A Geometric Method to Investigate Prolongation Structures for Differential Systems With Applications to Integrable Systems

A type of prolongation structure for several general systems is discussed. They are based on a set of one-forms in which the underlying structure group of the integrability condition corresponds to the Lie-algebra of SL (2,R), O(3), or SU(3). Each will be considered in turn and the latter two systems represent larger 3by3 cases. This geometric approach is applied to all three of these systems to obatin prolongation structures explicitly. In both 3by3 cases the prolongation structure is reduced to the situation of three smaller 2by2 problems. Many types of conservation laws can be obtained at different stages of the development, and at the end, a single result is developed to show how this can be done.

an n × n problem is reduced to a collection of n smaller (n − 1) × (n − 1) cases, and finally in the end to a set of 2 × 2 problems such as the SL(2, R) case at the beginning. Finally, to summarize at the end, a collection of conservation laws is developed from one of the results and some speculation as to how how this type relates to other procedures [9].
The procedure begins by associating a system of Pfaffian equations to the nonlinear system to be studied, namely, In (1), Ω is a traceless 2 × 2 matrix which consists of a family of one-forms, ω i . In the end, it is desired to express these one-forms in terms of independent variables, which are called x and t, the dependent variables and their derivatives. However, no specific equation need be assumed at the start. The underlying structure group is SL(2, R). Explicitly, the matrix of one-forms is given by The integrability conditions are expressed as the vanishing of a traceless 2 × 2 matrix of two-forms Θ given by In terms of components, the left-hand side of (3) has the form Substituting (2) into (3), a matrix which contains the integrability equations in terms of the basis one-forms is obtained. The three equations are Of use in the theorems which follow, it is also useful to have the dω i in (5) expressed explicitly in terms of the ϑ i as Therefore, by selecting a particular choice for the set {ω i }, the nonlinear equation of interest can be written simply as It is easy to see that the system is closed. Upon differentiating Θ and substituing (3), we obtain This implies that the exterior derivatives of the set of {ϑ i } are contained in the ring of two-forms Now the differential ideal can be prolonged by including the forms {α 1 , α 2 } given by (1).
Theorem 2.1. The differential one-forms have the following exterior derivatives, Hence, the exterior derivatives of the {α i } are contained in the ring of forms {ϑ i , α j }.
Corollary 2.1. The exterior derivatives of the α i in (9) can be expressed concisely in terms of the matrix elements of Θ and Ω as The Corollary is Theorem 2.1 after using the definitions of the matrices in (2) and (4).
The forms α 1 , α 2 given in (9) lead to a natural Riccati representation called α 3 , α 4 for this differential system. This arises by taking the following linear combinations of α 1 and α 2 , Two new functions, or pseudopotentials, y 3 and y 4 can be introduced which are defined by the transformation y 3 = y 2 y 1 , Then, from (12) and (13), the following Riccati forms result, These equations could also be thought of as Riccati equations for y 3 and y 4 .
The proof of this Theorem proceeds along exactly the same lines as Theorem 2.1. The SL(2, R) structure and the connection interpretation can be based on the forms α 3 , α 4 .
At this point, the process can be continued. Two more pseudopotentials y 5 and y 6 can be introduced and the differential system can be extended by including two more one-forms α 5 and α 6 . The specific structure of these forms is suggested by Theorem 2.2.
Theorem 2.3. Define two one-forms α 5 and α 6 as The exterior derivatives of the one-forms (17) are given by the expressions Results (18) specify the closure properties of the forms α 5 and α 6 . This theorem is proved along similar lines keeping in mind the results for dy 3 and dy 4 are obtained from (15).
A final extension can be made by adding two additional pseudopotentials y 7 and y 8 . At each stage, the ideal of forms is being enlarged and is found to be closed over the underlying subideal which does not contain the two new forms.
Theorem 2.4. Define the one-forms α 7 and α 8 by means of The exterior derivatives of the forms α 7 and α 8 are given by and are closed over the prolonged ideal.

O(3) Prolongations.
The first 3×3 problem we consider is formulated in terms of a set of one-forms ω i for i = 1, 2, 3.
The following system of Pfaffian equations is to be associated to a specific nonlinear equation In (21), Ω is a traceless 3 × 3 matrix of one-forms. The nonlinear equation to emerge is expressed as the vanishing of a traceless 3 × 3 matrix of two-forms Θ, These can be considered as integrability conditions for (21). The closure property, the gauge transformation and the gauge theoretic interpretation can also hold for 3 × 3 systems. Explicitly the matrix Ω is given by Using (23) in (21) and (22), the following prolonged differential system is obtained, By straightforward exterior differentiation of each α i and using (24), the following Theorem results.
The exterior derivatives are therefore contained in the ring spanned by {ϑ i } 3 1 and the {α j } 9 4 . In terms of Ω i , a corresponding matrix of two-forms Θ i can be defined in terms of the corresponding Ω i defined in (28), Proof: Differentiating Ω 1 and simplifying, we have Similar matrix expressions can be found for the cases in which Ω 1 is replaced by Ω 2 or Ω 3 , respectively.
Unlike Ω given in (23), the Ω i in (28) are not traceless. Define their traces to be These traces provide convenient ways of generating conservation laws on account of the following Theorem.

SU(3) Prolongations.
To formulate a 3 × 3 SU(3) problem, the set of Pfaffian equations The structure constants f lmn are totally anti-symmetric in l, m, n. The one-form Ω is expressed in terms of the λ i as Then the two-form Θ is written in the form The nonlinear equation to be solved has the form ϑ l = 0 for l = 1, · · · , 8.
It will be useful to display (35) and (36) by using the nonzero structure constants given in (34), since these may not be readily accessible. An explicit representation for the matrix Ω used here is given by Moreover, in order that the presentation be easier to follow, the eight forms dω i specified by (36) will be given explicitly, Substituting (37) into (33), the following system of one-forms {α i } 3 1 is obtained, The increase in complexity of this system makes it often necessay to resort to the use of symbolic manipulation to carry out longer calculations, and for the most part, only results are given..
Define the following set of 2 × 2 matrices of one-forms Ω i for i = 1, 2, 3 as Making use of the matrices Ω i defined in (42), the following result can be stated.
Based on the forms Ω i given in (42), we can differentiate to define the following matrices of two-forms ξ i given by An analogue of Theorem 3.3 can now be formulated.
The two-forms {ξ i } 3 1 therefore vanish for the solutions of the nonlinear equations (34) and of the coupled Riccati equations {α j = 0} 9 4 . As in the case of O(3), the one-forms Ω i are not traceless. Denoting the trace of Ω i by the following theorem then follows.
Theorem 4.4. The closure properties of the exterior derivatives of the traces τ i are given by, The one-forms (46) then can be used to generate conservation laws. Note that as in the O (3) case, a 3 × 3 problem Ω has been reduced to three separate 2 × 2 problems in terms of the matrices Ω l . Further prolongation can be continued beginning with the one-forms Ω l given in (42). To briefly outline the further steps in the procedure, a system of Pfaffians {α lj } are introduced which are based on the one-forms Ω lα The subscript l in y lj andα lj indicates that they belong to the sub-system defined by Ω l , and it is not summed over. Exterior differentiation ofα lj yields The {y lj } are pseudopotentials for the original nonlinear equation (34). Next, quadratic pseudopotentials are included as the homogeneous variables and the prolongation can be continued.

A Conservation Law and Summary.
These types of results turn out to be very useful for further study of integrable equations.
They can be used for generating infinite numbers of conservation laws. In addition, the results are independent of any further structure of the forms ω i in all cases. In fact, the one-form Ω need not be thought of as unique. This is due to the fact that Ω and Θ are form invariant under the gauge is an arbitrary 2 × 2 space-time dependent matrix of determinant one. The gauge transformation property holds in the 3 × 3 case as well. The pseudopotentials serve as potentials for conservation laws in a generalized sense. They can be defined under a choice of Pfaffian forms such as with the property that the exterior derivatives dα i are contained in the ring spanned by {α i } and This can be thought of as a generalization of the Frobenius Theorem for complete integrability of Pfaffian systems.
Theorem 2.2 implies that dσ i = 0 when i = 1, 2 for solutions of the original equation. If either of these forms is expressed as σ = I dx + J dt, then dσ = 0 imlies that Thus I is a conserved density and J a conserved current. Let us finally represent ω 1 = a 1 dx+b 1 dt, ω 2 = a 2 dx + b 2 dt and ω 3 = η dx + b 3 dt, where η is a parameter. The x-dependent piece of α 3 in (15) implies y 3,x + 2a 1 y 3 + a 2 y 2 3 − η = 0.
By substituting an asymptotic expansion around η = ∞ for y 3 in (53) of the form, a recursion for the Y n is obtained. Thus (53) becomes, n η −n Y n,x + 2a 1 n η −n Y n + a 2 ( n η −n Y n ) 2 − η = 0.
Expanding the power, collecting coefficients of η and equating coefficients of η to zero, we get, The consistency of solving α 3 by using just the x-part is guaranteed by complete integrability.
The x-part of the form σ 1 is expressed as Consequently, from (52) the n-th conserved density is The existence of links between the types of prolongation here and other types of prolongations which are based on closed differential systems is a subject for further work.