Symmetry, Integrability and Geometry: Methods and Applications Bäcklund Transformations for First and Second Painlevé Hierarchies

We give B\"acklund transformations for first and second Painlev\'e hierarchies. These B\"acklund transformations are generalization of known B\"acklund transformations of the first and second Painlev\'e equations and they relate the considered hierarchies to new hierarchies of Painlev\'e-type equations.


Introduction
One century ago Painlevé and Gambier have discovered the six Painlevé equations, PI-PVI. These equations are the only second-order ordinary differential equations whose general solutions can not be expressed in terms of elementary and classical special functions; thus they define new transcendental functions. Painlevé transcendental functions appear in many areas of modern mathematics and physics and they paly the same role in nonlinear problems as the classical special functions play in linear problems.
In recent years there is a considerable interest in studying hierarchies of Painlevé equations. This interest is due to the connection between these hierarchies of Painlevé equations and completely integrable partial differential equations. A Painlevé hierarchy is an infinite sequence of nonlinear ordinary differential equations whose first member is a Painlevé equation. Airault [1] was the first to derive a Painlevé hierarchy, namely a second Painlevé hierarchy, as the similarity reduction of the modified Korteweg-de Vries (mKdV) hierarchy. A first Painlevé hierarchy was given by Kudryashov [2]. Later on several hierarchies of Painlevé equations were introduced [3,4,5,6,7,8,9,10,11].
As it is well known, Painlevé equations possess Bäcklund transformations; that is, mappings between solutions of the same Painlevé equation or between solutions of a particular Painlevé equation and other second-order Painlevé-type equations. Various methods to derive these Bäcklund transformations can be found for example in [12,13,14,15]. Bäcklund transformations are nowadays considered to be one of the main properties of integrable nonlinear ordinary differential equations, and there is much interest in their derivation.
In the present article, we generalize known Bäcklund transformations of the first and second Painlevé equations to the first and second Painlevé hierarchies given in [6,11]. We give a Bäcklund transformation between the considered first Painlevé hierarchy and a new hierarchy of Painlevé-type equations. In addition, we give two new hierarchies of Painlevé-type equations related, via Bäcklund transformations, to the considered second Painlevé hierarchy. Then we derive auto-Bäcklund transformations for this second Painlevé hierarchy. Bäcklund transformations of the second Painlevé hierarchy have been studied in [6,16].

Bäcklund transformations for PI hierarchy
In this section, we will derive a Bäcklund transformation for the first Painlevé hierarchy (PI hierarchy) [6] n+1 j=2 γ j L j [u] = γx, (2.1) where the operator L j [u] satisfies the Lenard recursion relation The special case γ j = 0, 2 ≤ j ≤ n, of this hierarchy is a similarity reduction of the Schwarz-Korteweg-de Vries hierarchy [2,4]. Moreover its members may define new transcendental functions. The PI hierarchy (2.1) can be written in the following form [11] where R I is the recursion operator In [17,18], it is shown that the Bäcklund transformation defines a one-to-one correspondence between the first Painlevé equation We will show that there is a generalization of this Bäcklund transformation to all members of the PI hierarchy (2.3). Let Differentiating (2.7) and using (2.3), we find Substituting u = −y x into (2.7), we obtain the following hierarchy of differential equation for y where S I is the recursion operator The first member of the hierarchy (2.9) is the SD-I.e equation (2.6). Thus we shall call this hierarchy SD-I.e hierarchy. Therefor we have derived the Bäcklund transformation (2.7)-(2.8) between solutions u of the first Painlevé hierarchy (2.3) and solutions y of the SD-I.e hierarchy (2.9).
Example 2 (n = 3). The third member of the PI hierarchy (2.3) reads In this case, the Bäcklund transformation (2.7) has the form (2.14) Equations (2.8) and (2.14) give one-to-one correspondence between solutions u of (2.13) and solutions y of the following equation

Bäcklund transformations for second Painlevé hierarchy
In the present section, we will study Bäcklund transformations of the second Painlevé hierarchy (PII hierarchy) [6] ( where the operator L j [u] is defined by (2.2). The special case γ j = 0, 1 ≤ j ≤ n − 1, of this hierarchy is a similarity reduction of the modified Korteweg-de Vries hierarchy [2,4]. The members of this hierarchy may define new transcendental functions.
This hierarchy can be written in the following alternative form [11] R n II u + where R II is the recursion operator

A hierarchy of SD-I.d equation
As a first Bäcklund transformation for the PII hierarchy (3.1), we will generalize the Bäcklund transformation between the second Painlevé equation and the SD-I.d equation of Cosgrove and Scoufis [17,18]. Let where ǫ = ±1. Differentiating (3.2) and using (3.1), we find Now we will show that where the operator H j [p] satisfies the Lenard recursion relation Firstly, we will use induction to show that for any j = 1, 2, . . . , Thus Assume that it is true for j = k. Then Integration by parts gives Hence (3.8) can be written as Using (3. 3) to substitute u x and (3.7) to substitute u xx , (3.9) becomes 2R k+1 Since (3.5), and hence the proof by induction is finished. Now using (3.6) we find (3.10) Using (3.3) to substitute u x into (3.10) and then integrating, we obtain (3.4). Therefore (3.2) can be used to obtain the following quadratic equation for u Eliminating u between (3.3) and (3.11) gives a one-to-one correspondence between the second Painlevé hierarchy (3.1) and the following hierarchy of second-degree equations Therefore we have derived the Bäcklund transformation (3.2) and (3.11) between the PII hierarchy (3.1) and the new hierarchy (3.12).
Next we will give the explicit forms of the above results when n = 1, 2, 3.

A hierarchy of a second-order fourth-degree equation
In this subsection, we will generalize the Bäcklund transformation given in [23] between the second Painlevé equation and a second-order fourth-degree equation. Let Differentiating (3.26) and using (3.1), we find Equations (3.26) and (3.27) define a Bäcklund transformation between the second Painlevé hierarchy (3.1) and a new hierarchy of differential equations for y.
In order to obtain the new hierarchy, we will prove that where S II is the recursion operator First of all, we will use induction to prove that Using (3.27), we find Hence Thus (3.29) is true for j = 1.
Assume it is true for j = k. Then Using (3.30) to substitute u x and u xx and using (3.27) to substitute u 2 , we find the result. As a second step, we use (3.29) to find Thus using (3.30) to substitute u x and using (3.27) to substitute u 2 we find (3.28). Therefore (3.26) implies If α = 0, then substituting u from (3.31) into (3.27) we obtain the following hierarchy of differential equations for y If α = 0, then y satisfies the hierarchy The first member of the hierarchy (3.32) is a fourth-degree equation, whereas the other members are second-degree equations. Now we give some examples.