Exterior differential expression of the (1 + 1)-dimensional nonlinear evolution equation with Lax integrability

J. Math. Anal. Appl. 435 (2016) 735–745

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Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa

Exterior differential expression of the (1 + 1)-dimensional nonlinear evolution equation with Lax integrability Chuan-Qi Su a , Yi-Tian Gao a,∗ , Xin Yu a , Long Xue a,b , Yu-Jia Shen a a

Ministry-of-Education Key Laboratory of Fluid Mechanics and National Laboratory for Computational Fluid Dynamics, Beijing University of Aeronautics and Astronautics, Beijing 100191, China b Flight Training Base, Aviation University of Air Force, Fuxin, Liaoning, 123100, China

a r t i c l e

i n f o

Article history: Received 29 April 2015 Available online 21 October 2015 Submitted by S.A. Fulling Keywords: (1 + 1)-Dimensional nonlinear evolution equation with Lax integrability Exterior differential Complete integrability Frobenius theorem Darboux transformation

a b s t r a c t A (1 + 1)-dimensional nonlinear evolution equation with Lax integrability is investigated in this paper with the aid of differential forms and exterior differentials. Equivalent definition of the Lax integrability is given. Relation between the Lax integrability and complete integrability is discussed. Geometric interpretation of the Lax equation is presented. Gauge transformation and Darboux transformation are restated in terms of the differential forms and exterior differentials. If λI − S is a Darboux transformation between two Lax equations, the system that the matrix S should satisfy is given, where λ is the spectral parameter and I is the identity matrix. The completely integrable condition of that system is obtained according to the Frobenius theorem, and it is shown that S = HΛH −1 satisfies that completely integrable condition with H and Λ as two matrices. © 2015 Elsevier Inc. All rights reserved.

1. Introduction Differential forms and exterior differentials have been used in the application of calculus to manifolds [10, 13,28]. They have also appeared in the analysis of nonlinear evolution equations (NLEEs) [5,7]. Exterior differential is a version of differential that is independent of the coordinate system used, and is claimed to constitute a basic ingredient in searching for expressions of the NLEEs that are independent of the coordinate systems used to describe them [28]. With the help of differential forms and exterior differentials, certain integrability properties of the NLEEs have been investigated [5–7,10,13,28,29,34]. More on the NLEEs can be seen, e.g., in Refs. [19,30,31,36,37,39,40]. Some NLEEs have appeared as the compatibility conditions for systems of linear partial differential equations of the first order, and such NLEEs have been referred to as Lax integrable [7]. Using Lax pairs, people can construct the gauge transformations (GTs) [11,20,23], Darboux transformations (DTs) [3,21,38], * Corresponding author. E-mail address: [email protected] (Y.-T. Gao). http://dx.doi.org/10.1016/j.jmaa.2015.10.036 0022-247X/© 2015 Elsevier Inc. All rights reserved.

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and the existence of infinitely many conservation laws [2,22] of such NLEEs. For a given NLEE, it is difficult to determine whether it can be associated with a Lax pair [32,33]. One of the methods to test the Lax integrability is the prolongation structure (PS) method proposed via exterior differentials [32,33]. Complete integrability is another integrability property [10,13]. The definition of complete integrability for a distribution is as follows [10,13]: A distribution Δ on a manifold M is said to be completely integrable if there exists an integral manifold whose dimension is equal to the dimension of Δ, through each point that belongs to M . The Frobenius theorem gives necessary and sufficient condition for the complete integrability for a distribution, and it can be expressed in different but equivalent forms [10,13]. The exterior differential form of the Frobenius theorem states that a distribution Δ is completely integrable if and only if the ideal I(Δ) generated by Δ is a closed ideal, i.e., dI(Δ) ⊂ I(Δ) [10,13]. In this paper, we will investigate the (1 + 1)-dimensional NLEE with Lax integrability using the tool of differential forms and exterior differentials. The relation between complete integrability and Lax integrability will also be discussed. In Section 2, the Lax equation of the NLEE will be restated in the exterior differential form, an equivalent definition of Lax integrability will be given and some remarks on the PS method will be proposed. At the end of this section, a geometric interpretation of the Lax equation will be presented. In Section 3, the GT and DT will be investigated by virtue of the exterior differentials and Frobenius theorem. The conclusion will be presented in Section 4. 2. The exterior differential expression of Lax integrability 2.1. The exterior differential expression of Lax equation For a (1 + 1)-dimensional NLEE with Lax integrability, its Lax representation can be written in the following matrix form [1]: ψx = M (λ)ψ , ψt = N (λ)ψ ,

(1)

where x and t are the independent variables, the subscripts denote the partial differentials, λ is the spectral parameter, ψ is the eigenfunction associated with λ, while M and N are matrices whose elements are dependent on λ [1]. The corresponding NLEE can be obtained from the compatibility condition of system (1), which can be written as [1] Mt − Nx + [M, N ] = 0 ,

(2)

where [M, N ] = M N − N M . Define the system of 1-forms ω = M dx + N dt ,

(3)

where the symbol d denotes the exterior differentiation operator [7,10,26]. It should be noted that the partial differential only acts on the functions, while the operator d can also act on the differential forms. System (1) is equivalent to the following Pfaff system [7,10,26]: σ = dψ − ωψ = 0 .

(4)

Using the exterior differentiation operator d on (4), we have dσ = −dωψ + ω ∧ dψ ,

(5)

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737

where ∧ denotes the wedge product [10,13]. Combining (4) and (5), we get dσ = −dωψ + ω ∧ (σ + ωψ) = ω ∧ σ − (dω − ω ∧ ω)ψ .

(6)

Through the Frobenius theorem [10,13], Pfaff system (4) is completely integrable if and only if the following system is satisfied [26]: Ω = dω − ω ∧ ω = 0 ,

(7)

where Ω is a square matrix whose elements are 2-forms [10,13,28]. It can be verified that system (7) is equivalent to compatibility condition (2). Thus, system (7) also corresponds to the NLEE whose Lax representation is given by (1). On the relation between complete integrability and Lax integrability, we have the following proposition: Proposition 1. The (1 + 1)-dimensional NLEE whose Lax representation is given by (1) must be completely integrable. Proof. Differentiating system (7), we have dΩ = −dω ∧ ω + ω ∧ dω = −(Ω + ω ∧ ω) ∧ ω + ω ∧ (Ω + ω ∧ ω) = −Ω ∧ ω + ω ∧ Ω .

(8)

Thus, system (7) is completely integrable according to the Frobenius theorem. Because system (7) corresponds to the (1 + 1)-dimensional NLEE whose Lax representation is given by (1), the proof is completed.1 2 2.2. An equivalent definition of Lax integrability We replace the (1 + 1)-dimensional NLEE whose Lax representation is given by (1), by a system of 2-forms [27,32]. Hereby, we take the nonlinear Schrödinger equation (NLSE) [1], iut + uxx + 2|u|2 u = 0 ,

(9)

as an example to illustrate this procedure, where u is a complex function of x and t. Equation (9) can describe the propagation of light in the nonlinear optical fiber or Bose–Einstein condensate confined to highly anisotropic cigar-shaped trap [1,4,25]. Equation (9) has also appeared in the studies of small-amplitude gravity wave on the surface of deep inviscid water [24]. Let U ⊂ R2 be coordinated by x and t and V = U ×R4 ⊂ R6 be a fiber bundle [10,13] with U as the base manifold and V is coordinated by x, t, u, u ¯, p, p¯, where u ¯ and p¯ are complex conjugates of u and p respectively. Then, equation (9) can be represented by the following system of 2-forms which are defined on the manifold V : α1 = pdx ∧ dt − du ∧ dt , u ∧ dt , α2 = p¯dx ∧ dt − d¯ ¯dx ∧ dt , α3 = −idu ∧ dx + dp ∧ dt + 2u2 u u ∧ dx + d¯ p ∧ dt + 2¯ u2 udx ∧ dt . α4 = id¯ 1

(10)

The identity dΩ + Ω ∧ ω − ω ∧ Ω = 0, which appears in the proof of Proposition 1 is referred to as the Bianchi identity [10].

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If map s : U → V is a cross section of V with the property s∗ αi = 0 (i = 1, 2, 3, 4) ,

(11)

where s∗ denotes the pull back of the map s [10,13], it can be verified that u(x, t) is a solution of eq. (9). Conversely, for any given solution u(x, t) of eq. (9), the map s : U → V by (x, t) → (x, t, u(x, t), ¯x (x, t)) is a cross section of V which satisfies property (11). Without causing any conu ¯(x, t), ux (x, t), u fusion, we write property (11) just as {αi = 0} for simplicity. For any (1 + 1)-dimensional NLEE, the corresponding system of 2-forms {αi = 0} can be constructed as above. Two systems of 2-forms {αi = 0} and {βi = 0} are said to be equivalent if βi = fij αj , and the rank of matrix (fij ) is full [10,13], where βi ’s are 2-forms. It should be noted that the Einstein summation convention [27] is used here and below. With those preparations, we give the following equivalent definition of Lax integrability: Proposition 2. A (1 + 1)-dimensional NLEE is Lax integrable if and only if there exists a square matrix ω of 1-forms about dx and dt, such that the system Ω = dω − ω ∧ ω = 0 is equivalent to {αi = 0}. Proof. (i) Assume that the Lax representation for the (1 + 1)-dimensional NLEE is given by (1). If ω is set to be M dx + N dt, then ω is exactly what we wanted according to the chain of the equivalent relations: Ω = dω − ω ∧ ω = 0 ⇔ Mt − Nx + [M, N ] = 0 ⇔ (1 + 1)-dimensional NLEE whose Lax representation is given by (1) ⇔ {αi = 0} .

(12)

(ii) Assume that the square matrix of 1-forms about dx and dt is ω. Then, ω can be decomposed into the dx part and dt part, i.e., ω can be written in the form ω = M dx + N dt. It can be verified that M and N are just the Lax pair we are searching for. 2 Proposition 2 defines the (1 + 1)-dimensional NLEE whose Lax representation is given by (1) in terms of the differential forms and exterior differentials. 2.3. Some remarks on the PS method The PS method has been used to test the Lax integrability [8,9,32,33,35]. Just like what we have done above, we hereby replace the (1 + 1)-dimensional NLEE whose Lax representation is given by (1) by the system of 2-forms {αi = 0} which are defined on the manifold V . We introduce the Pfaff system on the vector bundle E = V × Rn [32,33], σi = dyi − Fi dx − Gi dt (i = 1, 2, · · · , n) ,

(13)

where yi ’s are called the pseudo-potentials [32,33] and they are the coordinates of Rn , n is a positive integer, σi ’s are 1-forms, Fi ’s and Gi ’s are functions on the vector bundle E. The PS method requires that the ideal generated by the system {αi = 0, σi = 0} is a closed ideal [32,33]. According to the Frobenius theorem and Proposition 1, we know that {αi = 0} must generate a closed ideal. Thus, the prolongation condition can be written as dσi = mji αj + nji ∧ σj (i = 1, 2, · · · , n) , where mji ’s are functions to be determined and nji ’s are 1-forms.

(14)

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Via the comparison between eqs. (4) and (13), it can be seen that the pseudo-potentials yi ’s correspond to the eigenfunction ψ, while Fi ’s and Gi ’s correspond to ω. What’s more, there is a correspondence between prolongation condition (14) and eq. (6). In fact, from that correspondence, we can see that prolongation condition (14) is to guarantee that Pfaff system (13) is completely integrable on the solution manifold {αi = 0} of the (1 + 1)-dimensional NLEE whose Lax representation is given by (1). Then, Lax equation (1) for the (1 + 1)-dimensional NLEE can be obtained with the complete integrability condition although the fact that the Lax integrability is a stronger integrability property than complete integrability [32,33]. 2.4. Geometric interpretation of Lax equation (4) Assume that there is a connection D defined on the vector bundle E = V ×Rn , and the sections s1 , . . . , sn form a frame of sections of E. Using the frame of sections S, where S is the transpose of the row vector (s1 , . . . , sn ), and the connection D, we can obtain an n × n connection matrix ω which is defined by the formula DS = ωS [10]. We will give the geometric interpretation of ω in this part. Elements of the connection matrix ω are functions on the manifold V . If the section s = η i si is a parallel section of E with η i ’s as functions on the manifold V , i.e., Ds = 0, then η i ’s should satisfy the following equations [10]: dη i + η j ωji = 0 (i = 1, 2, · · · , n) .

(15)

Let η = (η 1 , . . . , η n ), the above equations can be written in the matrix form dη + ηω = 0 .

(16)

The connection D on the vector bundle E can induce a connection D on the dual vector bundle E ∗ = V × (Rn )∗ , where (Rn )∗ denotes the dual space of Rn [10]. If we choose the dual frame of sections S ∗ = (s1∗ , . . . , sn∗ )T of E ∗ , i.e., si , sj∗  = δij , where  ,  represents the inner product in the vector bundles E and E ∗ , δij is the Kronecker delta, then we have the equation D S ∗ = −ωS ∗ [10]. That is to say, the induced connection matrix on the dual vector bundle E ∗ is −ω. If the section s∗ = θi si∗ is a parallel section of E ∗ with θi ’s as functions on the manifold V , i.e., D s∗ = 0, θi ’s should satisfy the following equation: dθi − θj ωij = 0 (i = 1, 2, · · · , n) .

(17)

If θ = (θ1 , . . . , θn ), the above equations can be written in the matrix form dθ − ωθ = 0 .

(18)

It can be seen that eq. (18) is exactly Lax equation (4). Then, Lax equation (4) can be interpreted as the parallel section equation on the dual vector bundle E ∗ with the connection matrix equal to −ω = −(M dx + N dt), while the eigenfunction ψ corresponds to the vector formed by the coordinates of parallel section of E ∗ under the dual frame of sections S ∗ . In the differential geometry, formula Ω = dω − ω ∧ ω represents the curvature matrix with ω as the connection matrix [10,13]. Then, Pfaff system (4) is completely integrable if and only if the curvature matrix Ω vanishes. If zero-curvature condition (7) is satisfied, then there exists n linearly-independent parallel sections, i.e., the Lax equation dψ = ωψ has n linearly-independent solutions. With those solutions, we can construct the multi-soliton solutions using the Darboux transformation [1].

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3. Exterior differential expression of the GT and DT Assume that we have the following two Lax equations, ψx = M ψ, ψt = N ψ ,

(19)

ψx = M  ψ  , ψt = N  ψ  ,

(20)

where ψ  is the eigenfunction associated with the spectral parameter λ, while M  and N  are matrices. If there exists a gauge transformation (GT) ψ  = T ψ with T as a matrix, which converts Lax equation (19) into (20), the gauge transformation T should satisfy the following system [5,34]: Tx = M  T − T M , Tt = N  T − T N .

(21)

It can be verified that system (21) can be written in the form ω  = T ωT −1 + dT · T −1 ,

(22)

where ω is defined by eq. (3), while ω  is M  dx + N  dt. Equation (22) is just the transformation formula of the connection matrix under the transformation of the basis of sections S  = T S [26]. The transformation formula for the curvature matrix Ω is [26] Ω = T ΩT −1 ,

(23)

where Ω equals dω − ω ∧ ω, while Ω is dω  − ω  ∧ ω  . System (21) is also equivalent to the following Pfaff system: dT − ω  T + T ω = 0 .

(24)

Because the curvature matrices Ω and Ω both vanish from zero-curvature condition (7), it can be proved that Pfaff system (24) is completely integrable according to the Frobenius theorem. That is to say, every (1 + 1)-dimensional NLEE with Lax integrability is gauge equivalent to one another. We can choose the KdV equation as an example [1], vt + vxxx + 6vvx = 0 ,

(25)

where v is a real function of x and t. Equation (25) is encountered in many physical areas such as the shallow water waves in the ocean [15,18], internal gravity waves in the lake of changing cross section [14] and ion-acoustic waves in the plasma [12]. Assume that the n-dimensional Lax representation for eq. (25) is dψ = ωψ, then eq. (25) is equivalent to the system Ω = dω − ω ∧ ω = 0. Thus, we have the following proposition: Proposition 3. The (1 + 1)-dimensional NLEE with Lax integrability is equivalent to the system T ΩT −1 = 0 for some n × n invertible matrix T whose elements are functions on the manifold V . Proof. If a (1 + 1)-dimensional NLEE with Lax integrability is equivalent to the system T ΩT −1 = 0, then the Lax equation corresponding to that NLEE is dψ  = ω  ψ  , where ω  equals T ωT −1 + dT · T −1 . Then, that NLEE is Lax integrable. Conversely, we have known that every (1 + 1)-dimensional NLEE with

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Lax integrability is gauge equivalent to eq. (25) via the above discussion. Then, there exists an invertible matrix T , such that the given (1 + 1)-dimensional NLEE with Lax integrability is equivalent to the system T ΩT −1 = 0. 2 Gauge transformations (GTs) of the same (1 + 1)-dimensional NLEE with Lax integrability are useful in the analysis of this kind of the NLEEs. We say that T is a GT if the two systems T ΩT −1 = 0 and Ω = 0 are equivalent, i.e., they correspond to the same (1 + 1)-dimensional NLEE with Lax integrability. The set of all GTs in fact forms a group, which we call the gauge group. Because GTs preserve solution manifold, gauge group can also be called the symmetry group. The Darboux transformation (DT) is a particular gauge transformation [16]. The set of all DTs also forms a group which is a subgroup of the gauge group. Proposition 4. For the (1 + 1)-dimensional NLEE whose Lax representation is given by (1), if there exists a DT of the form T = λI − S, then the n × n matrix S, which is independent on the spectral parameter λ, must satisfy the following equation: dS + [S, ω(S)] = 0 .

(26)

Note: If we expand ω(λ) = M (λ)dx + N (λ)dt into the power series of the spectral parameter λ as ω(λ) = m m i i i=0 ωi λ , then ω(S) which appears in eq. (26) represents i=0 ωi S , where m is a positive integer. m m Proof. Substituting ω(λ) = i=0 ωi λi , ω  (λ) = i=0 ωi λi , T = λI − S into eq. (24), and collecting the coefficients of the same power of the spectral parameter λ, we have  λm+1 : ωm = ωm ,

(27)

 = ωj−1 − Sωj + ωj S (j = 1, . . . , m) , λj : ωj−1

(28)

λ : dS + Sω0 − 0

ω0 S

= 0.

(29)

The relation between ω0 and ω0 can be derived from eqs. (27) and (28) as ω0 = ω0 +

m 

[ωk , S]S k−1 .

(30)

k=1

Substituting eq. (30) into eq. (29), we obtain eq. (26). 2 Equation (26) can be decomposed into the following system: Sx + [S, M (S)] = 0 , St + [S, N (S)] = 0 ,

(31)

where M (S)dx + N (S)dt equals ω(S). Proposition 5. Equation (26) is completely integrable if and only if the following condition is satisfied: dω(S) − ω(S) ∧ ω(S) = 0 .

(32)

Equivalently, system (31) is completely integrable if and only if Mt (S) − Nx (S) + [M (S), N (S)] = 0 .

(33)

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Proof. Define the Pfaff system τ = dS + [S, ω(S)] = dS + [S, M (S)]dx + [S, N (S)]dt = 0. Differentiating τ , we have dτ = [dS, M (S)] ∧ dx + [S, dM (S)] ∧ dx + [dS, N (S)] ∧ dt + [S, dN (S)] ∧ dt = [dS, M (S)] ∧ dx + [S, Mt (S) − Nx (S)]dt ∧ dx + [dS, N (S)] ∧ dt .

(34)

Substituting dS by τ − [S, M (S)]dx − [S, N (S)]dt into the above equation, we have dτ = [τ, M (S)] ∧ dx + [τ, N (S)] ∧ dt + [S, Mt (S) − Nx (S)]dt ∧ dx − [[S, N (S)], M (S)]dt ∧ dx − [[S, M (S)], N (S)]dx ∧ dt .

(35)

By virtue of the Jacobi identity [M (S), [N (S), S]] + [S, [M (S), N (S)]] + [N (S), [S, M (S)]] = 0 [10,13], the above equation can be rewritten as dτ = [τ, M (S)] ∧ dx + [τ, N (S)] ∧ dt + [S, Mt (S) − Nx (S) + [M (S), N (S)]]dt ∧ dx .

(36)

Thus, the Pfaff system τ = 0 is completely integrable if and only if the condition Mt (S) − Nx (S) + [M (S), N (S)] = 0 is satisfied according to the Frobenius theorem. 2 The matrix S can be constructed as follows [17]: Let Λ be a n ×n diagonal matrix whose diagonal elements are λ1 , . . . , λn , where λ1 , . . . , λn are complex spectral parameters. H represents the n × n invertible matrix (ψ1 , . . . , ψn ), where ψi ’s (i = 1, . . . , n) satisfy the equations dψi = ω(λi )ψi (i = 1, . . . , n). Define S as HΛH −1 , and we will prove that S = HΛH −1 satisfies condition (30). For that, two lemmas will be given. Lemma 1. S = HΛH −1 satisfies the equation dS i = ω(S)S i − S i ω(S) .

(37)

Proof. Differentiating H = (ψ1 , . . . , ψn ), we have dH = (dψ1 , . . . , dψn ) = (ω(λ1 )ψ1 , . . . , ω(λn )ψn ) m m m    =( ωi λi1 ψ1 , . . . , ωi λin ψn ) = ωi HΛi . i=0

i=0

(38)

i=0

The differential of S can be calculated by use of the above equation, dS = d(HΛH −1 ) = dHΛH −1 + HΛdH −1 = dHΛH −1 − HΛH −1 dHH −1 m m   i −1 −1 =( ωi HΛ )ΛH − HΛH ( ωi HΛi )H −1 i=0 m 

=(

i=0 m  ωi HΛi H −1 )HΛH −1 − HΛH −1 ( ωi HΛi H −1 )

i=0 m 

=(

i=0

i=0 m 

ωi S i )S − S(

ωi S i ) = ω(S)S − Sω(S) .

(39)

i=0

Then we get dS i = dSS i−1 + SdSS i−2 + · · · + S i−1 dS ,

(40)

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where dSS i−1 = (ω(S)S − Sω(S))S i−1 = ω(S)S i − Sω(S)S i−1 , SdSS

i−2

= S(ω(S)S − Sω(S))S

i−2

= Sω(S)S

i−1

− S ω(S)S 2

(41a) i−2

,

(41b)

.. . S i−1 dS = S i−1 (ω(S)S − Sω(S)) = S i−1 ωS − S i ω(S) .

(41c)

From eqs. (40), (41a)–(41c), we obtain eq. (37). 2 Lemma 2. S = HΛH −1 satisfies the equation m 

dωi S i −

i=0

m 

ωi ∧ ω(S)S i = 0 .

(42)

i=0

Proof. The compatibility condition of equation dψ = ω(λ)ψ is Ω(λ) = dω(λ) − ω(λ) ∧ ω(λ) = 0. Ω(λ) can be expanded into the power series of the spectral parameter λ as

Ω(λ) =

m 

dωi λi −

i=0

m 

ωi λi ∧ ω(λ) = 0 .

(43)

i=0

Let λ be equal to λ1 , . . . , λn respectively, we obtain the following identities m 

dωi λi1 =

i=0

m 

ωi ∧ ω(λ1 )λi1 =

i=0

m 

ωi ∧ ωj λi+j , 1

(44a)

ωi ∧ ωj λi+j n .

(44b)

i,j=0

.. . m 

dωi λin =

i=0

m 

ωi ∧ ω(λn )λin =

i=0

m  i,j=0

From eqs. (44a)–(44b), we can derive m 

dωi λi1 ψ1

=

i=0

m 

ωi ∧ ωj λi+j 1 ψ1 ,

(45a)

ωi ∧ ωj λi+j n ψn .

(45b)

i,j=0

.. . m 

dωi λin ψn

=

i=0

m  i,j=0

Equations (45a)–(45b) can be rewritten in the matrix form m  i=0

dωi HΛi =

m 

ωi ∧ ωj HΛi+j .

i,j=0

Multiplying H −1 on both sides of eq. (46), we can get eq. (42).

2

(46)

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Based on Lemmas 1 and 2, we will prove the following proposition: Proposition 6. S = HΛH −1 satisfies condition (32). Proof. Via the direct calculation m m m    dω(S) = d( ωi S i ) = dωi S i − ωi ∧ dS i i=0

=

=

m  i=0

i=0

m 

m 

i=0

=

dωi S i −

i=0 m 

m 

dωi S i −

i=0

i=0

ωi ∧ (ω(S)S i − S i ω(S)) ωi ∧ ω(S)S i +

m 

ωi ∧ S i ω(S)

i=0

ωi ∧ S i ω(S) = ω(S) ∧ ω(S) ,

(47)

i=0

the proof is completed, where the second step is due to Lemma 1, while the last step is due to Lemma 2. 2 4. Conclusion In this paper, we have investigated the (1 + 1)-dimensional NLEE with Lax integrability with the aid of differential forms and exterior differentials. We have proved that the (1 + 1)-dimensional NLEE with Lax integrability must be completely integrable according to the Frobenius theorem and claimed that a (1 + 1)-dimensional NLEE is Lax integrable if and only if there exists a square matrix ω of 1-forms about dx and dt, such that the system Ω = dω − ω ∧ ω = 0 is equivalent to {αi = 0}. Prolongation condition (14) of the PS method has been seen to guarantee that Pfaff system (13) is completely integrable on the solution manifold {αi = 0}. Lax representation of the (1 + 1)-dimensional NLEE with Lax integrability has been interpreted as the parallel section equation on the dual vector bundle E ∗ with the connection matrix equal to −ω = −(M dx + N dt). Every (1 + 1)-dimensional NLEE with Lax integrability has been seen to be gauge equivalent to one another. The set of all the GTs of the same (1 + 1)-dimensional NLEE with Lax integrability has been shown to form a group which we call the gauge group. The set of all the DTs of the same (1 + 1)-dimensional NLEE with Lax integrability has also been shown to form a group which is a subgroup of the gauge group. If T = λI − S is a DT, then the matrix S must satisfy the equation dS + [S, ω(S)] = 0. We have proved that such an equation is completely integrable if and only if ω(S) satisfies the condition dω(S) − ω(S) ∧ ω(S) = 0. Defining S as HΛH −1 , we have proved that S satisfies that condition. Acknowledgments We express our sincere thanks to the editors, referees and all the members of our discussion group for their valuable comments. This work has been supported by the National Natural Science Foundation of China under Grant No. 11272023, and by the Open Fund of State Key Laboratory of Information Photonics and Optical Communications (Beijing University of Posts and Telecommunications) under Grant No. IPOC2013B008. References [1] M.J. Ablowitz, P.A. Clarkson, Soliton, Nonlinear Evolution Equations and Inverse Scattering, Cambridge Univ. Press, Cambridge, 1991.

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