Conditional or Partial Integrability

For systems without single-valued expansions (Painlevé Property) , it is possible to constrain the arbitrary functions in the expansion so as to restore the single-valued behavior. Depending on the number of constraints the resulting expansion will represent a solution of reduced dimensions.

The systems of constraints are expressed as a system of partial differential equations for the previously arbitrary functions (data) in the expansion. For this system of partial differential equations we make the following conjecture.

Conjecture: The constraint equations are completely integrable.

To illustrate this point we will present an example from reference [11].

We consider the N dimensional Sine-Gordon equation [11]

$$-\bigtriangleup u = \sin u$$ (48)

where

$$\bigtriangleup = \sum\partial_{x_j}^2 = \nabla^t\nabla.$$

Using $V=e^{iu}$

$$-V\bigtriangleup V + \nabla V\cdot\nabla V = \frac12(V^3 - V)$$ (49)

The Painlevé expansion

$$V=\phi^{-2}\sum_{j=0}^\infty V_j \phi^j$$ (50)

is valid with arbitrary $V_2$ iff

$$\nabla\phi\cdot D \nabla\phi = 0$$ (51)

where

$$D_{ii} = \frac12 \sum_{l=1, l \ne i}^N\sum_{m=1, m \ne i}^N (\phi_{lm}^2 - \phi_{ll}\phi_{mm})$$ (52)

$$D_{ij} = \sum_{m=1}^N (\phi_{ij}\phi_{mm} - \phi_{im}\phi_{jm}).$$ (53)

The matrix $D$ is symmetric and equation (51) is invariant under arbitrary scalings and translations in the independent variables, and orthogonal changes of independent variables

$$\nabla = B\nabla^{\prime}$$

where

$$B^t = B^{-1}.$$

Using these properties it can be shown that the hypersurface $M$ defined by the level sets

$$M = \{ \hat x ; \phi(\hat x) = \phi_0\}$$

has the property that principal curvatures of $M$ as a manifold in $R^N$ , $K_j ; j=1,\cdots,N-1$ verify the condition

$$K_1K_2 + K_1K_3 + \cdots + K_{N-2}K_{N-1} = 0.$$ (54)

When $N=2$ the condition is trivial and (48) is integrable. When $N=3$ equation (51) is

$$\phi_t^2(\phi_{xx}\phi_{yy} - \phi_{xy}^2) + \phi_x^2(\phi_{t... ...y} - \phi_{yt}^2) + \phi_y^2(\phi_{tt}\phi_{xx} - \phi_{xt}^2)$$

$$+ 2\phi_x\phi_t(\phi_{ty}\phi_{yx} - \phi_{xt}\phi_{yy})$$

$$+ 2\phi_y\phi_t(\phi_{tx}\phi_{xy} - \phi_{yt}\phi_{xx})$$

$$+ 2\phi_x\phi_y(\phi_{xt}\phi_{yt} - \phi_{xy}\phi_{tt}) = 0 .$$ (55)

Equation (55) may be integrated by a Legendre transformation, $\xi_1=\phi_t,\; \xi_2=\phi_x, \; \xi_3=\phi_y$ and $t=W_{\xi_1},\; x=W_{\xi_2},\; y=W_{\xi_3}$ where

$$\phi(t,x,y)+W(\xi_1,\xi_2,\xi_3) = t\xi_1 + x\xi_2 + y\xi_3$$

and

$$W = W_0 + W_1$$

where $W_0$ and $W_1$ are homogeneous of degree zero and one, respectively. Again, the form of $\phi$ might be used to find integrable reductions.

When $N \geq 4$ it is not known if (51) is integrable. Our conjecture states that it is integrable for all $N$.