relate the considered hierarchies to new hierarchies of Painleve-type equations.

Principally tame regular sequences associated with the fourth Painlevé hierarchy with a large parameter

The notion of principally tame regular sequences is introduced for systems of polynomials with a weight vector. As an application, construction of formal solutions is discussed for the systems of nonlinear differential equations which belong to the fourth Painleve hierarchy with a large parameter.

Special polynomials associated with some hierarchies

New special polynomials associated with the rational solutions of analogue to the Painleve hierarchies are introduced. The Hirota relations for these special polynomials are found. Differential - difference hierarchies for finding special polynomials are presented. These formulae allow us to search the special polynomials associated with the hierarchy studied. The ordinary differential hierarchy for the Yablonskii - Vorob'ev polynomials is given.

A 2 + 1 non-isospectral integrable lattice hierarchy related to a generalized discrete second Painlevé hierarchy

Abstract In this article, by considering a 2 + 1 dimensional discrete non-isospectral linear problem, a new 2 + 1 dimensional integrable lattice hierarchy is constructed. It is shown that a generalization of the discrete second Painleve hierarchy can be obtained as a reduction. Other reductions include new 1 + 1 dimensional integrable lattice hierarchies.

On WKB analysis of higher order Painlevé equations with a large parameter

We announce a generalization of the reduction theorem for 0-parameter solutions of the traditional (i.e., second order) Painleve equations with a large parameter to those of some higher order Painleve equations, i.e.,each member of the Painleve hierarchies (P J ) (J = I, II-1 and II-2) discussed in [KKNT]. Thus the scope of applicability of the reduction theorem ([KT1, KT2]) has been substantially enlarged; only six equations were covered by our previous result, while the result reported here applies to infinitely many equations.

Painlevé Hierarchies and the Painlevé Test

In a series of recent papers, we derived several new hierarchies of higher-order analogues of the six Painleve equations. Here we consider one particular example of such a hierarchy, namely, a recently derived fourth Painleve hierarchy. We use this hierarchy to illustrate how knowing the Hamiltonian structures and Miura maps can allow finding first integrals of the ordinary differential equations derived. We also consider the implications of the second member of this hierarchy for the Painleve test. In particular, we find that the Ablowitz–Ramani–Segur algorithm cannot be applied to this equation. This represents a significant failing in what is now a standard test of singularity structure. We present a solution of this problem.

On the Fourth Painlevé Hierarchy

We use the inverse monodromy transform to find the fourth Painleve hierarchy. The second and third members of this hierarchy are given. Special and rational solutions of the second and third members for the P4 hierarchy are discussed. We apply the Painleve test to the second member of the fourth Painleve hierarchy.

On a new non-isospectral variant of the Boussinesq hierarchy

We give a new non-isospectral extension to 2 + 1 dimensions of the Boussinesq hierarchy. Such a non-isospectral extension of the third-order scattering problem xxx +U x +(V - ) = 0 has not been considered previously. This extends our previous results on one-component hierarchies in 2 + 1 dimensions associated to third-order non-isospectral scattering problems. We characterize our entire (2 + 1)-dimensional hierarchy and its linear problem using a single partial differential equation and its corresponding non-isospectral scattering problem. This then allows an alternative approach to the construction of linear problems for the entire (2 + 1)-dimensional hierarchy. Reductions of this hierarchy yield new integrable hierarchies of systems of ordinary differential equations together with their underlying linear problems. In particular, we obtain a `fourth Painleve hierarchy', i.e. a hierarchy of ordinary differential equations having the fourth Painleveequation as its first member. We also obtain a hierarchy having as its first member a generalization of an equation defining a new transcendent due to Cosgrove.