Equivalence Problem of the Painlevé Equations

The manuscript is devoted to the equivalence problem of the Painlevé equations. Conditions which are necessary and sufficient for second-order ordinary differential equations   , ,  y F x y y    to be equivalent to the first and second Painlevé equation under a general point transformation are obtained. A procedure to check these conditions is found.


Introduction
Many physical phenomena are described by differential equations.Ordinary differential equations play a significant role in the theory of differential equations.In the 19th century an important problem in analysis was the classification of ordinary differential equations [1][2][3][4].One type of classification problem is an equivalence problem: a system of equations is equivalent to another system of equations if there exists an invertible change of the independent and dependent variables (point transformations) which transforms one system into another.
The six Painlevé equations (PI-PVI) are nonlinear second-order ordinary differential equations which are studied in many fields of Physics.These equations and their solutions, the Painlevé transcendent, play an important role in many areas of mathematics.
The Painlevé equations belongs to the class of equations of the form y a x y y a x y y a x y y a x y This form is conserved with respect to any change of the independent and dependent variables1 In fact, since under the change of variable (3) derivatives are changed by the formulae Here subscript means a derivative, for example, Two quantities play a major role in the study of Equation (5): , .
Under a point transformation (3) these components are transformed as follows [2]: Here tilde means that a value corresponds to system (2): the coefficients b i are exchanged with a i , the variables t and u are exchanged with x and y, respectively.S. Lie showed that any equation with  and 2 0 L  is equivalent to the equation u .For the Painlevé equations 0   0 L 1 and . 2 R. Liouville [2] also found other relative invariants, for example, where For the Painlevé equations 5 and 1  [5].Up to now, the equivalence problem has been solved in a form more convenient for testing only for (PI) and (PII) equations, by using an explicit point change of variables was given in [6].
The manuscript is devoted to solving the problem of describing all second-order differential equations Necessary and sufficient conditions for an equation y F x y y   to be equivalent to (PI) and (PII) are obtained.As was noted, some of the necessary conditions are [5]: Other conditions are also expressed in terms of relations for the coefficients of Equation (5).
The method of the study is similar to [7][8][9].It uses analysis of compatibility of an over determined system of partial differential equations.

Equations Equivalent to the Painlevé Equations
This section studies Equation ( 5) which are equivalent to the first and second Painlevé equation (PI) and (PII).Since any equation of ( 1) belongs to the type of equation ( 2), the necessary condition for an equation y F x y y  0  to be equivalent to the first and second Painlevé equation (PI) and (PII) are that it has to be of the same type.Since 5 v  and 1 are relative invariants with respect to (3), they are also necessary condition.0 w 

The First Painlevé Equation (PI)
For obtaining sufficient conditions one has to find conditions for the coefficients which guarantee existence of the functions     , , ,  x y x y   transforming the coefficient of Equation ( 6) into the coefficients of equations (PI).
Also note that the the first Painlevé equation has the coefficients are a x y a x y a x y a x y y x Without loss of generality it is assumed that 1  .Since for Equation ( 8), the value 2 , and hence, the functions      Substituting these coefficients into (6), one obtains over determined system of partial differential equations.
From Equations (10)-( 12) one can find the derivatives Finding the derivatives: L 2u from the equation v 5 = 0, L 2tt from the equation w 1 = 0, and Since of ( 14), ( 15) and (18) all second order derivatives of the function x y  can be found, then one can compose the equations   , are reduced to the only equation Differentiating Equation ( 23) with respect to x, one obtains2

. K L
Because of (25), the function .Differentiating (25) with respect to x and y, one gets where Differentiating Equation ( 27) with respect to x and y one obtains the only equation Finding the function ∆ 1 from ( 27), and substituting it into (23), ( 16), (13) one gets Thus, the necessary and sufficient conditions for equation to be equivalent to the first Painlevé equation are: the equation has to be of the form (5) with the coefficients 21), ( 24), ( 28) and (32), where the functions  , Q t u are defined by Equations ( 22), ( 26), (29).The transformation is defined by ( 25) and (31).

The Second
Substituting these coefficients into (6), one obtains ov r determined system of partial differential equations.
  where From Equations ( 34)-(36) one can find the derivatives is defined by the formula where the function All these conditions are satisfied except the first one, which becomes Differentiating (47) with respect to y and excluding x by using (47), one obtains Differentiating (49) with respect to x and y, one gets, respectively, Remaining equations are obtained by differentiating (51) with respect to x and y.Excluding from them x and y these equations are reduced to the equation Thus, the necessary and sufficient conditions for an equation   , , y F x y y    which can be transformed to the second Painlevé equations are: this equation has to be of the form (5), where the coefficients satisfy the equations v 5 = 0, w 1 = 0, (45), ( 52

Conclusion
The necessary and sufficient conditions that an equation of the form

 
, ,  y F x y y     to be equivalent to the first and second Painlevé equation under a general point transformation are obtained.As was noted some of the necessary conditions are v 5 = 0 and w 1 = 0. Other conditions are also expressed in terms of relations for the coefficients of Equation (2).A procedure to check these conditions is found.Since intermediate calculations in the equivalence problem are cumbersome, computer algebra system have become an important computational tool.
with respect to point transformations (3) to the first and second Painlevé equation (PI) and (PII).Example of the first Painlevé equation (PI) is presented.
Painlevé Equation (PII) Similar to the first Painlevé equation one can study the second Painlevé equation.Painlevé equation (PII) has the coefficients are

y
Differentiating this equation with respect to x and y, and substituting Ψ y found from Equation (41), one gets

3 .
)-(54), where the functions   , K t u and Q t are defined by Equations (44) and (50).The transformation of the Equation (5) into the second Painlevé equation (PII) is defined by Equations (49) and (51).Example of the Results Example.The following equation is equivalent to the first Painlevé equation (PI)This equation has to be of the form(5) with the coefficients