The Schwarzian Derivative

The Schwarzian derivative

$$\{\phi;x\}={\partial\over\partial x}\left({\phi_{xx}\over\phi... ...ght)-{1\over 2}\left({\phi_{xx}\over\phi_x}\right)^2\eqno (8.1)$$

is a third-order, differential expression which is invariant in form under the Moebius group acting on the dependent variable. That is,

$$\left\{{a\phi+b\over c\phi+d} ;x\right\}\equiv\{\phi;x\} . \eqno (8.2)$$

Furthermore,

$$\{\phi;x\}=\{\psi;x\}+\{f;\psi\}\psi_x^2 ,\eqno (8.3)$$

$$\{\phi;x\}=h^{\prime 2}\{\phi;z\}+\{h;x\} ,\eqno (8.4)$$

where

        (i)$\phi=f(\psi)$ is an arbitrary change of dependent variable,

        (ii)$z=h(x)$ is an arbitrary change of independent variable.

The Schwarzian derivative appears in a variety of contexts, including conformal mapping of curvilinear polygons [10], [11], algebraic solutions in differential equations [10], [12], and the theory of automorphic functions [11].

In the preceding sections we have considered the Painlevé property and Bäcklund transformations for a number of integrable partial differential equations. Specifically, it is found that the equation for the ``singular surface" (in the presence of a Bäcklund transform) is most naturally expressed in terms of the Schwarzian derivative. These equations are invariant under the Moebius group. By subjecting them to the well-known [2] "linearizing" transformation

$$\phi=v_1/v_2 ,\eqno (8.5)$$

the Lax pairs for these equations are readily found.

In this section we investigate the Painlevé property associated with equations formulated in terms of the Schwarzian derivative. By this means we seek to identify classes of integrable partial differential equations.

To begin, consider the equation associated with the KdV-MKdV equations, (Secs. 2 and 3).

$$\phi_t/\phi_x+\sigma\{\phi;x\}=\lambda ,\eqno (8.6)$$

where

$$\{\phi;x\}={\partial\over\partial x}\left({\phi_{xx}\over\phi_x} \right)-{1\over 2}{\phi_{xx}^2\over\phi_x^2} .$$

This equation is homogeneous in $\phi$, and consists of two terms; one, $\phi_t/\phi_x$ invariant under arbitrary changes of dependent variable, $\phi=f(\psi)$; the other, a function of the Schwarzian derivative, invariant under the Moebius group,

$$\phi=(a\psi+b)/(c\psi+d) .\eqno (8.7)$$

In consequence, (8.6) is invariant under (8.7). It is to be noted that all the equations [(2.18), (3.8), (4.9), (5.9), (6.6), and (7.6)] that were found have this structure.

Following the procedure defined before, we find for (8.6) that

$$\phi=\psi^{-1}\sum_{j=0}^{\infty}\phi_j\psi^j\eqno (8.8)$$

with resonances at

$$j=-1,0,1 .\eqno (8.9)$$

The compatability conditions at $j=0$ and $j=1$ are satisfied and Eq. (8.6) has the Painlevé property about the "movable" singularity (8.8).

The Bäcklund transform for (8.6) is

$$\phi=\pi_0/\psi+\phi_1 .\eqno (8.10)$$

Direct calculation reveals that the most general form allowed for (8.10) is the transformation (8.7), the Moebius group.

Furthermore, every equation for the singular surface considered in this paper has an expansion of the form (8.8) and possesses the Painlevé property about singularities of this type. We note that the vanishing of $\phi_x$, in (8.1) introduces the possibility of movable essential singularities in these equations [e.g., (8.6)] and ``manifolds of indeterminacy" [3] in the corresponding a Bäcklund transforms [i.e., (2.13)]. Therefore, equations of this type cannot be said to possess the Painlevé property identically.

Nevertheless, it is possible to identify a wide class of equations whose only singularities of finite degree are of the form (8.8) and which possess the Painlevé property about these singularities

. It is fairly easy to show that an equation of the form

$$A\left({\phi_t\over\phi_x}\right)+B\bigl(\{\phi;x\}\bigr)=0 ,\eqno (8.11)$$

where $A(\phi_t/\phi_x)$ and $B\bigl(\{\phi;x\}\bigr)$ are constant coefficient multinomials in
$(\partial^k/\partial t^j\partial x^i)(\phi_t/\phi_x)$ and $(\partial^l/\partial t^m\partial x^n)\{\phi;x\}$, respectively, with $i+j=k$, $m+n=l$, $\max(k)<\max(l)+2$, will have these properties.

Indeed, any equation of the form

$$A(\phi_{x_1},\phi_{x_2},\ldots,\phi_{x_n})+B\bigl(\{\phi;x_1\}, \{\phi;x_2\},\ldots,\{\phi;x_n\}\bigr)=0 ,\eqno (8.12)$$

where (i) $A$ is a constant coefficient multinomial in

$${\partial^N\over\partial_{x_1}^{j_1}\cdots\partial_{x_n}^{j_n... ...{\phi_{x_1}\over\phi_{x_m}}\right) ,\qquad j_1+\cdots+j_n=N ,$$

and (ii) $B$ is a constant coefficient multinomial in

$$\null \vcenter{\openup\jot \ialign{\strut\hfil$\displaysty... ...dots+j_m=m ,\qquad\max(N)<\max(m)+2 ,\cr\crcr}} \eqno (8.13)$$

will have the Painlevé property about the movable poles (or order-- 1). (Note that we require the highest-order derivatives occur in expressions involving the Schwarzian derivative.)

We conclude this section with an example of an equation of the form (8.10) that is related to nonintegrable equations:

$${\phi_t\over\phi_x}+{\partial\over\partial x}\{\phi;x\}=0 .\eqno (8.14)$$

About the ``movable poles" this equation has a solution of the form:

$$\phi=\psi^{-1}\sum_{j=0}^{\infty}\phi_j\psi^j ,\eqno (8.15)$$

with resonances at

$$j=-1,0,1,2 .\eqno (8.16)$$

It can be readily verified that the compatability conditions are satisfied. (The solution has the Painlevé property about the movable poles.) On the other hand, if we attempt to ``integrate" (8.14) by

$$\phi=v_1/v_2 ,\qquad v_{xx}=av ,\eqno (8.17)$$

and

$$v_t=bv_x+cv ,\eqno (8.18)$$

we find

$$v_{xx}=\textstyle{1\over 2}wv ;\qquad v_t=w_xv_x-\textstyle{1\over 2}w_{xx}v , \eqno (8.19)$$

and

$$w_t+w_{xxxx}=2ww_{xx}+w_x^2 .\eqno (8.20)$$

Unfortunately, it does not appear to be possible to introduce a spectral parameter into (8.19) and (8.20) in such a manner that Eq. (8.19) will hot depend on this parameter. This restricts the type of solution that can be represented by (8.17) to be of a special type.

About the singularities of Eq. (8.20) we find that

$$w\simeq w_0/\psi^2+\cdots\eqno (8.21)$$

with resonances at

$$r=-1 ,\qquad\bigl(7\pm i\sqrt{11} \bigr)/2,8 .\eqno (8.22)$$

There exists a "Painlevé type" of expansion

$$w=\sum_{j=0}^{\infty} w_j\psi^{j-1} ,\eqno (8.23)$$

where the compatability condition at $j=8$ is satisfied identically; (8.23) represents a ``special" form of the solution since the arbitrary functions introduced at the resonances $r=\bigl(7\pm\sqrt{11} \bigr)/2$ are not included in (8.23).

Furthermore, by the substitution

$$V=\phi_{xx}/\phi_x ,\eqno (8.24)$$

we find from (8.14) that

$$V_t+{\partial\over\partial x}(V_{xxx}-V_x^2-V^2V_x)=0 .\eqno (8.25)$$

For this equation we find, by a leading order analysis,

$$V\sim V_0\phi^{-1} ,\eqno (8.26)$$

where

$$V_0=3\phi_x, -2\phi_x .$$

Letting

$$V_0=3\phi_x ,\eqno (8.27)$$

we find the resonances

$$r=-1, \bigl(7\pm i\sqrt{11} \bigr)/2, 4 .\eqno (8.28)$$

Again, the expansion is Painlevé at $r=4$ and the equation allows the ``special" Painlevé form of solution

$$V=\sum_{j=j0}^{\infty} V_j\phi^{j-1} .\eqno (8.29)$$

We note that the complex resonances in (8.22) and (8.28) are identical.

We now let

$$V_0=-2\phi_x\eqno (8.30)$$

and find the resonances

$$r=-1,2,4,5 .\eqno (8.31)$$

In this case, we find that the equation has the Painlevé expansion

$$V=\sum_{j=0}^{\infty} V_j\phi^{j-1} ,\eqno (8.32)$$

where $(\phi,V_2,V_4,V_5)$ are arbitrary.

By forming the Bäcklund transform

$$V=V_0/\phi+V_1\eqno (8.33)$$

from (8.32), we find that

$$\null \vcenter{\openup\jot \ialign{\strut\hfil$\displaysty... ...2\phi_x ,\cr V_1&=\phi_{xx}/\phi_x ,\cr\crcr}} \eqno (8.34)$$

$(V,V_1)$ satisfy (8.25) and

$${\phi_t\over\phi_x}+{\partial\over\partial x}\{\phi;x\}=0 .\eqno (8.35)$$

Equation (8.35) is just Eq. (8.14).

If one attempts to form a Bäcklund transform (8.33) for the expansion (8.29), there is found

$$V_0=3\phi_x ,\qquad V_1=-\textstyle{3\over 2}(\phi_{xx}/\phi_x) .\eqno (8.36)$$

$(V,V_1)$ satisfy (8.25) and

$$\phi_t=0 ,\qquad\{\phi;x\}=0 .\eqno (8.37)$$

Although Eq. (8.14) does not lead to ``completely integrable" equations, it is somewhat curious that it does determine equations with somewhat ``special" properties.

We feel that îfurther study of equations of the form (8.10) is warranted.