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 several examples from reference [11].

The first example is the Double Sine-Gordon equation

To apply the Painlevé analysis we set

and find

The expansion about the singular manifold takes the form

From the recursion relations

The arbitrary functions areand. At the resonance j=2 and k=0 we can cancel theterms by requiring thatsatisfy the constraint equation

This equation is identical to (21) and is integrable by a Legendre transformation [11]

and has the result that

Let

and (78) becomes

The complete solution of (81) is

whereis homogenous of degree zero inandis homogenous of degree one. This implies, using the definition of the Legendre transformation, that

and

The Legendre transformation is invertible whenWhen, thenand we have a traveling wave form that is integrable for (75). Some simple closed form solutions whenare

for arbitrary f.

In the cases wheresatisfies the constraint equation the expansion (77) becomes single-valued

whereis arbitrary.

Next, we consider the N dimensional elliptic Sine-Gordon equation [11]

where

Using

The Painlevé expansion

is valid with arbitraryiff

where

The matrix D is symmetric and equation (86) is invariant under arbitrary scalings and translations in the independent variables, and orthogonal changes of independent variables

where

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

has the property that principal curvatures of M as a manifold in,verify the condition

When N=2 the condition is trivial and (83) is integrable. When N=3 equation (86) is

Equation (90) may be integrated by a Legendre transformation,andwhere

and

whereandare homogenous of degree zero and one, respectively. Again, the form ofmight be used to find integrable reductions.

Whenit is not known if (86) is integrable. Our conjecture states that it is integrable for all N.