Geometric Methods 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 for which the underlying structure group of the integrability condition corresponds to the Lie algebra of , , and . Each will be considered in turn and the latter two systems represent larger cases. This geometric approach is applied to all of the three of these systems to obtain prolongation structures explicitly. In both cases, the prolongation structure is reduced to the situation of three smaller problems.
Geometric approaches have been found useful in producing a great variety of results for nonlinear partial differential equations . A specific geometric approach discussed here has been found to produce a very elegant, coherent, and unified understanding of many ideas in nonlinear physics by means of fundamental differential geometric concepts. In fact, relationships between a geometric interpretation of soliton equations, prolongation structure, Lax pairs, and conservation laws can be clearly realized and made use of. The interest in the approach, its generality, and the results it produces do not depend on a specific equation at the outset. The formalism in terms of differential forms  can encompass large classes of nonlinear partial differential equation, certainly the AKNS systems [3, 4], and it allows the production of generic expressions for infinite numbers of conservation laws. Moreover, it leads to the consequence that many seemingly different equations turn out to be related by a gauge transformation.
Here the discussion begins by studying prolongation structures for a system discussed first by Sasaki [5, 6] and Crampin  to present and illustrate the method. This will also demonstrate the procedure and the kind of prolongation results that emerge. It also provides a basis from which to work out larger systems since they can generally be reduced to problems. Of greater complexity are a pair of problems which will be considered next. It is shown how to construct an system based on three constituent one forms as well as an system composed of eight fundamental one forms. The former has not appeared. The problem is less well known than the problem; however, some work has appeared in [8, 9]. In the first of these, the solitons degenerate to the AKNS solitons in three ways, while in the latter nondegenerate case, they do not. All of the results are presented explicitly; that is, the coefficients of all the forms and their higher exterior derivatives are calculated and given explicitly. Maple is used to do this in the harder cases. The approach is quite unified, and so once the formalism is established for the system, the overall procedure can be carried over to the other Lie algebra cases as well.
Of course, the problems will yield more types of conservation laws. The prolongation structures of the problems will be observed to be reduceable to the same type as the system considered at the beginning. However, the Riccati representations become much more complicated . The generalization of this formalism to an problem then becomes straightforward after completing the study of these smaller cases . The prolongation structure of an problem is reduced to a collection of smaller cases and finally at the end to a set of problems such as the case at the beginning. Finally, to summarize at the end, a collection of conservation laws are developed from one of the results and some speculation as to how this type relates to other procedures [12, 13].
The procedure begins by associating a system of Pfaffian equations to the nonlinear system to be studied, namely, In (1), is a traceless matrix which consists of a family of one forms, . At the end, it is desired to express these one forms in terms of independent variables, which are called and , the dependent variables and their derivatives. However, no specific equation need be assumed at the start. The underlying structure group is . Explicitly, the matrix of one forms is given by the matrix The integrability conditions are expressed as the vanishing of a traceless matrix of two forms 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 in (5) expressed explicitly in terms of the as Therefore, by selecting a particular choice for the set , the nonlinear equation of interest can be written simply as It is easy to see that the system is closed. Upon differentiating and substituting (3), we obtain This implies that the exterior derivatives of the set of are contained in the ring of two forms .
Now the differential ideal can be prolonged by including the forms given by (1).
Theorem 1. The differential one forms have the following exterior derivatives: Hence, the exterior derivatives of the are contained in the ring of forms .
Corollary 2. The exterior derivatives of the in (9) can be expressed concisely in terms of the matrix elements of and as
The forms given in (9) lead to a natural Riccati representation called , for this differential system. This arises by taking the following linear combinations of and : Two new functions, or pseudopotentials, and can be introduced which are defined by the transformation Then, from (15), the following forms which have a Riccati structure result: These equations could also be thought of as Riccati equations for and .
Theorem 3. (i) The one forms , given by (17) have exterior derivatives
Consequently, results (18) are contained in the ring spanned by and .
(ii) Define the forms and appearing in (18). The exterior derivatives of and are found to be Therefore and .
The proof of this Theorem proceeds along exactly the same lines as Theorem 1. The structure and the connection interpretation can be based on the forms , .
At this point, the process can be continued. Two more pseudopotentials and can be introduced and the differential system can be extended by including two more one forms and . The specific structure of these forms is suggested by Theorem 3.
Theorem 4. Define two one forms and as The exterior derivatives of the one forms (20) are given by the expressions Results (21) specify the closure properties of the forms and . This theorem is proved along similar lines keeping in mind that the results for and are obtained from (17).
A final extension can be made by adding two additional pseudopotentials and . 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 5. Define the one forms and by means of The exterior derivatives of the forms and are given by and are closed over the prolonged ideal.
The first problem we consider is formulated in terms of a set of one forms for . The following system of Pfaffian equations is to be associated to a specific nonlinear equation In (24), is a traceless matrix of one forms. The nonlinear equation to emerge is expressed as the vanishing of a traceless matrix of two forms , These can be considered as integrability conditions for (24). The closure property, the gauge transformation, and the gauge theoretic interpretation can also hold for systems. Explicitly the matrix is given by Using (26) in (24) and (25), the following prolonged differential system is obtained: By straightforward exterior differentiation of each and using (27), the following Theorem results.
Theorem 6. The given in (27) have exterior derivatives which are contained in the ring spanned by and and given explicitly as
The one forms admit a series of Riccati representations which can be realized by defining six new one forms, In effect, the larger system is breaking up into several systems. To write the new one forms explicitly, introduce the new functions which are defined in terms of the original as follows: In terms of the functions defined in (30), the are given as
To state the next theorem, we need to define the following three matrices:
Theorem 7. The closure properties for the forms given in (31) can be summarized in the form The exterior derivatives are therefore contained in the ring spanned by and the .
In terms of , a corresponding matrix of two forms can be defined in terms of the corresponding defined in (32),
Theorem 8. The matrix of two forms defined by (34) is contained in the ring of forms spanned by coupled with either , , or when , respectively.
Proof. Differentiating and simplifying, we haveSimilar matrix expressions can be found for the cases in which is replaced by or , respectively.
To formulate a 3 3 problem, the set of Pfaffian equations, are associated with the nonlinear equation. In (38), is a traceless matrix consisting of a system of one forms. The nonlinear equation to be considered is expressed as the vanishing of a traceless matrix of two forms exactly as in (25) which constitute the integrability condition for (38).
A matrix representation of the Lie algebra for is introduced by means of generators for , which satisfy the following set of commutation relations: The structure constants are totally antisymmetric in . The one form is expressed in terms of the generators as Then the two forms can be written as The nonlinear equation to be solved has the form for .
It will be useful to display (40) and (41) by using the nonzero structure constants given in (39), since these may not be readily accessible. An explicit representation for the matrix used here is given by Moreover, in order that the presentation can be easier to follow, the eight forms specified by (41) will be given explicitly: Substituting (42) into (38), the following system of one forms is obtained, The increase in complexity of this system makes it often necessary to resort to the use of symbolic manipulation to carry out longer calculations, and for the most part, only results are given.
Theorem 10. The exterior derivatives of the in (44) are given by
In order to keep the notation concise, an abbreviation will be introduced. Suppose that is a system of forms, and are integers, then we make the abbreviation At this point, quadratic pseudopotentials can be introduced in terms of the homogeneous variables of the same form as (30). In the same way that (31) was produced, the following Pfaffian equations based on the set in (44) are obtained: These are coupled Riccati equations for the pairs of pseudopotentials , , and , respectively.
Define the following set of matrices of one forms for as Making use of the matrices defined in (48), the following result can be stated.
Theorem 11. The closure properties of the set of forms given by (47) can be expressed in terms of the as follows:
Based on the forms given in (48), we can differentiate to define the following matrices of two forms given by
Theorem 12. The forms defined in (50) are given explicitly by the matrices
From these results, it is concluded that is contained in the ring of and the , is contained in the ring of and , and is in the ring of and .
The two forms therefore vanish for the solutions of the nonlinear equations (39) and of the coupled Riccati equations . As in the case of , the one forms are not traceless. Denoting the trace of by the following theorem then follows.
Theorem 13. The closure properties of the exterior derivatives of the traces are given by
The one forms (52) then can be used to generate conservation laws. Note that as in the case, a problem has been reduced to three separate problems in terms of the matrices . Further prolongation can be continued beginning with the one forms given in (48). To briefly outline the further steps in the procedure, a system of Pfaffians is introduced which are based on the one forms The subscript in and indicates that they belong to the subsystem defined by , and it is not summed over. Exterior differentiation of yields The are pseudopotentials for the original nonlinear equation (39). Next, quadratic pseudopotentials are included as the homogeneous variables and the prolongation can be continued.
5. 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 in all cases. In fact, the one form need not be thought of as unique. This is due to the fact that and are form invariant under the gauge transformation and where, in the case, is an arbitrary space-time-dependent matrix of determinant one. The gauge transformation property holds in the 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 are contained in the ring spanned by and This can be thought of as a generalization of the Frobenius Theorem for complete integrability of Pfaffian systems.
Theorem 3 implies that when for solutions of the original equation. If either of these forms is expressed as , then implies that Thus is a conserved density and a conserved current. Let us finally represent , , and , where is a parameter. The -dependent piece of in (17) implies By substituting an asymptotic expansion around for in (59) of the form, a recursion for the is obtained. Thus (59) becomes, Expanding the power, collecting coefficients of , and equating coefficients of to zero, we get The consistency of solving by using just the part is guaranteed by complete integrability. The part of the form is expressed as Consequently, from (58) the 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.
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