Non-autonomous Henon-Heiles system from Painleve class

TL;DR

This paper demonstrates how to deform a separable Henon-Heiles system, linked to the 5th-order KdV, into non-autonomous Painleve-type systems using isomonodromic Lax representations, revealing new connections between integrable systems.

Contribution

It introduces a method to deform integrable Henon-Heiles systems into Painleve-type systems via isomonodromic Lax representations, expanding the understanding of their integrability properties.

Findings

Deformation of Henon-Heiles system into Painleve-type systems.

Connection established between KdV stationary flow and Painleve equations.

Use of isomonodromic Lax representation for non-autonomous systems.

Abstract

We show how to deform separable Henon-Heiles system with isospectral Lax representation, related with the stationary flow of the $5th$-order KdV, to respective non-autonomous systems of Painleve type with isomonodromic Lax representation.

Equations

$$\frac{\partial \xi }{\partial t_r}=X_{h_r}(\xi )=\pi d{h}_r\text{, }r=1,\dots ,n$$

$${\left\{h_r,h_s\right\}}_{\pi }:=\pi (d{h}_r,d{h}_s)=0\text{, }r,s=1,\dots ,n$$

$$\left[X_{h_r},X_{h_s}\right]=0r,s=1,\dots ,n.$$

$$\frac{\partial L(\lambda ;\xi )}{\partial t_k}=[U_k(\lambda ;\xi ),L(\lambda ;\xi )],$$

$$\frac{\partial H_r}{\partial t_s}-\frac{\partial H_s}{\partial t_r}+\{H_r,H_s\}=f_{rs}(t_1,...,t_n),r,s=1,\dots ,n$$

$$Y_{H_k}(\xi ,t)=\pi d{H}_k\text{, }k=1,\dots ,n$$

$$\frac{\partial Y_{H_r}}{\partial t_s}-\frac{\partial Y_{H_s}}{\partial t_r}+\left[Y_{H_s},Y_{H_r}\right]=0,r,s=1,\dots ,n,$$

$$\frac{\partial \xi }{\partial t_r}=Y_{H_r}(\xi ,t)=\pi d{H}_r\text{, }r=1,\dots ,n$$

$$\frac{\partial \Psi }{\partial \lambda }=L(\lambda ;\xi ,t)\Psi ,\frac{\partial \Psi }{\partial t_k}=U_k(\lambda ;\xi ,t)\Psi ,$$

$$\frac{\partial L(\lambda ;\xi ,t)}{\partial t_k}=[U(\lambda ;\xi ,t),L(\lambda ;\xi ,t)]+\frac{\partial U_k(\lambda ;\xi ,t)}{\partial \lambda }$$

$$h_1$$

$$h_2$$

$$\frac{\partial x_1}{\partial t_1}$$

$$\frac{\partial p_1}{\partial t_1}$$

$$\frac{\partial x_1}{\partial t_2}$$

$$\frac{\partial p_1}{\partial t_2}$$

$$L(\lambda )=\left(\begin{array}{cc}{p}_1\lambda +\frac{1}{2}{x}_2{p}_2& {\lambda }^2-x_1\lambda -\frac{1}{4}{x}_2^2\\ & \\ -2{\lambda }^3-2x_1{\lambda }^2-\left(2x_1^2+\frac{1}{2}{x}_2^2\right)\lambda +p_2^2+2\alpha x_2^{-2}& -p_1\lambda -\frac{1}{2}{x}_2{p}_2\end{array}\right),$$

$$U_1(\lambda )=\left(\begin{array}{cc}0& \frac{1}{2}\\ & \\ -\lambda -2x_1& 0\end{array}\right),U_2(\lambda )=\left(\begin{array}{cc}\frac{1}{2}{p}_1& \frac{1}{2}\lambda -\frac{1}{2}{x}_1\\ & \\ -{\lambda }^2-x_1\lambda -x_1^2-\frac{1}{2}{x}_2^2& -\frac{1}{2}{p}_1\end{array}\right).$$

$$V^{(k)}=\left(\begin{array}[]{c}V_{1}^{(k)}\\ V_{2}^{(k)}\end{array}\right)=R^{k}\left(\begin{array}[]{c}0\\ 1\end{array}\right),\ \ \ \ R=\left(\begin{array}[]{cc}x_{1}&1\\ \frac{1}{4}x_{2}^{2}&0\end{array}\right),\ \ \ \ \ k\in\mathbb{Z}.$$

$$H_1(t)$$

$$=\frac{1}{2}{p}_1^2+\frac{1}{2}{p}_2^2+x_1^3+\frac{1}{2}{x}_1{x}_2^2+c_3(t_1,t_2)(x_1^2+\frac{1}{4}{x}_2^2)+c_2(t_1,t_2)x_1+\alpha x_2^{-2},$$

$$H_2(t)$$

$$=\frac{1}{2}{x}_2{p}_1{p}_2-\frac{1}{2}{x}_1{p}_2^2-p_1+\frac{1}{16}{x}_2^4+\frac{1}{4}{x}_1^2{x}_2^2+\frac{1}{4}{c}_3(t_1,t_2)x_1{x}_2^2+\frac{1}{4}{c}_2(t_1,t_2)x_2^2-\alpha x_1{x}_2^{-2}.$$

$$c_3(t_1,t_2)=3t_2,c_2(t_1,t_2)=t_1+3t_2^2,f_{12}(t_1,t_2)=-c_2(t_1,t_2).$$

$$\frac{\partial x_1}{\partial t_1}=\frac{\partial H_1}{\partial p_1}=p_1,\frac{\partial x_2}{\partial t_1}=\frac{\partial H_1}{\partial p_2}=p_2,$$

$$\displaystyle\frac{\partial p_{1}}{\partial t_{1}}=-\frac{\partial H_{1}}{\partial x_{1}}=-3x_{1}^{2}-\frac{1}{2}x_{2}^{2}-6t_{2}x_{1}+t_{1}+3t_{2}^{2},\ \ \ \ \ \$$

$$\frac{\partial p_2}{\partial t_1}=-\frac{\partial H_1}{\partial x_2}=-x_1{x}_2-\frac{3}{2}{t}_2{x}_2+2\alpha x_2^{-3}.$$

$$\frac{\partial x_1}{\partial t_2}=\frac{\partial H_2}{\partial p_1}=\frac{1}{2}{x}_2{p}_2-1,\frac{\partial x_2}{\partial t_2}=\frac{\partial H_2}{\partial p_2}=\frac{1}{2}{x}_2{p}_1-x_1{p}_2,$$

$$\frac{\partial p_1}{\partial t_2}=-\frac{\partial H_2}{\partial x_1}=\frac{1}{2}{p}_2^2-\frac{1}{2}{x}_1{x}_2^2-\frac{3}{4}{t}_2{x}_2^2+\alpha x_2^{-2},$$

$$\frac{\partial p_2}{\partial t_2}=-\frac{\partial H_2}{\partial x_2}=-\frac{1}{2}{p}_1{p}_2-\frac{1}{4}{x}_2^3-\frac{1}{2}{x}_1^2{x}_2-\frac{3}{2}{t}_2{x}_1{x}_2-\frac{1}{2}\left(t_1+3t_2^2\right)x_2-2\alpha x_1{x}_2^{-3}.$$

$$L(\lambda ;t)=\left(\begin{array}{cc}{p}_1\lambda +\frac{1}{2}{x}_2{p}_2& {\lambda }^2-x_1\lambda -\frac{1}{4}{x}_2^2\\ & \\ -2{\lambda }^3-2(x_1+3t_2){\lambda }^2-\left(2x_1^2+\frac{1}{2}{x}_2^2+6x_1{t}_2+6t_2^2+2t_1\right)\lambda +p_2^2+2\alpha x_2^{-2}& -p_1\lambda -\frac{1}{2}{x}_2{p}_2\end{array}\right),$$

$$U_1(\lambda ;t)=\left(\begin{array}{cc}0& \frac{1}{2}\\ & \\ -\lambda -2x_1-3t_2& 0\end{array}\right),$$

$$U_2(\lambda ;t)=\left(\begin{array}{cc}\frac{1}{2}{p}_1& \frac{1}{2}\lambda -\frac{1}{2}{x}_1\\ & \\ -{\lambda }^2-(x_1+3t_2)\lambda -x_1^2-\frac{1}{2}{x}_2^2-3x_1{t}_2-3t_2^2-t_1& -\frac{1}{2}{p}_1\end{array}\right).$$

$$\frac{\partial L(\lambda ;t)}{\partial t_1}-[U_1(\lambda ;t),L(\lambda ;t)]$$

$$\frac{\partial L(\lambda ;t)}{\partial t_2}-[U_2(\lambda ;t),L(\lambda ;t)]$$

$$\frac{\partial L(\lambda ;t)}{\partial t_k}=[U(\lambda ;t),L(\lambda ;t)]+2\lambda \frac{\partial U_k(\lambda ;t)}{\partial \lambda },k=1,2,$$

Full text

There are two particular classes of second order nonlinear ordinary differential equations (ODE’s) playing important roles in modern physics and mathematics. To the first class belong separable equations with autonomous Hamiltonian representation. To the second class, belong Painlevé equations with non-autonomous (in principle) Hamiltonian representation. The separable equations can be expressed by so-called Lax representation in the form of isospectral deformation equations while the Painlevé equations can be expressed by Lax representation in the form of isomonodromic deformation equations.

Actually, separable equations belong to the class of Liouville integrable systems. A Liouville system on a $2n$-dimensional Poisson manifold $(M,\pi)$, where $\pi$ is a Poisson operator, is the set of dynamical equations of the form

[TABLE]

where $\xi\in M$ denotes points on $M$ and $h_{r}(\xi)$ are $n$ Poisson-commuting functions on $M$

[TABLE]

so that

[TABLE]

Since all the vector fields $X_{h_{r}\text{ }}$commute (3), the system (1), as a Pfaffian system, has a common, unique (local) solution $\xi(t_{1},\ldots,t_{n},\xi_{0})$ through each point $\xi_{0}\in M$ depending in general on all the evolution parameters $t_{k}$. Further, let $L(\lambda;\xi)$ and $U_{k}(\lambda;\xi)$  be a matrices that belong to some Lie algebra and which depend rationally on the independent $\lambda$ called a spectral parameter. The autonomous separable equations (1) can be represented by the Lax form

[TABLE]

which is called the isospectral deformation equation because the eigenvalues of the matrix $L$ are independent of all times $t_{k}$, $k=1,...,n.$

Now consider a set of $n$ non-autonomous Hamiltonians $H_{r}(\xi,t)$ satisfying the Frobenius condition

[TABLE]

instead of (2) ones, where $f_{rs}$ are functions of evolution parameters only. In consequence, the non-autonomous Hamiltonian vector fields

[TABLE]

satisfy the vector-field counterpart of (5)

[TABLE]

as $Y_{\{H_{r},H_{s}\}}=-\left[Y_{H_{r}},Y_{H_{s}}\right]$. Therefore, the set of non-autonomous Hamiltonian equations (the Pfaffian system)

[TABLE]

has again common solutions $\xi(t_{1},\ldots,t_{n},\xi_{0})$ through each point $\xi_{0}$ of $M$ [7, 10].

If the non-autonomous Hamiltonian equations (6) are of the Painlevé type then are represented by so-called Lax isomonodromic deformations. This means that their solutions can be obtained from a system of linear equations

[TABLE]

where matrices $L$ and $U$ have rational singularities in $\lambda$, for which the compatibility condition

[TABLE]

is equivalent to the corresponding Painlevé equation (6). The analytic continuation of a fundamental matrix solution for the first equation in the system (9) defines monodromy data that is independent of all $t_{k}$, what is ensured by the second equation, hence the system (10) is called an isomonodromy problem. Note also, that the isomonodromy representation (10) is only the necessary condition for the Painlevé property [6], so equations with representation (10) should be rather called of the Painlevé type.

The advantage of nonlinear separable ODE’s is their integrability by quadratures. As for Painlevé equations, although they are not integrable by quadratures, nevertheless they have solutions which are free of movable branch points and essential singularities. So, poles are the only singularities of the solutions which change their position if one varies the initial data. Thus, the solutions of the Painlevé ODE’s are ‘regular’ single-valued functions around movable poles (meromorphic in the solution domain), and as such are good candidates that define new special (transcendental) functions.

A significant progress in construction of new multi-component Painlevé equations took place since the modern theory of nonlinear integrable PDE’s has been born (the so-called soliton theory). It was found that the Painlevé equations are inseparably connected with the soliton systems with whom they share many properties (see [5, 11, 12, 13, 15] and references therein). The Painlevé equations are constructed under particular reductions of soliton PDE’s hierarchies.

In that short letter we would like to draw the attention of the reader onto alternative way of construction of alredy known and new Painlevé type ODE’s by an appropriate deformations of separable ODE’s. The method consists of few steps. First, consider a separable geodesic motion on an appropriate $n$-dimensional pseudo-Riemannian space $(Q,g)$ with a metric $g$ that is flat or of constant curvature. In Hamiltonian formalism on $M=T^{\ast}Q$, with such system one can relates $n$ geodesic Hamiltonians $E_{1},...,E_{n}$ in involution and $n$ Hamiltonian vector fields $X_{1},...,X_{n}$ that commute. Next, extend geodesic Hamiltonians $E_{i}\rightarrow\mathfrak{h}_{i}=E_{i}+W_{i},\ i=2,...,n$ by linear in momenta terms, generated by Killing vectors of $g$ in such a way that $\mathfrak{h}_{i}$ constitute a Lie algebra [14]. Then, add separable potentials $\mathfrak{h}_{i}\rightarrow h_{i}=E_{i}+W_{i}+V_{i}$ and prove for which ones there exists a non-autonomous deformation $h_{i}\rightarrow H_{i}(t_{1},...,t_{n})$ satisfying the Frobenius condition (5). The deformation procedure in the geodesic case $\mathfrak{h}_{i}\rightarrow H_{i}(t_{1},...,t_{n})$ is presented in [3]. The systematic work on the deformation procedure with nontrivial potentials is in progress. Finally, one should investigate the related deformation of Lax representation, based on the results from [4].

Here, we would like to show the simple illustration of the method on the example of one of the integrable cases of the celebrated Hénon-Heiles system and its deformation to non-autonomous system with isomonodromic Lax representation. Slightly different deformation of that system, coming from the similarity solutions of soliton equations was considered in [9].

Consider Liouville integrable extended Hénon-Heiles system on $M=\mathbb{R}^{4},$ generated by two Hamiltonian functions

[TABLE]

in involution, written in Cartesian coordinates $(x_{1},x_{2})$ and conjugate momenta $(p_{1},p_{2})$, where $E$ are geodesic parts of $h$, while $V(x)$ are separable potentials. By setting the parameter $\alpha$ equal to zero we get one of the integrable cases of the standard Hénon-Heiles system. The Hénon-Heiles Hamiltonian is $h_{1}$, so for the canonical form of the Poisson tensor $\{x_{i},p_{j}\}_{\pi}=\delta_{ij}$, the related autonomous evolution equations are

[TABLE]

What is important, equations (12) represent the stationary flow of the $5th$-order KdV [8]. Here $h_{2}$ is the first integral of (12) while the related equations

[TABLE]

represent the symmetry of (12). Evolution equations (12) and (13) have Lax representations (4), where [4]

[TABLE]

Let us remark that for the geodesic Hamiltonians $E_{1}$ and $E_{2}$ there exists infinite hierarchy of basic separable potentials, generated by the recursion formula [1, 2]

[TABLE]

The Hénon-Heiles potential is the one for $k=4$ and the additional term in (11) is the potential with $k=-1$. The Lax representation for the Hamiltonians with arbitrary linear combination of basic potentials the reader can find in [4].

Now, let us deform the original Hamiltonians (11) in the following way. First, subtract from $h_{2}$ the momentum $p_{1}$. Notice that $\{E_{1},p_{1}\}=0$, i.e. $W_{2}=-p_{1}$ is generated by the Killing vector $Z=(-1,0)^{T}$ of the Euclidean metric in $\mathbb{R}^{2}$. Second, add to both Hamiltonians the lower nontrivial positive separable potentials (14) with coefficients depending on evolution parameters, i.e. $c_{3}(t_{1},t_{2})V^{(3)}+c_{2}(t_{1},t_{2})V^{(2)}$. Actually, consider the following deformed Hamiltonians

[TABLE]

From the demand of the Frobenius condition (5) we immediately find that

[TABLE]

Hence, the related non-autonomous evolution equations are

[TABLE]

and

[TABLE]

The matrices $L(\lambda,t),U_{1}(\lambda,t)$ and $U_{2}(\lambda,t)$ with extra potential $3t_{2}V^{(3)}+(t_{1}+3t_{2}^{2})V^{(2)}$ are as follows [4]

[TABLE]

Now, because of explicit time dependence and the deformation of geodesic Hamiltonian $E_{2}$ by $W_{2}=-p_{1}$ term, we get

[TABLE]

and so, the non-autonomous evolution equations (LABEL:p1) and (17) have the following isomonodromic Lax representation

[TABLE]

or the (10) one after reparametrization of spectral parameter $\lambda\rightarrow\exp(2\lambda)$.

The presented non-autonomous system seems to belong to the $P_{II}$-hierarchy as the extended Hénon-Heiles evolution equations (12) represent the stationary flow of the $5th$-order KdV, but we could not find in the literature neither the system (15) nor its isomonodromy representation in explicit form.

The complete theory of such deformations, with many other examples and the classification of hierarchies, will be presented in subsequent articles.