The Singular Manifold Method

For systems with the Painlevé Property, Bäcklund transformations appear as truncations of expansions of a solution about its singular manifold. With reference to the Lax pair for a system, these Bäcklund transformations are equivalent to transformations of linear systems developed by Laplace, Moutard and Darboux. The Painlevé analysis leads naturally to a reformulation of these systems in terms of the Schwarzian derivative. Using the conformal invariance of the Schwarzian derivative and certain discrete symmetries, a canonical form of Bäcklund transformations can be defined [1,2].

The Sine-Gordon equation

$$u_{xt}= \sin(u)$$ (1)

has the Painlevé property in the variable $V=e^{iu}$ where

$$VV_{xt} - V_xV_t= \frac12(V^3 - V) .$$ (2)

The expansion

$$V=\phi^{-2}\sum_{j=0}^{\infty}V_j\phi^j$$

with resonances $j= -1,2$ and $V_0 = 4\phi_x\phi_t$ and $V_1=\phi_{xt}$. The Painlevé transformation is

$$V=-4\frac{\partial^2}{\partial xt}\ln\phi + V_2$$ (3)

where $V_2=\phi_{xt}^2/(\phi_x\phi_t)$.

The Schwarzian modified equations are

$$\Omega_1 = \{\phi;t\} + 2Z_{tt}/Z = \alpha$$ (4)

$$\Omega_2 = \{\phi;x\} + 2W_{xx}/W = \beta$$ (5)

where $Z^2= \phi_x/\phi_t$, $W^2 = \phi_t/\phi_x$, and $\alpha\beta=\frac14$. The identity $\phi_x(\partial/\partial x) \Omega_1 + \phi_t(\partial/\partial t)\Omega_2 = 0$ demonstrates the equivalence of the above two equations. To find the discrete symmetries, let $\Theta= -\phi_{xt}/\phi_t$ and $\Phi= -\phi_{xt}/\phi_x$ and get the system

$$\Theta_t + \frac12\Theta\Phi + \frac{\lambda}{2}\Theta/\Phi = 0$$ (6)

$$\Phi_x + \frac12\Theta\Phi + \frac1{2\lambda}\Phi/\Theta = 0.$$ (7)

The discrete symmetries of these equations imply the strong Bäcklund transformations for the Schwarzian equations.

$$\frac{\phi_{xt}}{\phi_t}\frac{\psi_{xt}}{\psi_t}=\frac1{\lambda}, \;\;\; \phi_x\psi_x=1$$ (8)

$$\frac{\phi_{xt}}{\phi_x}\frac{\psi_{xt}}{\psi_x}=\lambda, \;\;\; \phi_t\psi_t=1$$ (9)

$$\frac{\phi_{xt}}{\phi_t}\frac{\psi_{xt}}{\psi_t}=-\frac1{\lam... ...\;\; \frac{\phi_{xt}}{\phi_x}\frac{\psi_{xt}}{\psi_x}=-\lambda$$ (10)

For instance, $\psi_t=-\phi_t^{-1}$ and $\psi_x = -(1/\lambda)\phi_{xt}^ 2/(\phi_t^2\phi_x)$ imply by the condition $\psi_{xt}=\psi_{tx}$ that $\phi$ satisfies the Schwarzian equation, (57).

The Moebius group is a point symmetry and composition with the BT above generates the solutions

$$\phi_0=e^{\sigma t + x/\sigma}$$

$$\phi_1=\tanh(\sigma t/2 + x/(2\sigma))$$

$$\phi_2=\sinh(\sigma t + x/\sigma) + \sigma t -x/\sigma.$$