Introduction

In [1] the Painlevé property for partial differential equations was defined. Briefly, we say that a partial differential equation has the Painlevé property when the solutions of the pde are ``single valued" about the movable singularity manifold. To be precise, if the singularity manifold is determined by

$$\varphi(z_1,z_2,\ldots,z_n)=0 ,\eqno (1.1)$$

and $u=u(z_1,\ldots,z_n)$ is a solution of the pde, then we require that

$$u=\varphi^{\alpha}\sum_{j=0}^{\infty} u_j\varphi^u ,\eqno (1.2)$$

where $u_0\ne 0$, $\varphi=\varphi(z_1,\ldots,z_n)$, $u_j(z_1,\ldots, z_n)$ are analytic functions of ($z_j$) in a neighborhood of the manifold (1.1) and $\alpha$ is a negative, rational number. Substitution of (l.2) into the pde determines the allowed values of $\alpha$, and defines the recursion relations for $u_j$, $j=0,1,2, \ldots$ . When the Anzatz (1.2) is correct the pde is said to possess the Painlevé property and is conjectured to be integrable.

In [2] Bäcklund transformations were obtained by truncating the expansion (1.2) at the ``constant" level term. That is, we set

$$u=u_0\varphi^{^{-N}}+u_1\varphi^{^{-N+1}}+\cdots+u_{_N} ,\eqno (1.3)$$

and find, from the recursion relations for $u_j$, an overdetermined system of equations for $(\varphi,u_j,j=0,1,\ldots,N)$, where $u_{_N}$ will satisfy the (original) pde. Upon solving the overdetermined system it was found, for those equations considered, that $\varphi$ satisfied an equation formulated in terms of the Schwarzian derivative:

$$\{\varphi;x\}={\partial\over\partial x}\left({\varphi_{xx}\ov... ...r 2} \left({\varphi_{xx}\over\varphi_x}\right)^2 .\eqno (1.4)$$

The invariance of (1.4) under the Moebius group,

$$\varphi=(a\psi+b)/c\psi+d) ,\qquad\{\varphi;x\}=\{\psi;x\} ,\eqno (1.5)$$

motivates the substitution

$$\varphi=V_1/V_2 ,\eqno (1.6)$$

where $V_1$ and $V_2$ satisfy the same linear equation. From the resulting Wronskian relations the Lax pair may be found.

In [3] it is shown how study of the Caudrey-Dodd-Gibbon equation leads to the formulation of a class of equations, in terms of the Schwarzian derivative, that identically possess the Painlevé property. This class of equations contains the higher-order KdV, Caudrey-Dodd-Gibbon, and Kuperschmidt equations.

In this paper various equations of sine-Gordon type are considered. These equations are somewhat different from those studied previously in that they have a symmetric dependence on the independent variables (under Lorenz transformation). Only the ($1+1$) sine-Gordon one space-one time variable) equation is found to identically possess the Painlevé property. The method of ``singular manifold" analysis, i.e., Bäcklund transform and formulation in terms of the Schwarzian derivative, obtains, for this equation, the Lax pair. In addition, a connection to the sequence of higher-order KdV equations is found. That is, the ($1+1$) sine-Gordon equation is formulated in terms of ``minus one" functional of the Lenard recursion relations, where positive functionals determine the sequence of higher-order KdV equations. For the sine-Gordon equation we define a system of ``modified" equations that identically possess the Painlevé property. These ``modified" equations are related to the ``characteristic" initial value problem. Furthermore, we find, using the discrete symmetries of the modified equations, certain rational solutions of the sine-Gordon equation.

The double sine-Gordon and ($N+1$) sine-Gordon equations are found not to possess the Painlevé property. This would seem to answer various questions concerning the integrability of these equations [4,5,6,7,8,9,10]. However, if the ``singular manifold", $\varphi$, in the Ansatz (1.2) is restricted (to satisfy a subsidiary constraint) a type of ``partial" integrability can be defined for these equations. The known, exact solutions appear to satisfy the appropriate constraint in a more or less trivial manner. We conjecture that the class of exact solutions (for these equations) is more general. Hopefully, study of the ``constrained" dynamics will lead to their discovery.

In a recent paper, Oevel [11] states that the coupled KdV, or Hirota-Satsuma, equations ``do not seem to be 'completely integrable' in the usual sense". Analysis reveals that these equations identically possess the Painlevé property. Thus, if these equations are ``partially integrable" it is in a different sense from that defined above. The Painlevé (``singular manifold") analysis is presented in the Appendices.

We note that ``partial" integrability (of various types) for ordinary differential equations has been considered by several authors, i.e., Segur [12] and Tabor and Weiss [13].