Appendix B:    The Harry Dym Equation

From the KdV-MKdV equations we have obtained the equation for the singular surface:

$$\phi_t/\phi_x+\{\phi;x\}=\lambda ,\eqno (B1)$$

where

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

is the Schwarzian derivative.

By the transform properties of the Schwarzian derivative,

$$\{\phi;x\}=-\phi_x^2\{x;\phi\} \eqno (B2)$$

Now, under the change of variables;

$$x\rightarrow\phi ,\qquad t\rightarrow t ,\qquad\phi\rightarrow x , \eqno (B3)$$

we note that

$$\phi_x=1/x_{\phi} ,\qquad x_t=\phi_t/\phi_x .\eqno (B4)$$

Therefore, we find

$$x_{\phi}^2x_t=\lambda x_{\phi}^2+\{x;\phi\} ,\eqno (B5)$$

or, by expanding $\{x;\phi\}$,

$$x_t=\lambda-{1\over 2}{\partial^2\over\partial\phi^2}\left({1... ...tial\phi}\left( {1\over x_{\phi}}\right)\right]^2 .\eqno (B6)$$

Letting

$$v=1/x_{\phi} ,\eqno (B7)$$

we find that

$$v_t=v^3v_{\phi\phi\phi} .\eqno (B8)$$

Finally, letting

$$v=cu^{-1/2} ,\qquad -2c^3=1 ,\eqno (B9)$$

we find

$$u_t={\partial^3\over\partial\phi^3} u^{-1/2} ,\eqno (B10)$$

where

$$u=c^2/v^2-c^2x_{\phi}^2 .\eqno (B11)$$

This equation is called the Harry Dym equation and is known to be integrable [14].

Equation (B8) raises some interesting questions concerning the nature of the Painlevé property. In general, about the singular manifold, $\psi=0$,

$$V=\psi^{2/3}\sum_{j=0}^{\infty} V_j\psi^{j/3}\eqno (B12)$$

with resonances at

$$j=-1,2,4 .$$

We note that $\ln\psi$ terms do not arise in the expansion for Eq. (B8).

(B12) demonstrates that Eq. (B8) does not posses the Painlevé property, as it has been defined herein, although the system (B8) is (presumably) integrable. The Painlevé property seems to be a sufficient, but not necessary condition, for integrability and would appear to require reformulation for systems with nontransformable branch point behavior.

Finally, we note that Eqs. (B5) and (B6) are invariant under the Moebius group acting on the independent variable $\phi$. Hence, by restricting $\phi$ to a fundamental region in the complex $\phi$ plane it should be possible to define an $x=x(\phi)$ as an automorphic function [11]

$$x(\phi)=X\left({a\phi+b\over c\phi+d}\right)\eqno (B13)$$

of the Moebius group.