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Generating of Nonisospectral Integrable Hierarchies via the Lie-Algebraic Recursion Scheme

School of Mathematics, China University of Mining and Technology, Xuzhou 221116, China
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Author to whom correspondence should be addressed.
Mathematics 2020, 8(4), 621; https://doi.org/10.3390/math8040621
Received: 20 February 2020 / Revised: 10 April 2020 / Accepted: 11 April 2020 / Published: 17 April 2020
(This article belongs to the Section Mathematical Physics)

Abstract

In the paper, we introduce an efficient method for generating non-isospectral integrable hierarchies, which can be used to derive a great many non-isospectral integrable hierarchies. Based on the scheme, we derive a non-isospectral integrable hierarchy by using Lie algebra and the corresponding loop algebra. It follows that some symmetries of the non-isospectral integrable hierarchy are also studied. Additionally, we also obtain a few conserved quantities of the isospectral integrable hierarchies.
Keywords: non-isospectral integrable hierarchy; Lie algebra; Hamiltonian structure; symmetry; conserved quantity non-isospectral integrable hierarchy; Lie algebra; Hamiltonian structure; symmetry; conserved quantity

1. Introduction

We have known that there exist two main approaches for constructing nonlinear systems integrable by the inverse scattering transform: the one of the Lax representation ( L t = [ A , L ] ) and the one of the zero curvature representation ( U t V x + [ U , V ] = 0 ) [1,2]. In [3], the authors introduced the spectral transform technique to solve certain classes of nonlinear evolution equations, and gave a thorough account also of the non-isospectral deformations of KdV-like equations [4,5]. Magri once proposed one approach for generating integrable systems [6], which was called the Lax-pair method [7,8]. Based on it, Tu [9] proposed a method for generating integrable Hamiltonian hierarchies by making use of a trace identity, which was called the Tu scheme [10,11]. Through making use of the Tu scheme, some integrable systems and the corresponding Hamiltonian structures as well as other properties were obtained, such as the works in [12,13,14,15,16]. It is well known that many different methods for generating isospectral integrable equations have been proposed [17,18,19]. However, as far non-isospectral integrable equations are concerned, fewer works were presented, as far as we know. In [20,21], the author proposed a method of constructing its corresponding non-isospearal λ t = λ n ( n 0 ) hierarchy of evolution equations closely related to τ -symmetries. Generally speaking, integrable systems correspond to the isospectral ( λ t = 0 ) case, and mastersymmetries of integrable systems correspond to the non-isospectral λ t = λ n ( n 0 ) case. In [22], the author adopted the Lenard series method to obtain some non-isospectral integrable hierarchies under the case λ t = λ m + 1 M , and found that the same spectral problem can produce two different hierarchies of soliton evolution equations.
In this article, we apply an efficient scheme to generating non-isospectral integrable hierarchies of evolution equations under the case where λ t = j = 0 n k j ( t ) λ n j . Obviously, this case is a generalized expression for the case λ t = λ n [23,24]. By taking different values of the parameters in the non-isospectral integrable hierarchies, we can obtain many integrable equations, such as the coupled equations. Under obtaining non-isospectral integrable systems, their properties including Darboux transformations, exact solutions, and so on, could be investigated; a lot of such work has been done, such as the papers [25,26,27,28,29,30,31,32,33,34].

2. A Non-Isospectral Integrable Hierarchy

In this section, we derive a non-isospectral integrable hierarchy by using the Lie algebra, and obtain a Hamiltonian construction of the hierarchy via the trace identity proposed by Tu [9]. In the following, the steps for generating non-isospectral integrable hierarchies of evolution equations present
Step 1: Introducing the spectral problems
ψ x = U ψ , U = R + u 1 e 1 ( n ) + + u q e q ( n ) ,
ψ t = V ψ , V = A 1 e 1 ( n ) + + A p e p ( n ) ,
λ t = i 0 k i ( t ) λ N i i ,
where the potential functions u 1 , , u q S (the Schwartz space), and R ( n ) , e 1 ( n ) , , e p ( n ) G ˜ satisfy that
(a)
R, e 1 , , e p are linear independent,
(b)
R is pseudoregular,
(c)
deg ( R ( n ) ) deg ( e i ( n ) ) , i = 1 , 2 , , p .
Step 2: Solving the following stationary zero curvature equation for A i , i = 1 , 2 , , p :
V x = U λ λ t + [ U , V ] .
It follows that one can get the compatibility condition of Equations (1) and (2)
U u u t + U λ λ t V x + [ U , V ] = 0 .
Equation (4) can be broken down into
V + , x ( n ) + U λ λ t , + ( n ) + [ U , V + ( n ) ] = V , x ( n ) U λ λ t , ( n ) [ U , V ( n ) ] ,
where
λ t , + ( m ) = λ N i m λ t λ t , ( m ) = i = μ m k i ( t ) λ N i m N i i + x , x = 0 , 1 , , N i 1 ; m < n .
Step 3: We search for a modified term Δ n so that, for
V ( n ) = ( λ N i n V ) + + Δ n = : V + ( n ) + Δ n ,
V x ( n ) + U λ λ t , + ( n ) + [ U , V ( n ) ] = B 1 e 1 + + B q e q ,
where B i ( i = 1 , 2 , , q ) C .
Step 4: The non-isospectral integrable hierarchies of evolution equations could be deduced via the non-isospectral zero curvature equation
U u u t + U λ λ t , + ( n ) V x ( n ) + [ U , V ( n ) ] = 0 .
Step 5: The Hamiltonian structures of the hierarchies Equation (7) are sought out according to the trace identity given by Tu [9]. We will show the specific calculation process in the following:
A basis of the Lie algebra A is given by
A = s p a n { h , e , f }
with h = 1 2 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 , e = 1 2 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0 , f = 1 2 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0 , and the corresponding loop algebra is taken by
A ˜ = s p a n { h ( n ) , e ( n ) , f ( n ) } ,
where h ( n ) = h λ 2 n , e ( n ) = e λ 2 n 1 , f ( n ) = f λ 2 n 1 . It is easy to find that the commutator of A ˜ is as follows:
[ h ( n ) , e ( m ) ] = f λ 2 n + 2 m 1 = f ( m + n ) , [ h ( n ) , f ( m ) ] = e ( m + n ) , [ e ( n ) , f ( m ) ] = h ( m + n 1 ) , m , n Z ,
where the gradations of h ( n ) , e ( n ) , and f ( n ) are given by
deg h ( n ) = 2 n , deg e ( n ) = 2 n 1 , deg f ( n ) = 2 n 1 , n Z .
Consider the following non-isospectral problems based on A ˜
ψ x = U ψ , U = h ( 1 ) + q e ( 1 ) + r f ( 1 ) = 1 2 λ 2 ( r + q ) λ 0 0 ( r q ) λ λ 2 0 0 0 0 λ 2 ( r + q ) λ 0 0 ( r q ) λ λ 2 ,
ψ t = V ψ , V = a h ( 0 ) + b e ( 1 ) + c f ( 1 ) = 1 2 a ( b + c ) λ 0 0 ( b + c ) λ a 0 0 0 0 a ( b + c ) λ 0 0 ( b + c ) λ a ,
where i ¯ 2 = 1 , a = i 0 a i λ 2 i , b = i 0 b i λ 2 i , c = i 0 c i λ 2 i .
It follows that we have
U λ λ t = 1 2 2 λ r + q 0 0 r q 2 λ 0 0 0 0 2 λ r + q 0 0 r q 2 λ i 0 k i ( t ) λ 2 i + 1 = i 0 k i ( t ) [ 2 h ( 1 i ) + q e ( 1 i ) + r f ( 1 i ) ] .
Furthermore, the following equation can be derived by taking λ t = i 0 k i ( t ) λ 1 2 i with Equation (6),
a i x = q c i + 1 r b i + 1 + 2 k i + 1 ( t ) , b i x = c i + 1 r a i + k i ( t ) q , c i x = b i + 1 q a i + k i ( t ) r ,
that is,
a i x = q b i x r c i x q 2 k i ( t ) + r 2 k i ( t ) + 2 k i + 1 ( t ) , c i + 1 = b i x + r a i q k i ( t ) , b i + 1 = c i x + q a i r k i ( t ) .
We take the initial values
b 0 = k 0 1 q , c 0 = k 0 1 r .
Then, Equation (11) admits that
a 0 = 2 k 1 ( t ) x + β 0 ( t ) ,
b 1 = 2 k 1 ( t ) q x , c 1 = 2 k 1 ( t ) r x ,
a 1 = k 1 ( t ) x ( q 2 r 2 ) + 2 k 2 ( t ) x + β 1 ( t ) ,
b 2 = k 1 ( t ) ( r + 2 x r x ) + q x ( k 1 ( t ) q 2 k 1 ( t ) r 2 + 2 k 2 ( t ) ) ,
c 2 = k 1 ( t ) ( q + 2 x q x ) + r x ( k 1 ( t ) q 2 k 1 ( t ) r 2 + 2 k 2 ( t ) ) ,
where β 0 ( t ) β 1 ( t ) = 0 is an integral constant. Denoting that
V + ( n ) = i = 0 n ( a i h ( n i ) + b i e ( n + 1 i ) + c i f ( n + 1 i ) ) ,
V ( n ) = i = n + 1 ( a i h ( n i ) + b i e ( n + 1 i ) + c i f ( n + 1 i ) ) ,
λ t , + ( n ) = i = 0 n K i ( t ) λ 2 n 2 i + 1 , λ t , ( n ) = i = n + 1 K i ( t ) λ 2 n 2 i + 1 ,
By using Equations (8) and (9), the gradations of the left-hand side of Equation (6) are derived as:
deg V + ( n ) = : ( 0 , 1 , 1 ) 0 , deg U λ λ t , + ( n ) = : ( 2 , 1 , 1 ) 1 , deg ( [ U , V + ( n ) ] ) = : ( 2 , 1 , 1 ; 0 , 1 , 1 ) 1 ,
which signifies that the minimum gradation of the left-hand side of Equation (6) is zero. Similarly, the gradations of the right-hand side of Equation (6) are also obtained as follows:
deg V ( n ) = : ( 2 , 1 , 1 ) 1 , deg U λ λ t , ( n ) = : ( 0 , 1 , 1 ) 0 , deg ( [ U , V ( n ) ] ) = : ( 2 , 1 , 1 ; 2 , 1 , 1 ) 1 ,
which indicates that the maximum gradation of the right-hand side of Equation (6) is 1. By taking these terms which have the gradations 0 and 1, one has
V , x ( n ) U λ λ t , ( n ) [ U , V ( n ) ] = b n + 1 f ( 1 ) c n + 1 e ( 1 ) q c n + 1 h ( 0 ) + r b n + 1 h ( 0 ) 2 K n + 1 ( t ) h ( 0 ) ,
that is,
V + , x ( n ) + U λ λ t , + ( n ) + [ U , V + ( n ) ] = b n + 1 f ( 1 ) c n + 1 e ( 1 ) q c n + 1 h ( 0 ) + r b n + 1 h ( 0 ) 2 K n + 1 ( t ) h ( 0 ) .
In what follows, we takes modified term Δ n = a n h ( 0 ) so that, for V ( n ) = V + ( n ) a n h ( 0 ) to obtain the non-isospectral integrable hierarchies, we have from Equation (13) that
V x ( n ) + U λ λ t , + ( n ) + [ U , V ( n ) ] = ( c n + 1 + r a n ) e ( 1 ) + ( b n + 1 + q a n ) f ( 1 ) .
Thus, Equation (7) admits the non-isospectral integrable hierarchy
u t n = q r t n = c n + 1 r a n b n + 1 q a n = b n x K n ( t ) q c n x K n ( t ) r = 0 0 c n b n K n ( t ) q r = : J 1 c n b n K n ( t ) q r ,
or
u t n = q r t n = r 1 r b n + 1 + ( 1 r 1 q ) c n + 1 2 r K n + 1 ( t ) x q 1 q c n + 1 + ( 1 + q 1 r ) b n + 1 2 q K n + 1 ( t ) x = 1 r 1 q r 1 r q 1 q 1 + q 1 r c n + 1 b n + 1 + 2 K n + 1 ( t ) x r q = : J 2 c n + 1 b n + 1 + 2 K n + 1 ( t ) x r q ,
where
J 1 = 0 0 , J 2 = 1 r 1 q r 1 r q 1 q 1 + q 1 r .
From Equation (11), we infer that
c n + 1 b n + 1 = r 1 r + r 1 q q 1 r q 1 q c n b n + K n ( t ) r 1 ( q 2 + r 2 ) q q 1 ( q 2 + r 2 ) r + 2 K n + 1 ( t ) x r q = : L c n b n + K n ( t ) Q + 2 K n + 1 ( t ) x R ,
where
L = r 1 r + r 1 q q 1 r q 1 q , Q = r 1 ( q 2 + r 2 ) q q 1 ( q 2 + r 2 ) r , R = r q .
Therefore, Equation (13) can be written as
u t n = q r t n = J 1 L n K 0 1 r K 0 1 q + J 1 i = 0 n 1 ( L i K n 1 i ( t ) Q ) + 2 J 1 i = 0 n 1 L i K n i ( t ) x R K n ( t ) q r = Φ n K 0 q r + i = 0 n 1 Φ i J 1 K n 1 i ( t ) Q + 2 i = 0 n 1 K n i ( t ) Φ i x q x r K n ( t ) q r ,
where
Φ = J 1 L J 1 1 = q x 1 q + q 2 q x 1 r q r + r x 1 q + q r r x 1 r r 2 .
When n = 1 , the non-isospectral integrable hierarchy Equation (16) becomes
q t = K 1 ( 2 x q x + q ) , r t = K 1 ( 2 x r x + r ) .
When n = 2 , the non-isospectral integrable hierarchy Equation (16) reduces to
q t = K 1 ( q 3 x q r 2 x + r + 2 r x x ) x + 2 K 2 ( q x ) x + K 2 ( 2 x q x + q ) , r t = K 1 ( r 3 x + r q 2 x + q + 2 q x x ) x + K 2 ( 2 x r x + r )
Furthermore, we focus on a format of Hamiltonian constructure of the hierarchy Equation (16) via the trace identity proposed by Tu [9]. Denoting the trace of the square matrices A and B by < A , B > = t r ( A B ) .
Equations (8) and (9) admit that
< V , U q > = b λ 2 , < V , U r > = c λ 2 , < V , U λ > = c r λ + 2 a λ b q λ ,
which can be substituted into the trace identity
δ δ u ( < V , U λ > ) = λ γ λ λ γ < V , U q > < V , U r >
gives rise to
δ δ u ( c r λ + 2 a λ b q λ ) = λ γ λ b λ 2 + γ c λ 2 + γ .
It follows that one can get the following equation by comparing the two sides of the above formula
δ δ u ( 2 a n q b n + r c n ) = ( 2 2 n + γ ) b n c n .
Inserting the initial values of Equations (11) into (21), we obtain γ = 0 . Hence, we have
b n c n = δ H n δ u = 0 1 1 0 c n b n = : M 1 c n b n ,
where
H n = 2 i ¯ a n q b n r c n 2 n 2 , M 1 1 = 0 1 1 0 , M 1 = 0 1 1 0 .
Hence, the hierarchy Equations (13) and (14) can be written as
u t n = q r t n = J 1 M 1 1 δ H n δ u K n ( t ) q r = J 2 M 1 1 δ H n + 1 δ u + 2 K n + 1 ( t ) x r q .
It is remarkable that, when K n ( t ) = K n + 1 ( t ) = 0 , Equation (22) is the Hamiltonian structure of the corresponding isospectral integrable hierarchy of Equation (16).

3. Discussion on Symmetries and Conserved Quatities

In this section, we consider the K symmetries and τ symmetries of the hierarchy Equation (16), and obtain some conserved quantities of the hierarchy Equation (16) from the obtained symmetries. The way to find K symmetries and τ symmetries comes from Li and Zhu [14], who applied the isospectral and non-isospectral integrable AKNS hierarchy to construct K symmetries and τ symmetries which constitute an infinite-dimensional Lie algebra. In the following, we show the specific process.
One can find that the Φ presented in Equation (17) satisfies
Φ [ Φ f ] g Φ [ Φ g ] f = Φ { Φ [ f ] g Φ [ g ] f } ,
for f , g S . Therefore, Φ is the hereditary symmetry of Equation (16). In addition, we can also prove the following relation holding:
Proposition 1.
Φ [ K 0 ] = [ K 0 , Φ ] ,
where K 0 = q x r x = u t 0 .
In fact,
Φ [ K 0 ] = q x 1 q + q 1 q x q x 1 r q 1 r x r x 1 q + r 1 q x r x 1 r r 1 r x ,
and thus
Φ [ K 0 ] = q x x 1 q + ( q 2 ) x + q x 1 q x q x x 1 r ( q r ) x q x 1 r x r x x 1 q + ( q r ) x + r x 1 q x r x x 1 r ( r 2 ) x r x 1 r x ,
K 0 Φ = 0 0 q x 1 q + q 2 q x 1 r q r + r x 1 q + q r r x 1 r r 2 = q x x 1 q + 3 q q x + q 2 2 q x x 1 r 2 q x r q r x q r 2 + r x x 1 q + 2 r x q + r q x + q r r x x 1 r 3 r r x r 2 .
Φ K 0 = q x 1 q + q 2 2 q x 1 r q r 2 + r x 1 q + q r r x 1 r r 2
We therefore verified that Equation (23) is correct. Owing to the Φ is a hereditary symmetry, one finds
Φ [ K m ] = [ K m , Φ ] ,
which means Φ is a strong symmetry, where K m = Φ m q x r x .
Proposition 2.
Φ [ x u ] + Φ ( x u ) ( x u ) Φ = H I ,
where u = q x r x , H = 0 0 and I is an identity matrix.
In fact,
Φ [ x u ] = A B C D ,
where
A = q x 1 q + x q x x 1 q + 2 x q x q + q x 1 x q x , B = ( q x 1 r + x q x x 1 r + x q x r + x q r x + q x 1 x r x ) , C = r x 1 q + x r x x 1 q + x r x q + x r q x + r x 1 x q x , D = ( r x 1 r + x r x x 1 r + 2 x r x r + r x 1 x r x ) .
Φ ( x u ) = x q 2 + x q q x q x 1 ( q + x q x ) x q r + + x 2 x r q x + q x 1 ( r + x r x ) x q r + + x 2 + x q r x r x 1 ( q + x q x ) x r 2 x r r x + r x 1 ( r + x r x ) ,
( x u ) Φ = x q x x 1 q + 3 x q q x + x q 2 x 2 x q x x 1 r 2 x r q x x q r x x q r x 2 + x r x x 1 q + 2 x q r x + x r q x + x q r x r x x 1 r 3 x r r x x r 2 ,
where
( x u ) [ σ ] = d d ϵ ϵ = 0 x ( q + ϵ σ 1 ) x x ( q + ϵ σ 2 ) x = x σ 1 σ 2 ( x u ) = x 0 0 x .
Thus, Equation (24) holds.
Proposition 3.
[ K 1 , x u ] = [ Φ u , x u ] = H u + K 1 ,
where u = q x r x , H = 0 0 and K 1 = Φ u .
In fact,
Φ u = r x x + 1 2 q x ( q 2 r 2 ) q r r x + q 2 q x q x x + 1 2 r x ( q 2 r 2 ) + q r q x r 2 r x ,
( Φ u ) = 1 2 ( q 2 r 2 ) + 3 q q x + q 2 r r x 2 q r ( q r ) x 2 + q r + ( q r ) x 1 2 ( q 2 r 2 ) 3 r r x r 2 + q q x ,
( Φ u ) x q x x r x = 1 2 ( q 2 r 2 ) ( x q x ) + 3 x q q x 2 + q 2 ( x q x ) x r r x q x + 2 ( x r x ) q r ( x r x ) x r x ( q r ) x 2 ( x q x ) + q r ( x q x ) + x q x ( q r ) x + 1 2 ( q 2 r 2 ) ( x r x ) 3 x r r x 2 r 2 ( x r x ) + x r x q q x ,
Then, we have
( x u ) [ Φ u ] = x ( r x x + 1 2 q x ( q 2 r 2 ) q r r x + q 2 q x ) x ( q x x + 1 2 r x ( q 2 r 2 ) + q r q x r 2 r x ) ,
[ Φ u , x u ] = ( Φ u ) [ x u ] ( x u ) [ Φ u ] = 0 0 q x r x + r x x + 1 2 q x ( q 2 r 2 ) q r r x + q 2 q x q x x + 1 2 r x ( q 2 r 2 ) + q r q x r 2 r x = H u + K 1 .
We therefore verified that Equation (25) is correct.
Proposition 4.
[ K m , K n ] = 0 , m , n = 0 , 1 , 2 ,
where K m = Φ m u , K n = Φ n u .
Proposition 5.
[ Φ m x u , x u ] = m Φ m 1 ( x u ) .
The proofs of Propositions 4 and 5 were presented in [23].
From the above results, one have
[ Φ m x u , Φ n x u ] = ( m n ) Φ m + n 1 ( x u ) , m = 0 , 1 , 2 , ; n = 0 , 1 , 2 , .
We can find that { Φ n u , Φ m x u } can not constitute a Lie algebra from Equation (25). However, { Φ n u , n = 0 , 1 , 2 , } and { Φ n x u , n = 0 , 1 , 2 , } constitute the infinite-dimensional Lie algebra, respectively based on the above analysis.
Now, we will derive some conserved quantities of Tu isospectral hierarchy
u t n = q r t n = Φ n q x r x .
Definition 1
([14,16]). If we have known the integrable hierarchy u t = K n ( u ) , then the v satisfied the following equation:
d v d t + K * v = 0 ,
which is called the conserved covariance, where K is the linearized operator of K, and K * denotes a conjugate operator of K .
Proposition 6
([13,16]). If σ is a symmetry of Equation u t = K n ( u ) , v is its conserved covariance, then we have
v σ d x = < v , σ > ,
which is independent of time t, that is, d d t < v , σ > = 0 .
Definition 2
([13,14,16]). If F f = < v , f > , for f S , then v is called the gradient of the functional F, which is denoted by v = δ F δ u .
Proposition 7
([16]). If v = v * , then v is the gradient of the following functional
F = 0 1 < v ( λ u ) , u > d λ .
According to the symbols above, we can deduce:
Proposition 8
([13,14]). If I is a conserved quality of the hierarchy u t = K n ( u ) , and the conserved covariance v satisfies
I K n = < v , K n > ,
then one has
I t + < v , K n > = 0 ,
that is,
v t + K n * v + v K n = 0 .
Therefore, we deduce the following conserved quantities related to the integrable hierarchy u t = K n ( u )
I m = 0 1 < x 1 K m ( λ u ) , u > d λ .
Moreover, a few conserved quantities are also derived for the integrable hierarchy Equation (27) as follows:
K 0 = Φ 0 u = q x r x = 0 1 1 0 r x q x ,
I 0 = 0 1 < x 1 K 0 ( λ u ) , u > d λ = 0 1 < [ 0 1 1 0 q x λ r x λ ] T , q r > d λ = ( q x r r x q ) d x ,
K 1 = Φ u = r x x + 1 2 q x ( q 2 r 2 ) q r r x + q 2 q x q x x + 1 2 r x ( q 2 r 2 ) + q r q x r 2 r x = 0 1 1 0 q x x 1 2 r x ( q 2 r 2 ) q r q x + r 2 r x r x x + 1 2 q x ( q 2 r 2 ) q r r x + q 2 q x ,
I 1 = 0 1 < [ 0 1 1 0 r x x λ + 1 2 q x ( q 2 r 2 ) λ 3 q r r x λ 3 + q 2 q x λ 3 q x x λ + 1 2 r x ( q 2 r 2 ) λ 3 + q r q x λ 3 r 2 r x λ 3 ] T , q r > d λ = [ 1 2 ( q q x x + r r x x ) + 1 8 ( q 2 r 2 ) ( q x r r x q ) ] d x ,
I k = < 0 1 1 0 K k ( λ u ) , q r > d λ .

Author Contributions

Formal analysis, Y.Z.; Funding acquisition, Y.Z.; Investigation, H.W.; Methodology, Y.Z.; Writing original draft, H.W.; Writing review and editing, H.W. All authors have read and agreed to the published version of the manuscript.

Funding

This work was supported by the National Natural Science Foundation of China (Grant No. 11971475).

Conflicts of Interest

The authors declare no conflict of interest.

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