IJMMS International Journal of Mathematics and Mathematical Sciences 1687-0425 0161-1712 Hindawi Publishing Corporation 504645 10.1155/2013/504645 504645 Research Article Geometric Methods to Investigate Prolongation Structures for Differential Systems with Applications to Integrable Systems Bracken Paul Krieg Aloys Department of Mathematics University of Texas Edinburg TX 78541-2999 USA utpa.edu 2013 31 3 2013 2013 08 01 2013 13 02 2013 2013 Copyright © 2013 Paul Bracken. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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,), 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 . 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 2×2SL(2,) 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 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,) 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 . The generalization of this formalism to an n×n problem then becomes straightforward after completing the study of these smaller cases . 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,) 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. <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M26"><mml:mi>S</mml:mi><mml:mi>L</mml:mi><mml:mo mathvariant="bold">(</mml:mo><mml:mn>2</mml:mn><mml:mo mathvariant="bold">,</mml:mo><mml:mi>ℝ</mml:mi><mml:mo mathvariant="bold">)</mml:mo></mml:math></inline-formula> Prolongations

The procedure begins by associating a system of Pfaffian equations to the nonlinear system to be studied, namely, (1)αi=0,αi=dyi-Ωijyj,i,j=1,2. In (1), Ω is a traceless 2×2 matrix which consists of a family of one forms, ωi. 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,). Explicitly, the matrix of one forms is given by the 2×2 matrix (2)Ω=(ω1ω2ω3-ω1). The integrability conditions are expressed as the vanishing of a traceless 2×2 matrix of two forms Θ given by (3)Θ=dΩ-ΩΩ. In terms of components, the left-hand side of (3) has the form (4)Θ=(Θij)=(ϑ1ϑ2ϑ3-ϑ1). Substituting (2) into (3), a matrix which contains the integrability equations in terms of the basis one forms is obtained. The three equations are (5)ϑ1=dω1-ω2ω3,ϑ2=dω2-2ω1ω2,ϑ3=dω3+2ω1ω3. Of use in the theorems which follow, it is also useful to have the dωi in (5) expressed explicitly in terms of the ϑi as (6)dω1=ϑ1+ω2ω3,dω2=ϑ2+2ω1ω2,dω3=ϑ3-2ω1ω3. Therefore, by selecting a particular choice for the set {ωi}, the nonlinear equation of interest can be written simply as (7)Θ=0,ϑi=0,i=1,2,3. It is easy to see that the system is closed. Upon differentiating Θ and substituting (3), we obtain (8)dΘ=ΩΘ-ΘΩ. This implies that the exterior derivatives of the set of {ϑi} are contained in the ring of two forms {ϑi}.

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

Theorem 1.

The differential one forms (9)α1=dy1-ω1y1-ω2y2,α2=dy2-ω3y1+ω1y2, have the following exterior derivatives: (10)dα1=ω1α1+ω2α2-y1ϑ1-y2ϑ2,dα2=-ω1α2+ω3α1+y2ϑ1-y1ϑ3. Hence, the exterior derivatives of the {αi} are contained in the ring of forms {ϑi,αj}.

Proof.

The forms (9) follow by substituting (2) into (1). Differentiating α1 in (9) it is found that (11)dα1=-y1dω1+ω1dy1-y2dω2+ω2dy2. Obtaining dy1 and dy2 from (9) and dω1, dω2 from (6), dα1 becomes (12)dα1=-y1ϑ1-y1ω2ω3+ω1α1+y2ω1ω2-y2ϑ2-2y2ω1ω2+ω2α2+y1ω2ω3-y2ω2ω1=ω1α1+ω2α2-y1ϑ1-y2ϑ2. Similarly, beginning with α2, (13)dα2=-y1dω3+ω3dy1+y2dω1-ω1dy2=-y1ϑ3+2y1ω1ω3+ω3α1+y1ω3ω1+y2ω3ω2+y2ϑ1+y2ω2ω3-ω1α2-y1ω1ω3=-ω1α2+ω3α1+y2ϑ1-y1ϑ3, as required.

Corollary 2.

The exterior derivatives of the αi in (9) can be expressed concisely in terms of the matrix elements of Θ and Ω as (14)dαi=-Θijyj+Ωijαj,i,j=1,2.

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: (15)y12α3=y1α2-y2α1=y1dy2-y2dy1-y12ω3+2y1y2ω1+y22ω2,y22α4=y2α1-y1α2=y2dy1-y1dy2-y22ω2-2y1y2ω1+y12ω3. Two new functions, or pseudopotentials, y3 and y4 can be introduced which are defined by the transformation (16)y3=y2y1,y4=y1y2. Then, from (15), the following forms which have a Riccati structure result: (17)α3=dy3-ω3+2y3ω1+y32ω2,α4=dy4-ω2-2y4ω1+y42ω3. 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 (18)dα3=2y3ϑ1+y32ϑ2-ϑ3+2α3(ω1+y3ω2),dα4=-2y4ϑ1-ϑ2+y42ϑ3+2α4(-ω1+y4ω3). Consequently, results (18) are contained in the ring spanned by {ϑi} and {αj}.

(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 (19)dσ1=ϑ1+y3ϑ2+α3ω2,dσ2=-ϑ1+y3ϑ2+α4ω2. Therefore dσ10 and dσ20mod{ϑi,αj}.

The proof of this Theorem proceeds along exactly the same lines as Theorem 1. The SL(2,) 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 (20)α5=dy5-ω1-y3ω2,α6=dy6+ω1-y4ω3. The exterior derivatives of the one forms (20) are given by the expressions (21)dα5=-ϑ1-y3ϑ3-α3ω2,dα6=ϑ1-y4ϑ3-α4ω3. 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 (22)α7=dy7-e-2y5ω2,α8=dy8-e-2y6ω3. The exterior derivatives of the forms α7 and α8 are given by (23)dα7=2e-2y5α5ω2-e-2y5ϑ2,dα8=2e-2y6α6ω3-e-2y6ϑ3, and are closed over the prolonged ideal.

3. <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M117"><mml:mi>O</mml:mi><mml:mo mathvariant="bold">(</mml:mo><mml:mn>3</mml:mn><mml:mo mathvariant="bold">)</mml:mo></mml:math></inline-formula> Prolongations

The first 3×3 problem we consider is formulated in terms of a set of one forms ωi for i=1,2,3. The following system of Pfaffian equations is to be associated to a specific nonlinear equation (24)αi=0,αi=dyi-Ωijyj,i,j=1,2,3. 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, (25)Θ=0,Θ=dΩ-ΩΩ. 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 (26)Ω=(0-ω1ω2ω10-ω3-ω2ω30). Using (26) in (24) and (25), the following prolonged differential system is obtained: (27)dy1=α1-y2ω1+y3ω2,dω1=ϑ1-ω2ω3,dy2=α2+y1ω1-y3ω3,dω2=ϑ2-ω3ω1,dy3=α3-y1ω2+y2ω3,dω3=ϑ3-ω1ω2. By straightforward exterior differentiation of each αi and using (27), the following Theorem results.

Theorem 6.

The αi given in (27) have exterior derivatives which are contained in the ring spanned by {ϑi}13 and {αj}13 and given explicitly as (28)dα1=y2ϑ1-y3ϑ2-ω1α2+ω2α3,dα2=-y1ϑ1+y3ϑ3+ω1α1-ω3α3,dα3=y1ϑ2-y2ϑ3-ω2α1+ω3α2.

The one forms {αi}13 admit a series of Riccati representations which can be realized by defining six new one forms, (29)y12α4=y1α2-y2α1,y12α5=y1α3-y3α1,y22α6=y2α1-y1α2,y22α7=y2α3-y3α2,y32α8=y3α1-y1α3,y32α9=y3α2-y2α3. 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 {yj}49 which are defined in terms of the original {yi}13 as follows: (30)y4=y2y1,y5=y3y1,y6=y1y2,y7=y3y1,y8=y1y3,y9=y2y3. In terms of the functions defined in (30), the {αj}49 are given as (31)α4=dy4-(1+y42)ω1+y4y5ω2+y5ω3,α5=dy5-y4y5ω1+(1+y52)ω2-y4ω3,α6=dy6+(1+y62)ω1-y7ω2-y6y7ω3,α7=dy7+y6y7ω1+y6ω2-(1+y72)ω3,α8=dy8+y9ω1-(1+y82)ω2+y8y9ω3,α9=dy9-y8ω1-y8y9ω2+(1+y92)ω3.

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

Theorem 7.

The closure properties for the forms {αi}49 given in (31) can be summarized in the form (33)d(α4α5)=(-(1+y42)ϑ1+y4y5ϑ2+y5ϑ3-y4y5ϑ1+(1+y52)ϑ2-y4ϑ3)+Ω1(α4α5),d(α6α7)=((1+y62)ϑ1-y7ϑ2-y6y7ϑ3y6y7ϑ1+y6ϑ2-(1+y72)ϑ3)+Ω2(α6α7),d(α8α9)=(y1ϑ1-(1+y82)ϑ2+y8y9ϑ3-y8ϑ1-y8y9ϑ2+(1+y92)ϑ3)+Ω3(α8α9). The exterior derivatives are therefore contained in the ring spanned by {ϑi}13 and the {αj}49.

In terms of Ωi, a corresponding matrix of two forms Θi can be defined in terms of the corresponding Ωi defined in (32), (34)Θi=dΩi-ΩiΩi,i=1,2,3.

Theorem 8.

The 2×2 matrix of two forms Θi defined by (34) is contained in the ring of forms spanned by {ϑi}13 coupled with either {α4,α5}, {α6,α7}, or {α8,α9} when i=1,2,3, respectively.

Proof.

Differentiating Ω1 and simplifying, we have(35)Θ1=dΩ1-Ω1Ω1=(2y4ϑ1-y5ϑ2+2α4ω1-α5ω2-y4ϑ2-ϑ3-α4ω2y5ϑ1+ϑ3+α5ω1y4ϑ1-2y5ϑ2+α4ω1-2α5ω2).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 Ωi in (32) are not traceless. Define their traces to be (36)κi=trΩi,i=1,2,3. 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 (37)13dκ1=y4ϑ1-y5ϑ2-ω1α4+ω2α5,13dκ2=-y6ϑ1+y7ϑ3+ω1α6-ω3α7,13dκ3=y8ϑ2-y9ϑ3-ω2α8+ω3α9. Results (37) are contained in the ring spanned by {ϑi}13 and {αj}49.

4. <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M172"><mml:mi>S</mml:mi><mml:mi>U</mml:mi><mml:mo mathvariant="bold">(</mml:mo><mml:mn>3</mml:mn><mml:mo mathvariant="bold">)</mml:mo></mml:math></inline-formula> Prolongations

To formulate a 3 × 3 SU(3) problem, the set of Pfaffian equations, (38)αi=0,αi=dyi-Ωijyj,i,j=1,2,3, 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 λj for j=1,,8, which satisfy the following set of commutation relations: (39)[λl,λm]=2i  flmnλn. The structure constants flmn are totally antisymmetric in l,m,n. The one form Ω is expressed in terms of the generators λi as (40)Ω=l=18  ωlλl. Then the two forms Θ can be written as (41)Θ=l=18  ϑlλl,ϑl=dωl-iflmnωmωn. The nonlinear equation to be solved has the form ϑl=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 (42)Ω=(ω3+13ω8ω1-iω2ω4-iω5ω1+iω2-ω3+13ω8ω6-iω7ω4+iω5ω6+iω7-23ω8). Moreover, in order that the presentation can be easier to follow, the eight forms dωi specified by (41) will be given explicitly: (43)dω1=ϑ1+2iω2ω3+iω4ω7-iω5ω6,dω2=ϑ2-2iω1ω3+iω4ω6+iω5ω7,dω3=ϑ3+2iω1ω2+iω4ω5-iω6ω7,dω4=ϑ4-iω1ω7-iω2ω6-iω3ω5+3iω5ω8,dω5=ϑ5+iω1ω6-iω2ω7+iω3ω4-3iω4ω8,dω6=ϑ6-iω1ω5+iω2ω4+iω3ω7+3iω7ω8,dω7=ϑ7+iω1ω4+iω2ω5-iω3ω6-3iω6ω8,dω8=ϑ8+3iω4ω5+3iω6ω7. Substituting (42) into (38), the following system of one forms {αi}13 is obtained, (44)α1=dy1-(ω3+13ω8)y1-(ω1-iω2)y2-(ω4-iω5)y3,α2=dy2-(ω1+iω2)y1+(ω3-13ω8)y2-(ω6-iω7)y3,α3=dy3-(ω4+iω5)y1-(ω6+iω7)y2+23ω8y3. 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 αi in (44) are given by (45)dα1=-y2(ϑ1-iϑ2)-y1ϑ3-y3(ϑ4-iϑ5)-13y1ϑ8+(ω3+13ω8)α1+(ω1-iω2)α2+(ω4-iω5)α3,dα2=-y1(ϑ1+iϑ2)+y2ϑ3-13y2ϑ8-y3(ϑ6-iϑ7)+(ω1+iω2)α1-(ω3-13ω8)α2+(ω6-iω7)α3,dα3=-y1(ϑ4+iϑ5)-y2(ϑ6+iϑ7)+23y3ϑ8+(ω4+iω5)α1+(ω6+iω7)α2-23ω8α3.

In order to keep the notation concise, an abbreviation will be introduced. Suppose that {γi} is a system of forms, and i,j are integers, then we make the abbreviation (46)γij±=γi±iγj. 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 {αi}13 in (44) are obtained: (47)α4=dy4-ω12++2y4ω3+y42ω12--y5ω67-+y4y5ω45-,α5=dy5-ω45++y5(ω3+3ω8)+y52ω45--y4ω67++y4y5ω12-,α6=dy6-ω12--2y6ω3+y62ω12+-y7ω45-+y6y7ω67-,α7=dy7-ω67+-y7(ω3-3ω8)+y72ω67--y6ω45++y6y7ω12+,α8=dy8-ω45--y8(ω3+3ω8)+y82ω45+-y9ω12-+y8y9ω67+,α9=dy9-ω67-+y9(ω3-3ω8)+y92ω67+-y8ω12++y8y9ω45+. 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 Ωi for i=1,2,3 as (48)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+). Making use of the matrices Ωi defined in (48), the following result can be stated.

Theorem 11.

The closure properties of the set of forms {αi}49 given by (47) can be expressed in terms of the Ωi as follows: (49)d(α4α5)=((y42-1)ϑ1-i(y42+1)ϑ2+2y4ϑ3+y4y5ϑ45--y5ϑ67-y4y5ϑ12-+y5ϑ3+(y42-1)ϑ4-i(y52+1)ϑ5-y4ϑ67++3y5ϑ8)+Ω1(α4α5),d(α6α7)=((y62-1)ϑ1+i(y62+1)ϑ2-2y6ϑ3-y7ϑ45-+y6y7ϑ67-y6y7ϑ12+-y7ϑ3-y6ϑ45++(y72-1)ϑ6-i(y72+1)ϑ7+3y7ϑ8)+Ω2(α6α7),d(α8α9)=(-y9ϑ12--y8ϑ3+(y82-1)ϑ4+i(y82+1)ϑ5+y8y9ϑ67+-3y8ϑ8-y8ϑ12++y9ϑ3+y8y9ϑ45++(y92-1)ϑ6+i(y92+1)ϑ7-3y9ϑ8)+Ω3(α8α9).

Based on the forms Ωi given in (48), we can differentiate to define the following matrices of two forms ξi given by (50)ξi=dΩi-ΩiΩi.

Theorem 12.

The forms ξi defined in (50) are given explicitly by the matrices (51)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-y7  ϑ12++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).

From these results, it is concluded that ξ1 is contained in the ring of {α4,α5} and the {ϑl}, ξ2 is contained in the ring of {α6,α7} and {ϑl}, and ξ3 is in the ring of {α8,α9} and {ϑl}.

The two forms {ξi}13 therefore vanish for the solutions of the nonlinear equations (39) and of the coupled Riccati equations {αj=0}49. As in the case of O(3), the one forms Ωi are not traceless. Denoting the trace of Ωi by (52)τi=trΩi, the following theorem then follows.

Theorem 13.

The closure properties of the exterior derivatives of the traces τi are given by (53)13dτ1=ω12-α4+ω45-α5-y4ϑ12--ϑ3-y5ϑ45--13ϑ8,13dτ2=ω12+α6+ω67-α7-y6ϑ12++ϑ3-y7ϑ67--13ϑ8,13dτ3=ω45+α8+ω67+α9-y8ϑ45+-y9ϑ67++23ϑ8.

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 Ωl. Further prolongation can be continued beginning with the one forms Ωl 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 Ωl(54)α~lj=0,α~lj=dylj-(Ωl)jkylk. The subscript l in ylj and α~lj indicates that they belong to the subsystem defined by Ωl, and it is not summed over. Exterior differentiation of α~lj yields (55)dα~lj=(Ωl)jkα~lk-(Θ~)jkylk. The {ylj} 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 ωi 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 Ωω=dA  A-1+AΩA-1 and ΘΘ=AΘA-1 where, in the SL(2,) 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 (56)αi=0,αi=dyi+Fidx+Gidt, with the property that the exterior derivatives dαi are contained in the ring spanned by {αi} and {ϑl}(57)dαi=j  Aijαj+l  Γilϑl. This can be thought of as a generalization of the Frobenius Theorem for complete integrability of Pfaffian systems.

Theorem 3 implies that dσi=0 when i=1,2 for solutions of the original equation. If either of these forms is expressed as σ=dx+𝒥dt, then dσ=0 implies that (58)t-𝒥t=0. Thus 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 (59)y3,x+2a1y3+a2y32-η=0. By substituting an asymptotic expansion around η= for y3 in (59) of the form, (60)y3=0  η-nYn, a recursion for the Yn is obtained. Thus (59) becomes, (61)n  η-nYn,x+2a1n  η-nYn+a2(n  η-nYn)2-η=0. Expanding the power, collecting coefficients of η, and equating coefficients of η to zero, we get (62)Y1,x+2a1Y1=1,Yn,x+2a1Yn+a2k=1n-1  Yn-kYk=0.   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 (63)(σ1)x=(a1+n=1  η-nYna2)dx. Consequently, from (58) the nth conserved density is (64)n=a2Yn.

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.

Estabrook F. B. Wahlquist H. D. Classical geometries defined by exterior differential systems on higher frame bundles Classical and Quantum Gravity 1989 6 3 263 274 MR983923 10.1088/0264-9381/6/3/008 ZBL0672.53035 Cartan H. Differential Forms 2006 Mineola, NY, USA Dover Ablowitz M. J. Kaup D. J. Newell A. C. Segur H. Method for solving the sine-Gordon equation Physical Review Letters 1973 30 1262 1264 MR0406175 10.1103/PhysRevLett.30.1262 Ablowitz M. J. Kaup D. K. Newell A. C. Segur H. Nonlinear-evolution equations of physical significance Physical Review Letters 1973 31 125 127 MR0406176 10.1103/PhysRevLett.31.125 ZBL1243.35143 Sasaki R. Pseudopotentials for the general AKNS system Physics Letters A 1979 73 2 77 80 10.1016/0375-9601(79)90439-0 MR592379 Sasaki R. Geometric approach to soliton equations Proceedings of the Royal Society. London 1980 373 1754 373 384 10.1098/rspa.1980.0154 MR601916 ZBL0451.35080 Crampin M. Solitons and SL(2,) Physics Letters A 1978 66 3 170 172 10.1016/0375-9601(78)90646-1 MR598750 Kaup D. J. Yang J. The inverse scattering transform and squared eigenfunctions for a degenerate 3×3 operator Inverse Problems 2009 25 10 105010 10.1088/0266-5611/25/10/105010 MR2545979 ZBL1178.35382 Kaup D. J. Van Gorder R. A. The inverse scattering transform and squared eigenfunctions for the nondegenerate 3×3 operator and its soliton structure Inverse Problems 2010 26 5 055005 10.1088/0266-5611/26/5/055005 MR2647147 ZBL1204.47011 Bracken P. Intrinsic formulation of geometric integrability and associated Riccati system generating conservation laws International Journal of Geometric Methods in Modern Physics 2009 6 5 825 837 10.1142/S0219887809003771 MR2555479 ZBL1188.53090 Kaup D. J. Van Gorder R. A. Squared eigenfunctions and the perturbation theory for the nondegenerate N × N operator: a general outline Journal of Physics A 2010 43 434019 Wahlquist H. D. Estabrook F. B. Prolongation structures of nonlinear evolution equations Journal of Mathematical Physics 1975 16 1 7 MR0358111 10.1063/1.522396 Bracken P. On two-dimensional manifolds with constant Gaussian curvature and their associated equations International Journal of Geometric Methods in Modern Physics 2012 9 3 1250018 10.1142/S0219887812500181 MR2917322 ZBL1252.35129