Geometric Methods to Investigate Prolongation Structures for Differential Systems with Applications to Integrable Systems

Abstract

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 𝑆𝐿(2,ℝ), 𝑂(3), and 𝑆𝑈(3). Each will be considered in turn and the latter two systems represent larger 3×3 cases. This geometric approach is applied to all of the three of these systems to obtain prolongation structures explicitly. In both 3×3 cases, the prolongation structure is reduced to the situation of three smaller 2×2 problems.

1. Introduction

Geometric approaches have been found useful in producing a great variety of results for nonlinear partial differential equations [1]. 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 [2] 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 2×2𝑆𝐿(2,ℝ) system discussed first by Sasaki [5, 6] and Crampin [7] 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 2×2 problems. Of greater complexity are a pair of 3×3 problems which will be considered next. It is shown how to construct an 𝑂(3) system based on three constituent one forms as well as an 𝑆𝑈(3) system composed of eight fundamental one forms. The former has not appeared. The 3×3 problem is less well known than the 2×2 problem; however, some work has appeared in [8, 9]. In the first of these, the 3×3 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 𝑆𝐿(2,ℝ) system, the overall procedure can be carried over to the other Lie algebra cases as well.

Of course, the 3×3 problems will yield more types of conservation laws. The prolongation structures of the 3×3 problems will be observed to be reduceable to the same type as the 2×2 system considered at the beginning. However, the Riccati representations become much more complicated [10]. The generalization of this formalism to an 𝑛×𝑛 problem then becomes straightforward after completing the study of these smaller cases [11]. The prolongation structure of an 𝑛×𝑛 problem is reduced to a collection of 𝑛 smaller (𝑛−1)×(𝑛−1) cases and finally at 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 are developed from one of the results and some speculation as to how this type relates to other procedures [12, 13].

2. 𝑆𝐿(2,ℝ) Prolongations

The procedure begins by associating a system of Pfaffian equations to the nonlinear system to be studied, namely, 𝛼_𝑖=0,𝛼_𝑖=𝑑𝑦_𝑖−Ω_𝑖𝑗𝑦_𝑗,𝑖,𝑗=1,2. (1) In (1), Ω is a traceless 2×2 matrix which consists of a family of one forms, 𝜔_𝑖. At 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 2×2 matrix 𝜔Ω=¹𝜔₂𝜔₃−𝜔₁. (2) The integrability conditions are expressed as the vanishing of a traceless 2×2 matrix of two forms Θ given by Θ=𝑑Ω−Ω∧Ω. (3) In terms of components, the left-hand side of (3) has the form Θ=Θ_𝑖𝑗=𝜗₁𝜗₂𝜗₃−𝜗₁. (4) Substituting (2) into (3), a matrix which contains the integrability equations in terms of the basis one forms is obtained. The three equations are 𝜗₁=𝑑𝜔₁−𝜔₂∧𝜔₃,𝜗₂=𝑑𝜔₂−2𝜔₁∧𝜔₂,𝜗₃=𝑑𝜔₃+2𝜔₁∧𝜔₃. (5) 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 𝑑𝜔₁=𝜗₁+𝜔₂∧𝜔₃,𝑑𝜔₂=𝜗₂+2𝜔₁∧𝜔₂,𝑑𝜔₃=𝜗₃−2𝜔₁∧𝜔₃. (6) Therefore, by selecting a particular choice for the set {𝜔_𝑖}, the nonlinear equation of interest can be written simply as Θ=0,𝜗_𝑖=0,𝑖=1,2,3. (7) It is easy to see that the system is closed. Upon differentiating Θ and substituting (3), we obtain 𝑑Θ=Ω∧Θ−Θ∧Ω. (8) This implies that the exterior derivatives of the set of {𝜗_𝑖} are contained in the ring of two forms {𝜗_𝑖}.

Now the differential ideal can be prolonged by including the forms {𝛼₁,𝛼₂} given by (1).

Theorem 1. The differential one forms 𝛼₁=𝑑𝑦₁−𝜔₁𝑦₁−𝜔₂𝑦₂,𝛼₂=𝑑𝑦₂−𝜔₃𝑦₁+𝜔₁𝑦₂, (9) have the following exterior derivatives: 𝑑𝛼₁=𝜔₁∧𝛼₁+𝜔₂∧𝛼₂−𝑦₁𝜗₁−𝑦₂𝜗₂,𝑑𝛼₂=−𝜔₁∧𝛼₂+𝜔₃∧𝛼₁+𝑦₂𝜗₁−𝑦₁𝜗₃. (10) Hence, the exterior derivatives of the {𝛼_𝑖} are contained in the ring of forms {𝜗_𝑖,𝛼_𝑗}.

Proof. The forms (9) follow by substituting (2) into (1). Differentiating 𝛼₁ in (9) it is found that 𝑑𝛼₁=−𝑦₁𝑑𝜔₁+𝜔₁∧𝑑𝑦₁−𝑦₂𝑑𝜔₂+𝜔₂∧𝑑𝑦₂. (11) Obtaining 𝑑𝑦₁ and 𝑑𝑦₂ from (9) and 𝑑𝜔₁, 𝑑𝜔₂ from (6), 𝑑𝛼₁ becomes 𝑑𝛼₁=−𝑦₁𝜗₁−𝑦₁𝜔₂∧𝜔₃+𝜔₁∧𝛼₁+𝑦₂𝜔₁∧𝜔₂−𝑦₂𝜗₂−2𝑦₂𝜔₁∧𝜔₂+𝜔₂∧𝛼₂+𝑦₁𝜔₂∧𝜔₃−𝑦₂𝜔₂∧𝜔₁=𝜔₁∧𝛼₁+𝜔₂∧𝛼₂−𝑦₁𝜗₁−𝑦₂𝜗₂. (12) Similarly, beginning with 𝛼₂, 𝑑𝛼₂=−𝑦₁𝑑𝜔₃+𝜔₃∧𝑑𝑦₁+𝑦₂𝑑𝜔₁−𝜔₁∧𝑑𝑦₂=−𝑦₁𝜗₃+2𝑦₁𝜔₁∧𝜔₃+𝜔₃∧𝛼₁+𝑦₁𝜔₃∧𝜔₁+𝑦₂𝜔₃∧𝜔₂+𝑦₂𝜗₁+𝑦₂𝜔₂∧𝜔₃−𝜔₁∧𝛼₂−𝑦₁𝜔₁∧𝜔₃=−𝜔₁∧𝛼₂+𝜔₃∧𝛼₁+𝑦₂𝜗₁−𝑦₁𝜗₃, (13) as required.

Corollary 2. The exterior derivatives of the 𝛼_𝑖 in (9) can be expressed concisely in terms of the matrix elements of Θ and Ω as 𝑑𝛼_𝑖=−Θ_𝑖𝑗𝑦_𝑗+Ω_𝑖𝑗∧𝛼_𝑗,𝑖,𝑗=1,2. (14)

The corollary is Theorem 1 after using the definitions of the matrices in (2) and (4).

The forms 𝛼₁,𝛼₂ given in (9) lead to a natural Riccati representation called 𝛼₃, 𝛼₄ for this differential system. This arises by taking the following linear combinations of 𝛼₁ and 𝛼₂: 𝑦²₁𝛼₃=𝑦₁𝛼₂−𝑦₂𝛼₁=𝑦₁𝑑𝑦₂−𝑦₂𝑑𝑦₁−𝑦²₁𝜔₃+2𝑦₁𝑦₂𝜔₁+𝑦²₂𝜔₂,𝑦²₂𝛼₄=𝑦₂𝛼₁−𝑦₁𝛼₂=𝑦₂𝑑𝑦₁−𝑦₁𝑑𝑦₂−𝑦²₂𝜔₂−2𝑦₁𝑦₂𝜔₁+𝑦²₁𝜔₃. (15) Two new functions, or pseudopotentials, 𝑦₃ and 𝑦₄ can be introduced which are defined by the transformation 𝑦₃=𝑦₂𝑦₁,𝑦₄=𝑦₁𝑦₂. (16) Then, from (15), the following forms which have a Riccati structure result: 𝛼₃=𝑑𝑦₃−𝜔₃+2𝑦₃𝜔₁+𝑦²₃𝜔₂,𝛼₄=𝑑𝑦₄−𝜔₂−2𝑦₄𝜔₁+𝑦²₄𝜔₃. (17) These equations could also be thought of as Riccati equations for 𝑦₃ and 𝑦₄.

Theorem 3. (i) The one forms 𝛼₃, 𝛼₄ given by (17) have exterior derivatives 𝑑𝛼₃=2𝑦₃𝜗₁+𝑦²₃𝜗₂−𝜗₃+2𝛼₃∧𝜔₁+𝑦₃𝜔₂,𝑑𝛼₄=−2𝑦₄𝜗₁−𝜗₂+𝑦²₄𝜗₃+2𝛼₄∧−𝜔₁+𝑦₄𝜔₃. (18) Consequently, results (18) are contained in the ring spanned by {𝜗_𝑖} and {𝛼_𝑗}.
(ii) Define the forms 𝜎₁=𝜔₁+𝑦₃𝜔₂ and 𝜎₂=−𝜔₁+𝑦₄𝜔₃ appearing in (18). The exterior derivatives of 𝜎₁ and 𝜎₂ are found to be 𝑑𝜎₁=𝜗₁+𝑦₃𝜗₂+𝛼₃∧𝜔₂,𝑑𝜎₂=−𝜗₁+𝑦₃𝜗₂+𝛼₄∧𝜔₂. (19) Therefore 𝑑𝜎₁≡0 and 𝑑𝜎₂≡0mod{𝜗_𝑖,𝛼_𝑗}.

The proof of this Theorem proceeds along exactly the same lines as Theorem 1. The 𝑆𝐿(2,ℝ) structure and the connection interpretation can be based on the forms 𝛼₃, 𝛼₄.

At this point, the process can be continued. Two more pseudopotentials 𝑦₅ and 𝑦₆ can be introduced and the differential system can be extended by including two more one forms 𝛼₅ and 𝛼₆. The specific structure of these forms is suggested by Theorem 3.

Theorem 4. Define two one forms 𝛼₅ and 𝛼₆ as 𝛼₅=𝑑𝑦₅−𝜔₁−𝑦₃𝜔₂,𝛼₆=𝑑𝑦₆+𝜔₁−𝑦₄𝜔₃. (20) The exterior derivatives of the one forms (20) are given by the expressions 𝑑𝛼₅=−𝜗₁−𝑦₃𝜗₃−𝛼₃∧𝜔₂,𝑑𝛼₆=𝜗₁−𝑦₄𝜗₃−𝛼₄∧𝜔₃. (21) Results (21) specify the closure properties of the forms 𝛼₅ and 𝛼₆. This theorem is proved along similar lines keeping in mind that the results for 𝑑𝑦₃ and 𝑑𝑦₄ are obtained from (17).

A final extension can be made by adding two additional pseudopotentials 𝑦₇ and 𝑦₈. 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 𝛼₇ and 𝛼₈ by means of 𝛼₇=𝑑𝑦₇−𝑒^−2𝑦5𝜔₂,𝛼₈=𝑑𝑦₈−𝑒^−2𝑦6𝜔₃. (22) The exterior derivatives of the forms 𝛼₇ and 𝛼₈ are given by 𝑑𝛼₇=2𝑒^−2𝑦5𝛼₅∧𝜔₂−𝑒^−2𝑦5𝜗₂,𝑑𝛼₈=2𝑒^−2𝑦6𝛼₆∧𝜔₃−𝑒^−2𝑦6𝜗₃, (23) and are closed over the prolonged ideal.

3. 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 𝛼_𝑖=0,𝛼_𝑖=𝑑𝑦_𝑖−Ω_𝑖𝑗𝑦_𝑗,𝑖,𝑗=1,2,3. (24) In (24), 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, Θ=0,Θ=𝑑Ω−Ω∧Ω. (25) These can be considered as integrability conditions for (24). 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 Ω=0−𝜔₁𝜔₂𝜔₁0−𝜔₃−𝜔₂𝜔₃0. (26) Using (26) in (24) and (25), the following prolonged differential system is obtained: 𝑑𝑦₁=𝛼₁−𝑦₂𝜔₁+𝑦₃𝜔₂,𝑑𝜔₁=𝜗₁−𝜔₂∧𝜔₃,𝑑𝑦₂=𝛼₂+𝑦₁𝜔₁−𝑦₃𝜔₃,𝑑𝜔₂=𝜗₂−𝜔₃∧𝜔₁,𝑑𝑦₃=𝛼₃−𝑦₁𝜔₂+𝑦₂𝜔₃,𝑑𝜔₃=𝜗₃−𝜔₁∧𝜔₂. (27) By straightforward exterior differentiation of each 𝛼_𝑖 and using (27), the following Theorem results.

Theorem 6. The 𝛼_𝑖 given in (27) have exterior derivatives which are contained in the ring spanned by {𝜗_𝑖}³₁ and {𝛼_𝑗}³₁ and given explicitly as 𝑑𝛼₁=𝑦₂𝜗₁−𝑦₃𝜗₂−𝜔₁∧𝛼₂+𝜔₂∧𝛼₃,𝑑𝛼₂=−𝑦₁𝜗₁+𝑦₃𝜗₃+𝜔₁∧𝛼₁−𝜔₃∧𝛼₃,𝑑𝛼₃=𝑦₁𝜗₂−𝑦₂𝜗₃−𝜔₂∧𝛼₁+𝜔₃∧𝛼₂. (28)

The one forms {𝛼_𝑖}³₁ admit a series of Riccati representations which can be realized by defining six new one forms, 𝑦²₁𝛼₄=𝑦₁𝛼₂−𝑦₂𝛼₁,𝑦²₁𝛼₅=𝑦₁𝛼₃−𝑦₃𝛼₁,𝑦²₂𝛼₆=𝑦₂𝛼₁−𝑦₁𝛼₂,𝑦²₂𝛼₇=𝑦₂𝛼₃−𝑦₃𝛼₂,𝑦²₃𝛼₈=𝑦₃𝛼₁−𝑦₁𝛼₃,𝑦²₃𝛼₉=𝑦₃𝛼₂−𝑦₂𝛼₃. (29) In effect, the larger 3×3 system is breaking up into several 2×2 systems. To write the new one forms explicitly, introduce the new functions {𝑦_𝑗}⁹₄ which are defined in terms of the original {𝑦_𝑖}³₁ as follows: 𝑦₄=𝑦₂𝑦₁,𝑦₅=𝑦₃𝑦₁,𝑦₆=𝑦₁𝑦₂,𝑦₇=𝑦₃𝑦₁,𝑦₈=𝑦₁𝑦₃,𝑦₉=𝑦₂𝑦₃. (30) In terms of the functions defined in (30), the {𝛼_𝑗}⁹₄ are given as 𝛼₄=𝑑𝑦₄−1+𝑦²₄𝜔₁+𝑦₄𝑦₅𝜔₂+𝑦₅𝜔₃,𝛼₅=𝑑𝑦₅−𝑦₄𝑦₅𝜔₁+1+𝑦²₅𝜔₂−𝑦₄𝜔₃,𝛼₆=𝑑𝑦₆+1+𝑦²₆𝜔₁−𝑦₇𝜔₂−𝑦₆𝑦₇𝜔₃,𝛼₇=𝑑𝑦₇+𝑦₆𝑦₇𝜔₁+𝑦₆𝜔₂−1+𝑦²₇𝜔₃,𝛼₈=𝑑𝑦₈+𝑦₉𝜔₁−1+𝑦²₈𝜔₂+𝑦₈𝑦₉𝜔₃,𝛼₉=𝑑𝑦₉−𝑦₈𝜔₁−𝑦₈𝑦₉𝜔₂+1+𝑦²₉𝜔₃. (31)

To state the next theorem, we need to define the following three matrices: Ω₁=2𝑦₄𝜔₁−𝑦₅𝜔₂−𝑦₄𝜔₂−𝜔₃𝑦₅𝜔₁+𝜔₃𝑦₄𝜔₁−2𝑦₅𝜔₂,Ω₂=−2𝑦₆𝜔₁+𝑦₇𝜔₃𝜔₂+𝑦₆𝜔₃−𝑦₇𝜔₁−𝜔₆−𝑦₆𝜔₁+2𝑦₇𝜔₃,Ω₃=2𝑦₈𝜔₂−𝑦₉𝜔₃−𝜔₁−𝑦₉𝜔₃𝜔₁+𝑦₉𝜔₂𝑦₈𝜔₂−2𝑦₉𝜔₃. (32)

Theorem 7. The closure properties for the forms {𝛼_𝑖}⁹₄ given in (31) can be summarized in the form 𝑑𝛼₄𝛼₅=−1+𝑦²₄𝜗₁+𝑦₄𝑦₅𝜗₂+𝑦₅𝜗₃−𝑦₄𝑦₅𝜗₁+1+𝑦²₅𝜗₂−𝑦₄𝜗₃+Ω₁∧𝛼₄𝛼₅,𝑑𝛼₆𝛼₇=1+𝑦²₆𝜗₁−𝑦₇𝜗₂−𝑦₆𝑦₇𝜗₃𝑦₆𝑦₇𝜗₁+𝑦₆𝜗₂−1+𝑦²₇𝜗₃+Ω₂∧𝛼₆𝛼₇,𝑑𝛼₈𝛼₉=𝑦₁𝜗₁−1+𝑦²₈𝜗₂+𝑦₈𝑦₉𝜗₃−𝑦₈𝜗₁−𝑦₈𝑦₉𝜗₂+1+𝑦²₉𝜗₃+Ω₃∧𝛼₈𝛼₉. (33) The exterior derivatives are therefore contained in the ring spanned by {𝜗_𝑖}³₁ and the {𝛼_𝑗}⁹₄.

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 (32), Θ_𝑖=𝑑Ω_𝑖−Ω_𝑖∧Ω_𝑖,𝑖=1,2,3. (34)

Theorem 8. The 2×2 matrix of two forms Θ_𝑖 defined by (34) is contained in the ring of forms spanned by {𝜗_𝑖}³₁ coupled with either {𝛼₄,𝛼₅}, {𝛼₆,𝛼₇}, or {𝛼₈,𝛼₉} when 𝑖=1,2,3, respectively.

Proof. Differentiating Ω₁ and simplifying, we haveΘ₁=𝑑Ω₁−Ω₁∧Ω₁=2𝑦₄𝜗₁−𝑦₅𝜗₂+2𝛼₄∧𝜔₁−𝛼₅∧𝜔₂−𝑦₄𝜗₂−𝜗₃−𝛼₄∧𝜔₂𝑦₅𝜗₁+𝜗₃+𝛼₅∧𝜔₁𝑦₄𝜗₁−2𝑦₅𝜗₂+𝛼₄∧𝜔₁−2𝛼₅∧𝜔₂. (35)Similar matrix expressions can be found for the cases in which Ω₁ is replaced by Ω₂ or Ω₃, respectively.

Unlike Ω given in (26), the Ω_𝑖 in (32) are not traceless. Define their traces to be 𝜅_𝑖=trΩ_𝑖,𝑖=1,2,3. (36) These traces provide convenient ways of generating conservation laws on account of the following Theorem.

Theorem 9. The exterior derivatives of traces (36) are given by 13𝑑𝜅₁=𝑦₄𝜗₁−𝑦₅𝜗₂−𝜔₁∧𝛼₄+𝜔₂∧𝛼₅,13𝑑𝜅₂=−𝑦₆𝜗₁+𝑦₇𝜗₃+𝜔₁∧𝛼₆−𝜔₃∧𝛼₇,13𝑑𝜅₃=𝑦₈𝜗₂−𝑦₉𝜗₃−𝜔₂∧𝛼₈+𝜔₃∧𝛼₉. (37) Results (37) are contained in the ring spanned by {𝜗_𝑖}³₁ and {𝛼_𝑗}⁹₄.

4. 𝑆𝑈(3) Prolongations

To formulate a 3 × 3 𝑆𝑈(3) problem, the set of Pfaffian equations, 𝛼_𝑖=0,𝛼_𝑖=𝑑𝑦_𝑖−Ω_𝑖𝑗𝑦_𝑗,𝑖,𝑗=1,2,3, (38) are associated with the nonlinear equation. In (38), Ω is a traceless 3×3 matrix consisting of a system of one forms. The nonlinear equation to be considered is expressed as the vanishing of a traceless 3×3 matrix of two forms Θ exactly as in (25) which constitute the integrability condition for (38).

A 3×3 matrix representation of the Lie algebra for 𝑆𝑈(3) is introduced by means of generators 𝜆_𝑗 for 𝑗=1,…,8, which satisfy the following set of commutation relations: 𝜆_𝑙,𝜆_𝑚=2𝑖𝑓_𝑙𝑚𝑛𝜆_𝑛. (39) The structure constants 𝑓_𝑙𝑚𝑛 are totally antisymmetric in 𝑙,𝑚,𝑛. The one form Ω is expressed in terms of the generators 𝜆_𝑖 as Ω=⁸_𝑙=1𝜔_𝑙𝜆_𝑙. (40) Then the two forms Θ can be written as Θ=⁸_𝑙=1𝜗_𝑙𝜆_𝑙,𝜗_𝑙=𝑑𝜔_𝑙−𝑖𝑓_𝑙𝑚𝑛𝜔_𝑚∧𝜔_𝑛. (41) The nonlinear equation to be solved has the form 𝜗_𝑙=0 for 𝑙=1,…,8.

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 Ω used here is given by 𝜔Ω=³+1√3𝜔₈𝜔₁−𝑖𝜔₂𝜔₄−𝑖𝜔₅𝜔₁+𝑖𝜔₂−𝜔₃+1√3𝜔₈𝜔₆−𝑖𝜔₇𝜔₄+𝑖𝜔₅𝜔₆+𝑖𝜔₇−2√3𝜔₈. (42) Moreover, in order that the presentation can be easier to follow, the eight forms 𝑑𝜔_𝑖 specified by (41) will be given explicitly: 𝑑𝜔₁=𝜗₁+2𝑖𝜔₂∧𝜔₃+𝑖𝜔₄∧𝜔₇−𝑖𝜔₅∧𝜔₆,𝑑𝜔₂=𝜗₂−2𝑖𝜔₁∧𝜔₃+𝑖𝜔₄∧𝜔₆+𝑖𝜔₅∧𝜔₇,𝑑𝜔₃=𝜗₃+2𝑖𝜔₁∧𝜔₂+𝑖𝜔₄∧𝜔₅−𝑖𝜔₆∧𝜔₇,𝑑𝜔₄=𝜗₄−𝑖𝜔₁∧𝜔₇−𝑖𝜔₂∧𝜔₆−𝑖𝜔₃∧𝜔₅+√3𝑖𝜔₅∧𝜔₈,𝑑𝜔₅=𝜗₅+𝑖𝜔₁∧𝜔₆−𝑖𝜔₂∧𝜔₇+𝑖𝜔₃∧𝜔₄−√3𝑖𝜔₄∧𝜔₈,𝑑𝜔₆=𝜗₆−𝑖𝜔₁∧𝜔₅+𝑖𝜔₂∧𝜔₄+𝑖𝜔₃∧𝜔₇+√3𝑖𝜔₇∧𝜔₈,𝑑𝜔₇=𝜗₇+𝑖𝜔₁∧𝜔₄+𝑖𝜔₂∧𝜔₅−𝑖𝜔₃∧𝜔₆−√3𝑖𝜔₆∧𝜔₈,𝑑𝜔₈=𝜗₈+√3𝑖𝜔₄∧𝜔₅+√3𝑖𝜔₆∧𝜔₇. (43) Substituting (42) into (38), the following system of one forms {𝛼_𝑖}³₁ is obtained, 𝛼₁=𝑑𝑦₁−𝜔₃+1√3𝜔₈𝑦₁−𝜔₁−𝑖𝜔₂𝑦₂−𝜔₄−𝑖𝜔₅𝑦₃,𝛼₂=𝑑𝑦₂−𝜔₁+𝑖𝜔₂𝑦₁+𝜔₃−1√3𝜔₈𝑦₂−𝜔₆−𝑖𝜔₇𝑦₃,𝛼₃=𝑑𝑦₃−𝜔₄+𝑖𝜔₅𝑦₁−𝜔₆+𝑖𝜔₇𝑦₂+2√3𝜔₈𝑦₃. (44) 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 𝛼_𝑖 in (44) are given by 𝑑𝛼₁=−𝑦₂𝜗₁−𝑖𝜗₂−𝑦₁𝜗₃−𝑦₃𝜗₄−𝑖𝜗₅−1√3𝑦₁𝜗₈+𝜔₃+1√3𝜔₈∧𝛼₁+𝜔₁−𝑖𝜔₂∧𝛼₂+𝜔₄−𝑖𝜔₅∧𝛼₃,𝑑𝛼₂=−𝑦₁𝜗₁+𝑖𝜗₂+𝑦₂𝜗₃−1√3𝑦₂𝜗₈−𝑦₃𝜗₆−𝑖𝜗₇+𝜔₁+𝑖𝜔₂∧𝛼₁−𝜔₃−1√3𝜔₈∧𝛼₂+𝜔₆−𝑖𝜔₇∧𝛼₃,𝑑𝛼₃=−𝑦₁𝜗₄+𝑖𝜗₅−𝑦₂𝜗₆+𝑖𝜗₇+2√3𝑦₃𝜗₈+𝜔₄+𝑖𝜔₅∧𝛼₁+𝜔₆+𝑖𝜔₇∧𝛼₂−2√3𝜔₈∧𝛼₃. (45)

In order to keep the notation concise, an abbreviation will be introduced. Suppose that {𝛾_𝑖} is a system of forms, and 𝑖,𝑗 are integers, then we make the abbreviation 𝛾_𝑖𝑗±=𝛾_𝑖±𝑖𝛾_𝑗. (46) 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 {𝛼_𝑖}³₁ in (44) are obtained: 𝛼₄=𝑑𝑦₄−𝜔₁₂₊+2𝑦₄𝜔₃+𝑦²₄𝜔₁₂₋−𝑦₅𝜔₆₇₋+𝑦₄𝑦₅𝜔₄₅₋,𝛼₅=𝑑𝑦₅−𝜔₄₅₊+𝑦₅𝜔₃+√3𝜔₈+𝑦²₅𝜔₄₅₋−𝑦₄𝜔₆₇₊+𝑦₄𝑦₅𝜔₁₂₋,𝛼₆=𝑑𝑦₆−𝜔₁₂₋−2𝑦₆𝜔₃+𝑦²₆𝜔₁₂₊−𝑦₇𝜔₄₅₋+𝑦₆𝑦₇𝜔₆₇₋,𝛼₇=𝑑𝑦₇−𝜔₆₇₊−𝑦₇𝜔₃−√3𝜔₈+𝑦²₇𝜔₆₇₋−𝑦₆𝜔₄₅₊+𝑦₆𝑦₇𝜔₁₂₊,𝛼₈=𝑑𝑦₈−𝜔₄₅₋−𝑦₈𝜔₃+√3𝜔₈+𝑦²₈𝜔₄₅₊−𝑦₉𝜔₁₂₋+𝑦₈𝑦₉𝜔₆₇₊,𝛼₉=𝑑𝑦₉−𝜔₆₇₋+𝑦₉𝜔₃−√3𝜔₈+𝑦²₉𝜔₆₇₊−𝑦₈𝜔₁₂₊+𝑦₈𝑦₉𝜔₄₅₊. (47) These are coupled Riccati equations for the pairs of pseudopotentials (𝑦₄,𝑦₅), (𝑦₆,𝑦₇), and (𝑦₈,𝑦₉), respectively.

Define the following set of 2×2 matrices of one forms Ω_𝑖 for 𝑖=1,2,3 as Ω₁=^{−2𝜔3−2𝑦4𝜔12−−𝑦5𝜔45−𝜔67−−𝑦4𝜔45−}_{𝜔67+−𝑦5𝜔12−−𝜔3−√3𝜔8−𝑦4𝜔12−−2𝑦5𝜔45−},Ω₂=^{2𝜔3−2𝑦6𝜔12+−𝑦7𝜔67−𝜔45−−𝑦6𝜔67−}_{𝜔45+−𝑦7𝜔12+𝜔3−√3𝜔8−𝑦6𝜔12+−2𝑦7𝜔67−},Ω₃=^{𝜔3+√3𝜔8−2𝑦8𝜔45+−𝑦9𝜔67+𝜔12−−𝑦8𝜔67+}_{𝜔12+−𝑦9𝜔45+−𝜔3+√3𝜔8−𝑦8𝜔45+−2𝑦9𝜔67+}. (48) Making use of the matrices Ω_𝑖 defined in (48), the following result can be stated.

Theorem 11. The closure properties of the set of forms {𝛼_𝑖}⁹₄ given by (47) can be expressed in terms of the Ω_𝑖 as follows: 𝑑𝛼₄𝛼₅=𝑦²₄−1𝜗₁−𝑖𝑦²₄+1𝜗₂+2𝑦₄𝜗₃+𝑦₄𝑦₅𝜗₄₅₋−𝑦₅𝜗₆₇₋𝑦₄𝑦₅𝜗₁₂₋+𝑦₅𝜗₃+𝑦²₄−1𝜗₄−𝑖𝑦²₅+1𝜗₅−𝑦₄𝜗₆₇₊+√3𝑦₅𝜗₈+Ω₁∧𝛼₄𝛼₅,𝑑𝛼₆𝛼₇=𝑦²₆−1𝜗₁+𝑖𝑦²₆+1𝜗₂−2𝑦₆𝜗₃−𝑦₇𝜗₄₅₋+𝑦₆𝑦₇𝜗₆₇₋𝑦₆𝑦₇𝜗₁₂₊−𝑦₇𝜗₃−𝑦₆𝜗₄₅₊+𝑦²₇−1𝜗₆−𝑖𝑦²₇+1𝜗₇+√3𝑦₇𝜗₈+Ω₂∧𝛼₆𝛼₇,𝑑𝛼₈𝛼₉=−𝑦₉𝜗₁₂₋−𝑦₈𝜗₃+𝑦²₈−1𝜗₄+𝑖𝑦²₈+1𝜗₅+𝑦₈𝑦₉𝜗₆₇₊−√3𝑦₈𝜗₈−𝑦₈𝜗₁₂₊+𝑦₉𝜗₃+𝑦₈𝑦₉𝜗₄₅₊+𝑦²₉−1𝜗₆+𝑖𝑦²₉+1𝜗₇−√3𝑦₉𝜗₈+Ω₃∧𝛼₈𝛼₉. (49)

Based on the forms Ω_𝑖 given in (48), we can differentiate to define the following matrices of two forms 𝜉_𝑖 given by 𝜉_𝑖=𝑑Ω_𝑖−Ω_𝑖∧Ω_𝑖. (50)

Theorem 12. The forms 𝜉_𝑖 defined in (50) are given explicitly by the matrices 𝜉₁=2𝜔₁₂₊∧𝛼₄+𝜔₄₅₋∧𝛼₅𝜔₄₅₋∧𝛼₄−𝑦₄𝜗₄₅₋+𝜗₆₇₋−2𝑦₄𝜗₁₂₋−2𝜗₃−𝑦₅𝜗₄₅₋𝜔₁₂₋∧𝛼₄+2𝜔₄₅₋∧𝛼₅𝜔₁₂₋∧𝛼₅−𝑦₅𝜗₁₂₋+𝜗₆₇₊−𝑦₄𝜗₁₂₋−𝜗₃−2𝜗₄₅₋−√3𝜗₈,𝜉₂=2𝜔₁₂₊∧𝛼₆+𝜔₆₇₋∧𝛼₇𝜔₆₇₋∧𝛼₆+𝜗₄₅₋−𝑦₆𝜗₆₇₋−2𝑦₆𝜗₁₂₊+𝜗₃−𝑦₇𝜗₆₇₋𝜔₁₂₊∧𝛼₆+2𝜔₆₇₋∧𝛼₇−𝑦₆𝜗₁₂₊𝜔₁₂₊∧𝛼₇−𝑦₇𝜗₁₂₊+𝜗₄₅₊+𝜗₃−2𝑦₇𝜗₆₇₋−√3𝜗₈,𝜉₃=2𝜔₄₅₊∧𝛼₈+𝜔₆₇₊∧𝛼₉+𝜗₃𝜔₆₇₊∧𝛼₈+𝜗₁₂₋−𝑦₈𝜗₆₇₊−2𝑦₈𝜗₄₅₊−𝑦₉𝜗₆₇₊+√3𝜗₈𝜔₄₅₊∧𝛼₈+2𝜔₆₇₊∧𝛼₉−𝜗₃𝜔₄₅₊∧𝛼₉+𝜗₁₂₊−𝑦₉𝜗₄₅₊−𝑦₈𝜗₄₅₊−2𝑦₉𝜗₆₇₊+√3𝜗₈. (51)
From these results, it is concluded that 𝜉₁ is contained in the ring of {𝛼₄,𝛼₅} and the {𝜗_𝑙}, 𝜉₂ is contained in the ring of {𝛼₆,𝛼₇} and {𝜗_𝑙}, and 𝜉₃ is in the ring of {𝛼₈,𝛼₉} and {𝜗_𝑙}.
The two forms {𝜉_𝑖}³₁ therefore vanish for the solutions of the nonlinear equations (39) and of the coupled Riccati equations {𝛼_𝑗=0}⁹₄. As in the case of 𝑂(3), the one forms Ω_𝑖 are not traceless. Denoting the trace of Ω_𝑖 by 𝜏_𝑖=trΩ_𝑖, (52) the following theorem then follows.

Theorem 13. The closure properties of the exterior derivatives of the traces 𝜏_𝑖 are given by 13𝑑𝜏₁=𝜔₁₂₋∧𝛼₄+𝜔₄₅₋∧𝛼₅−𝑦₄𝜗₁₂₋−𝜗₃−𝑦₅𝜗₄₅₋−1√3𝜗₈,13𝑑𝜏₂=𝜔₁₂₊∧𝛼₆+𝜔₆₇₋∧𝛼₇−𝑦₆𝜗₁₂₊+𝜗₃−𝑦₇𝜗₆₇₋−1√3𝜗₈,13𝑑𝜏₃=𝜔₄₅₊∧𝛼₈+𝜔₆₇₊∧𝛼₉−𝑦₈𝜗₄₅₊−𝑦₉𝜗₆₇₊+2√3𝜗₈. (53)

The one forms (52) 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 (48). To briefly outline the further steps in the procedure, a system of Pfaffians {𝛼_𝑙𝑗} is introduced which are based on the one forms Ω_𝑙𝛼_𝑙𝑗=0,𝛼_𝑙𝑗=𝑑𝑦_𝑙𝑗−Ω_𝑙𝑗𝑘𝑦_𝑙𝑘. (54) The subscript 𝑙 in 𝑦_𝑙𝑗 and 𝛼_𝑙𝑗 indicates that they belong to the subsystem defined by Ω_𝑙, and it is not summed over. Exterior differentiation of 𝛼_𝑙𝑗 yields 𝑑𝛼_𝑙𝑗=Ω_𝑙𝑗𝑘∧𝛼_𝑙𝑘−Θ_𝑗𝑘𝑦_𝑙𝑘. (55) The {𝑦_𝑙𝑗} 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 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 transformation Ω→𝜔^′=𝑑𝐴𝐴⁻¹+𝐴Ω𝐴⁻¹ and Θ→Θ^′=𝐴Θ𝐴⁻¹ where, in the 𝑆𝐿(2,ℝ) case, 𝐴 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 𝛼_𝑖=0,𝛼_𝑖=𝑑𝑦_𝑖+𝐹_𝑖𝑑𝑥+𝐺_𝑖𝑑𝑡, (56) with the property that the exterior derivatives 𝑑𝛼_𝑖 are contained in the ring spanned by {𝛼_𝑖} and {𝜗_𝑙}𝑑𝛼_𝑖=_𝑗𝐴_𝑖𝑗∧𝛼_𝑗+_𝑙Γ_𝑖𝑙𝜗_𝑙. (57) This can be thought of as a generalization of the Frobenius Theorem for complete integrability of Pfaffian systems.

Theorem 3 implies that 𝑑𝜎_𝑖=0 when 𝑖=1,2 for solutions of the original equation. If either of these forms is expressed as 𝜎=ℐ𝑑𝑥+𝒥𝑑𝑡, then 𝑑𝜎=0 implies that 𝜕ℐ−𝜕𝑡𝜕𝒥𝜕𝑡=0. (58) Thus ℐ is a conserved density and 𝒥 a conserved current. Let us finally represent 𝜔₁=𝑎₁𝑑𝑥+𝑏₁𝑑𝑡, 𝜔₂=𝑎₂𝑑𝑥+𝑏₂𝑑𝑡, and 𝜔₃=𝜂𝑑𝑥+𝑏₃𝑑𝑡, where 𝜂 is a parameter. The 𝑥-dependent piece of 𝛼₃ in (17) implies 𝑦_3,𝑥+2𝑎₁𝑦₃+𝑎₂𝑦²₃−𝜂=0. (59) By substituting an asymptotic expansion around 𝜂=∞ for 𝑦₃ in (59) of the form, 𝑦₃=^∞₀𝜂^−𝑛𝑌_𝑛, (60) a recursion for the 𝑌_𝑛 is obtained. Thus (59) becomes, _𝑛𝜂^−𝑛𝑌_𝑛,𝑥+2𝑎_1𝑛𝜂^−𝑛𝑌_𝑛+𝑎_2𝑛𝜂^−𝑛𝑌_𝑛²−𝜂=0. (61) Expanding the power, collecting coefficients of 𝜂, and equating coefficients of 𝜂 to zero, we get 𝑌_1,𝑥+2𝑎₁𝑌₁=1,𝑌_𝑛,𝑥+2𝑎₁𝑌_𝑛+𝑎₂^𝑛−1_𝑘=1𝑌_𝑛−𝑘𝑌_𝑘=0. (62) The consistency of solving 𝛼₃ by using just the 𝑥 part is guaranteed by complete integrability. The 𝑥 part of the form 𝜎₁ is expressed as 𝜎_1𝑥=𝑎₁+^∞_𝑛=1𝜂^−𝑛𝑌_𝑛𝑎₂𝑑𝑥. (63) Consequently, from (58) the 𝑛th conserved density is ℐ_𝑛=𝑎₂𝑌_𝑛. (64)

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.