EXAMPLES AND COUNTER-EXAMPLES

To illustrate the nature of the Painlevé Property it is worthwhile to examine a few examples of equations with and without the Painlevé Property.

A simple case of an equation with the Painlevé Property is Burgers' Equation.

$$u_t + uu_x = u_{xx}$$ (3)

It is not difficult to find the Psi series

$$u = \phi^{-1}\sum_{j=0}^\infty u_j\phi^j$$ (4)

is valid for (3). Examination of the recursion relations for the $u_j$ obtains a system of the form

$$(j-2)(j+1)\phi_x^2 u_j = F_j(u_{j-1},\cdots,u_0,\phi_t,\phi_x,\cdots).$$

For (4) to be valid $F_2$ must vanish identically. Evaluation of the recursions obtains

$$j = 0,\; u_0 = -2\phi_x$$

$$j = 1,\; \phi_t + u_1\phi_x = \phi_{xx}$$

$$j = 2,\; \partial_x(\phi_t + u_1\phi_x - \phi_{xx}) = 0.$$

The relation (compatibility condition) at $j=2$ is satisfied identically and the expansion (4) is valid with arbitrary functions $\phi$ and $u_2$.

The Korteweg-de Vries equation

$$u_t + u_{xxx} + 3uu_x = 0$$ (5)

has singularities of the form

$$u = \phi^{-2}\sum_{j=0}^\infty u_j \phi^j$$ (6)

with arbitrary functions $\phi$, $u_4$ and $u_6$. The KdV equation has the Painlevé Property about singularities of the form (6).

The Schwarzian KdV equation [8]

$$\frac{\psi_t}{\psi_x} + \{ \psi;x\} = \lambda$$ (7)

,where

$$\{\psi;x\} = \frac{\psi_{xxx}}{\psi_x} - \frac32(\frac{\psi_{xx}}{ \psi_x})^2$$

is the Schwarzian derivative , has singularities of the form

$$\psi = \phi^{-1}\sum_{j=0}^\infty \psi_j\phi^j$$ (8)

with arbitrary $\phi, u_0, u_1$ if the non-characteristic condition $\psi_x \simeq \phi_x \ne 0$ is verified. If $\psi_x = 0$ the expansion about the characteristic manifold has the form

$$\psi = f(t) + \phi^3\sum_{j=0}^\infty\psi_j\phi^j$$ (9)

,where $\phi = x - x_0(t)$ and $\psi_j =\psi_j(t)$.

The Schwarzian KdV equation has single valued expansions about both characteristic and noncharacteristic manifolds. The Painlevé Property requires all movable singularity manifolds to be single valued, whether characteristic or not. The above result runs counter to the observation of Ward [12,13] that direct consideration of expansions about characteristic manifolds cannot be allowed in the definition of the Painlevé Property since, for linear systems, arbitrarily bad singularities propagate along characteristics. For general systems, expansions about characteristics, when they exist, introduce certain arbitrary data [14]. If the data is bad, the expansion is still required to be a single valued functional of that data. In this sense, expansion (9) is a single valued functional of the data $f(t),\; x_0(t)$, however multiple valued that data as a function of $t$ may be. Of course, the same observation applies to the non-characteristic expansion (8). The Painlevé Property is a statement of how the solutions behave as functionals of the data in a neighborhood of a singularity manifold and not a statement about the data itself. The following example will illustrate this point.

A derivative Schwarzian equation

$$\frac{\psi_t}{\psi_x} + \frac{\partial}{\partial x}\{\psi;x\} = 0$$ (10)

has non-characteristic singularities of the form

$$\psi = \phi^{-1}\sum_{j=0}^\infty \psi_j\phi^j$$ (11)

,where $\phi,\psi_0,\psi_1,\psi_2$ are arbitrary. Therefore, (10) has a single valued expansion depending on the maximum number of arbitrary functions allowed for by the order of the equation. However, about the characteristic manifold where $\psi_x = 0$

$$\psi=f(t)+\phi^4\sum_{j=0}^\infty\sum_{k=0}^\infty\psi_{jk} \phi^j\phi^{k\alpha}$$ (12)

,where $\alpha = \frac72 + \imath \sqrt{11}/2$ and $\phi = x - x_0(t)$. The expansions (12) are highly multiple valued as functionals of $\phi$.

In general, to verify that an equation has the Painlevé Property it is necessary to show that all the allowed singularities are single valued (as functionals of the data). This requirement is often overlooked and has lead to some wrong conclusions.

It is thought that the Clarkson equation

$$u_t^2 = 2uu_x^2 - (1+u^2)u_{xx}$$ (13)

has only meromorphic psi-series and has Painlevé Property [16,17]. However, this is not the case. Consider the points where

$$u^2 + 1 = 0.$$ (14)

Rewriting the above we have

$$u_t^2 - 2iu_x^2 = (u-i)(2u_x^2 - (u+i)u_{xx})$$

and letting $G = u + i$

$$G_t^2 - 2iG_x^2 = (G-2i)(2G_x^2 - GG_{xx}).$$

To leading order

$$G = 2i + G_0\phi^{\alpha} + \cdots$$

where $\Re \alpha \geq 0$.

By substitution in the above

$$\alpha^2G_0^2(\phi_t^2-2i\phi_x^2)\phi^{2\alpha-2} =$$

$$\alpha(\alpha+1)G_0^3\phi_x^2\phi^{3\alpha-2} - 2i\alpha(\alpha-1)G_0^2\phi_x^2\phi^{2\alpha-2}.$$

Since $\Re \alpha \geq 0$ we have

$$\alpha = 2i\phi_x^2/\phi_t^2.$$

For this leading order the balance equations are

$$G_t^2 + 2i(GG_{xx} - G_x^2) = 0$$

and the resonances

$$G = 2i + G_0\phi^\alpha + G_1\phi^{\alpha+r}$$

obtains

$$\alpha(\alpha+r)\phi_t^2 + i\{\alpha(\alpha-1) + (\alpha + r)(r - \alpha - 1)\}\phi_x^2 = 0.$$

Using the leading order

$$r = -1, 0 .$$

Thus $G_0$ is arbitrary. From the above, the Clarkson equation has a movable essential singularity and does not have the Painlevé Property.

An example of an equation with a non-constant resonance is the Rand equation [18]

$$u^2u_{xxx} = 3u_t^3$$ (15)

It has the leading order $u = u_0\phi^{\alpha} + \cdots$ where it can be shown that

$$(\alpha - 1)(\alpha - 2) = 3(\phi_t^3/\phi_x^3)\alpha^2.$$

This quadratic equation for $\alpha$ determines the leading order. Of course $\alpha$ is a non-constant functional of $\phi_t$ and $\phi_x$.

The resonance condition

$$u = u_0\phi^{\alpha} + u_1\phi^{\alpha + r} + \cdots$$

easily determines the resonances

$$r = -1, 0, 4 - 3\alpha .$$

It is the case here that one resonance, $4 - 3\alpha$, is a functional of the singular manifold, $\phi$.

In the preceding paragraphs our intent is to illustrate both the definition of the Painlevé Property and the variety of singularities revealed by the functional Psi series. This approach is capable of substantial generalization. In this paper we will, for the most part, describe the applications to integrable systems.