AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 10.1155/2014/235159 235159 Research Article Conservation Laws and Self-Consistent Sources for an Integrable Lattice Hierarchy Associated with a Three-by-Three Discrete Matrix Spectral Problem Li Yu-Qing 1 Yin Bao-Shu 2,3 Dong Huanhe 1 College of Mathematics and Systems Science Shandong University of Science and Technology Qingdao 266590 China sdust.edu.cn 2 Institute of Oceanology China Academy of Sciences, Qingdao 266071 China 3 Key Laboratory of Ocean Circulation and Wave Chinese Academy of Sciences Qingdao 266071 China 2014 16102014 2014 25 06 2014 14 08 2014 19 10 2014 2014 Copyright © 2014 Yu-Qing Li and Bao-Shu Yin. 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 lattice hierarchy with self-consistent sources is deduced starting from a three-by-three discrete matrix spectral problem. The Hamiltonian structures are constructed for the resulting hierarchy. Liouville integrability of the resulting equations is demonstrated. Moreover, infinitely many conservation laws of the resulting hierarchy are obtained.

1. Introduction

Nonlinear integrable systems of the discrete version, treated as models of some physical phenomena, have attracted more and more attention in recent years. A well-known result is that a hierarchy of soliton equations can be generated through the isospectral compatibility condition of a pair of spectral problems . By using the discrete trace identity the Hamiltonian forms of the soliton equations can be constructed . Various methods have been developed to search for new integrable systems , integrable coupling systems , soliton solutions , and so on. However, the work of searching for new integrable systems associated with higher order matrix spectral problems is few. In [8, 9], the well-known method is used for the matrix spectral problems with 3 × 3 matrixes. In studying the integrability of discrete systems, the conservation laws play important roles. From the Lax pair of lattice soliton equations conservation laws can be deduced directly .

With the development of soliton theory, people began to focus on the soliton equations with self-consistent sources. Soliton equations with self-consistent sources are often used to express interactions between different solitary waves and are relevant to some problems of hydrodynamics, solid state physics, plasma physics, and so on. Many integrable coupling systems with self-consistent sources in continuous cases are obtained . In [15, 16], integrable discrete systems with self-consistent sources are given.

In the present paper, first, a new three-by-three discrete matrix spectral problem is proposed. By means of constructing a proper continuous time evolution equation and using the discrete zero curvature equation a hierarchy of lattice models is derived. Then the Hamiltonian forms of the resulting hierarchy are worked out by using the discrete trace identity. Further, the Liouville integrability of the discrete systems is demonstrated. Infinitely many conservation laws and self-consistent sources for the integrable systems are also obtained.

2. A New Three-by-Three Discrete Matrix Spectral Problem and Related Integrable Lattice Hierarchy

We first recall some presentations on a discrete integrable system. For a lattice function f n = f ( n ) , the shift operator E , the inverse of E , and the operator E k are defined by (1) E f n = f ( n + 1 ) , E - 1 f n = f ( n - 1 ) , E k f n = f ( n + k ) , k Z . Let u n = u ( n , t ) be the potential vector. The variational derivative, the Gateaux derivative, the inner product, and the Poisson bracket are defined by (2) H ~ n = n z H n , δ H ~ n δ u n = m z E - m ( H n u n + m ) , (3) P ( u n ) [ v n ] = ε P ( u n + ε v n ) | ε = 0 , f n , g n = n z ( f n , g n ) R 3 , (4) { f n , g n } J = δ f n δ u n , J δ g n δ u n = n z ( δ f n δ u n , J δ g n δ u n ) R 3 , where P can be a vector function or an operator, f n and g n are vector functions, ( f n , g n ) R 3 denotes the standard inner product of f n and g n in the Euclidean space R 3 , and J is a Hamiltonian operator. A system of evolution equations u n t m = K m ( u n ) is called a Hamiltonian system, if there is a Hamiltonian operator J and a sequence of conserved functionals H ~ n ( m ) , m = 1,2 , , such that (5) u n t m = K m ( u n ) = J δ H ~ n ( m ) δ u n . The functional H ~ n ( m ) is called a Hamiltonian functional of the system, and we say that the system possesses Hamiltonian structures. As to a discrete Hamiltonian system, if there are infinitely many involutive conserved functionals, we say it is a Liouville integrable discrete Hamiltonian system.

In this paper, we consider the 3 × 3 discrete matrix spectral problem: (6) E φ n = U n ( u n , λ ) φ n , U n ( u n , λ ) = ( 0 1 0 0 0 1 p n r n w n + λ ) , in which u n = ( p n , r n , w n ) T is the potential, p n = p ( n , t ) , r n = r ( n , t ) , and w n = w ( n , t ) are real functions defined over Z × R , λ is a spectral parameter, λ t = 0 , and φ n = ( φ 1 ( n ) , φ 2 ( n ) , φ 3 ( n ) ) T is the eigenfunction.

To get a hierarchy of lattice models associated with (6), first we solve the stationary discrete zero curvature equation: (7) ( E Γ n ) U n - U n Γ n = 0 , where (8) Γ n = ( a n b n c n d n e n f n g n h n - a n - e n ) . Equation (8) gives (9) p n c n + 1 = d n , r n c n + 1 + a n + 1 = e n , b n + 1 - f n + c n + 1 ( w n + λ ) = 0 , p n f n + 1 = g n , r n f n + 1 + d n + 1 - h n = 0 , f n + 1 ( w n + λ ) + e n + 1 + a n + e n = 0 , - r n d n - ( w n + λ ) g n = p n ( a n + a n + 1 + e n + 1 ) , - r n ( a n + 1 + e n + 1 + e n ) + g n + 1 = p n b n + h n ( w n + λ ) , - r n f n + h n + 1 = p n c n + ( a n + 1 + e n + 1 - a n - e n ) ( w n + λ ) . Substituting the expansions (10) a n = m = 0 a n ( m ) λ - m , b n = m = 0 b n ( m ) λ - m , c n = m = 0 c n ( m ) λ - m , d n = m = 0 d n ( m ) λ - m , e n = m = 0 e n ( m ) λ - m , f n = m = 0 f n ( m ) λ - m , g n = m = 0 g n ( m ) λ - m , h n = m = 0 h n ( m ) λ - m into (9), we get the recursion relation (11) p n c n + 1 ( m ) = d n ( m ) , a n + 1 ( m ) - e n ( m ) + r n c n + 1 ( m ) = 0 , b n + 1 ( m ) - f n ( m ) + w n c n + 1 ( m ) = - c n + 1 ( m + 1 ) , g n ( m ) - p n f n + 1 ( m ) = 0 , r n f n + 1 ( m ) + d n + 1 ( m ) - h n ( m ) = 0 , w n f n + 1 ( m ) + ( e n + 1 ( m ) + e n ( m ) + a n ( m ) ) = - f n + 1 ( m + 1 ) , - p n ( a n ( m ) + a n + 1 ( m ) ) - w n g n ( m ) - p n e n + 1 ( m ) - r n d n ( m ) = g n ( m + 1 ) , - r n ( a n + 1 ( m ) + e n ( m ) + e n + 1 ( m ) ) - w n h n ( m ) + g n + 1 ( m ) - p n b n ( m ) = h n ( m + 1 ) , h n + 1 ( m ) + w n ( a n ( m ) - a n + 1 ( m ) + e n ( m ) - e n + 1 ( m ) ) - p n c n ( m ) - r n f n ( m ) = a n + 1 ( m + 1 ) - a n ( m + 1 ) + e n + 1 ( m + 1 ) - e n ( m + 1 ) and the initial requirement (12) a n + 1 ( 0 ) + e n + 1 ( 0 ) = a n ( 0 ) + e n ( 0 ) , g n ( 0 ) = h n ( 0 ) = d n ( 0 ) = f n + 1 ( 0 ) = c n + 1 ( 0 ) = 0 .

The initial values are taken as b n ( 0 ) = c n ( 0 ) = f n ( 0 ) = 0 , so we can get e n ( 0 ) = a n + 1 ( 0 ) , a n ( 0 ) = e n + 1 ( 0 ) , and then we take the initial value a n ( 0 ) = e n ( 0 ) = - 1 / 3 . Note that the definition of the inverse operator of D = ( E - 1 ) does not yield any arbitrary constant in computing a n ( m ) and e n ( m ) , m 1 . Thus, the recursion relation (11) uniquely determines a n ( m ) , b n ( m ) , c n ( m ) , d n ( m ) , e n ( m ) , f n ( m ) , g n ( m ) , and h n ( m ) , m 1 , and the first few quantities are given by (13) a n ( 1 ) = b n ( 1 ) = c n ( 1 ) = d n ( 1 ) = e n ( 1 ) = 0 , f n ( 1 ) = 1 , g n ( 1 ) = p n , h n ( 1 ) = r n , a n ( 2 ) = 0 , b n ( 2 ) = 1 , c n ( 2 ) = 1 , d n ( 2 ) = p n , e n ( 2 ) = r n , f n ( 2 ) = - w n , g n ( 2 ) = - p n w n , h n ( 2 ) = p n + 1 - w n r n , . Let (14) V n ( m ) = ( i = 0 m a n ( i ) λ m - i i = 0 m b n ( i ) λ m - i i = 0 m c n ( i ) λ m - i i = 0 m d n ( i ) λ m - i i = 0 m e n ( i ) λ m - i i = 0 m f n ( i ) λ m - i i = 0 m g n ( i ) λ m - i i = 0 m h n ( i ) λ m - i - i = 0 m a n ( i ) λ m - i - i = 0 m e n ( i ) λ m - i ) , m m m m m m m m m m m m m m m m m m m m m m v v c c c c c c m 0 , and then take a modification (15) η n ( m ) = ( f n - 1 ( m + 1 ) c n ( m + 1 ) 0 0 f n ( m + 1 ) 0 0 0 0 ) , m 0 . Now we set (16) V n [ m ] = V n ( m ) + η n ( m ) , m 0 . Then we introduce the auxiliary spectral problems associated with the spectral problem (6): (17) φ n t m = V n [ m ] φ n , m 0 . The compatibility conditions of (6) and (17) are (18) U n t m = ( E V n [ m ] ) U n - U n V n [ m ] , m 0 , which give rise to the following hierarchy of integrable lattice equations: (19) p n t m = g n ( m + 1 ) - p n f n - 1 ( m + 1 ) , r n t m = h n ( m + 1 ) - r n f n ( m + 1 ) - p n c n ( m + 1 ) , w n t m = ( a n + 1 ( m + 1 ) - a n ( m + 1 ) ) + ( e n + 1 ( m + 1 ) - e n ( m + 1 ) ) , ( m 0 ) .

So the discrete spectral problem (6) and (17) constitute the Lax pairs of (19), and (19) are a hierarchy of Lax integrable lattice equations. It is easy to verify that the first lattice equation in (19), when m = 1 , under t 1 t , is (20) p n t = p n ( w n - 2 - w n ) , r n t = r n ( w n - 1 - w n ) + ( p n + 1 - p n ) , w n t = r n + 1 - r n . The Lax pair of (20) is (6) and the time evolution law for φ n is as follows: (21) φ n t = V n [ 1 ] φ n = ( - λ 3 - w n - 2 + 2 3 1 0 0 - λ 3 + 2 3 - w n - 1 1 p n r n 2 3 λ + 2 3 ) φ n .

Now we would like to derive the Hamiltonian structures for (19).

Set R n = Γ n U n - 1 ; through a direct calculation, we get (22) Tr ( R n U n λ ) = g n p n , Tr ( R n U n p n ) = a n p n , Tr ( R n U n r n ) = d n p n , Tr ( R n U n w n ) = g n p n . By the discrete trace identity  (23) δ δ u n i n Z Tr ( R n U n λ ) = λ - ε ( λ ) λ ε Tr ( R n U n u n i ) , i = 1,2 , 3 , we have (24) ( δ δ p n , δ δ r n , δ δ w n ) T n Z ( g n p n ) = λ - ε ( λ ) λ ε ( a n p n d n p n g n p n ) , where ε is a constant to be found.

By substituting (25) a n = m = 0 a n ( m ) λ - m , d n = m = 0 d n ( m ) λ - m , g n = m = 0 g n ( m ) λ - m into (24) and equating the coefficients of λ - m - 1 , we have (26) ( δ δ p n , δ δ r n , δ δ w n ) T n Z ( g n ( m + 1 ) - m p n ) = ( a n ( m ) p n d n ( m ) p n g n ( m ) p n ) , m > 0 . Now we can rewrite (19) as follows: (27) ( p n r n w n ) t m = J ( a n ( m ) p n d n ( m ) p n g n ( m ) p n ) , where (28) J = ( J 11 J 12 J 13 J 21 J 22 J 23 J 31 J 32 J 33 ) , J 11 = p n ( E - 1 - E + E - 2 - E 2 ) p n , J 12 = p n ( E - 2 + E - 1 - E - 1 ) r n , J 13 = p n ( E - 2 - 1 ) w n , J 21 = r n ( E - 1 + 1 - E - E 2 ) p n , J 22 = r n ( E - 1 - E ) r n + p n E - 1 w n - w n E p n , J 23 = E p n - p n E - 2 - r n ( 1 - E - 1 ) w n , J 31 = w n ( 1 - E 2 ) p n , J 32 = E 2 p n - p n E - 1 + w n ( 1 - E ) r n , J 33 = E r n - r n E - 1 . It is easy to verify that the operator J is a Hamiltonian operator. So the lattice systems (19) can be rewritten as the hierarchy of discrete Hamiltonian equation (27). Set (29) H ~ n ( m ) = n Z g n ( m + 1 ) - m p n , m 1 , (30) δ H ~ n ( m ) δ u n = ϕ n δ H ~ n ( m - 1 ) δ u n . From the recursion relation (11) we can get the recursion operator ϕ n in (30): (31) ϕ n = ( ϕ 11 ϕ 12 ϕ 13 ϕ 21 ϕ 22 ϕ 23 ϕ 31 ϕ 32 ϕ 33 ) , ϕ 11 = 1 p n ( E 2 - 1 ) - 1 w n ( 1 - E 2 ) p n + 1 p n ( E + 1 ) - 1 r n ( E p n + p n E - 1 ) - 1 r n ( E - 1 ) p n , ϕ 12 = 1 p n ( E 2 - 1 ) - 1 [ E 2 p n - p n E - 1 + w n ( 1 - E ) r n ] - 1 p n ( E + 1 ) - 1 r n ( E p n + p n E - 1 ) - 1 × ( p n E - 1 w n - w n E p n ) , ϕ 13 = 1 p n ( E 2 - 1 ) - 1 ( E r n - r n E - 1 ) - 1 p n ( E + 1 ) - 1 r n ( E p n + p n E - 1 ) - 1 ( E p n - p n E - 2 ) , ϕ 21 = - ( E p n + p n E - 1 ) - 1 r n ( E - 1 ) p n , ϕ 22 = ( E p n + p n E - 1 ) - 1 ( p n E - 1 w n - r n E p n ) , ϕ 23 = ( E p n + p n E - 1 ) - 1 ( E p n - p n E - 2 ) , ϕ 31 = - p n - E p n - E 2 p n , ϕ 32 = - E r n - r n , ϕ 33 = - w n .

Therefore, we have (32) M = J ϕ n = ( 0 0 p n ( 1 - E - 2 ) 0 E p n - p n E - 1 r n ( 1 - E - 1 ) ( E 2 - 1 ) p n ( E - 1 ) r n 0 ) . It is easy to verify that M * = - M ; moreover, we can prove that (33) { H ~ n ( m ) , H ~ n ( l ) } J = 0 , m , l 1 , ( H ~ n ( m ) ) t l = δ H ~ n ( m ) δ u n , u n t l = δ H ~ n ( m ) δ u n , J δ H ~ n ( l ) δ u n = { H ~ n ( m ) , H ~ n ( l ) } J = 0 , m , l 1 . So we get the following.

Proposition 1.

{ H ~ n ( m ) } m 1 defined by (29) forms an infinite set of conserved functionals of the hierarchy (19), and H ~ n ( m ) , m 1 , are involution in pairs with respect to the Poisson bracket (4).

Theorem 2.

The lattice equations in (19) are all discrete Liouville integrable Hamiltonian systems.

3. Infinitely Many Conservation Laws

We can get the following alternative form from (6) and (21): (34) φ 3 ( n + 1 ) = p n φ 3 ( n - 2 ) + r n φ 3 ( n - 1 ) + ( w n + λ ) φ 3 ( n ) , (35) φ 3 t ( n ) = p n φ 3 ( n - 2 ) + r n φ 3 ( n - 1 ) + 2 3 ( 1 + λ ) φ 3 ( n ) . Set (36) θ n = θ ( n ) = φ 3 ( n ) φ 3 ( n + 1 ) , and we can obtain (37) - ( ln θ n ) t = ( E - 1 ) [ p n θ n - 2 θ n - 1 + r n θ n - 1 + 2 3 ( 1 + λ ) ] . Equation (34) can be written as follows: (38) λ θ n = 1 - p n θ n - 2 θ n - 1 θ n - r n θ n - 1 θ n - w n θ n . Then, expanding θ n in the power series of 1 / λ : (39) θ n = j = 1 θ n ( j ) λ - j , and substituting it into (38), we can obtain all the coefficients θ n ( j ) . Substitute them into (37) and due to θ n ( 1 ) = 1 and λ t = 0 , we can get the following fact: (40) [ k = 1 ( - 1 ) k 1 k ( j = 1 θ n ( j + 1 ) λ j ) k ] t = ( E - 1 ) × [ p n j = 1 θ n - 2 ( j ) λ j j = 1 θ n - 1 ( j ) λ j + r n j = 1 θ n - 1 ( j ) λ j + 2 3 ( 1 + λ ) ] , from which an infinite number of conservation laws can be determined by equating the powers of 1 / λ . The following are the first three of them: (41) w n t = ( E - 1 ) r n , ( - r n - 1 2 w n 2 ) t = ( E - 1 ) ( - w n - 1 r n + p n ) , ( p n - r n w n - 1 - w n r n + 1 3 w n 3 ) t = ( E - 1 ) [ r n ( r n - 1 + w n - 1 2 ) - p n ( w n - 1 + w n - 2 ) ] . We can get other conservation laws in the hierarchy (19) similarly.

4. Self-Consistent Sources for the Lattice Hierarchy (<xref ref-type="disp-formula" rid="EEq15">19</xref>)

In this section, we will construct the lattice hierarchy (19) with self-consistent sources. Consider the auxiliary linear problems (42) E ( φ 1 j φ 2 j φ 3 j ) = U n ( u n , λ j ) ( φ 1 j φ 2 j φ 3 j ) , ( φ 1 j φ 2 j φ 3 j ) t m = V n [ m ] ( u n , λ j ) ( φ 1 j φ 2 j φ 3 j ) , and, based on the results in , we show the following equation: (43) δ H ~ n ( m ) δ u n + j = 1 N δ λ j δ u n = 0 , where (44) δ λ j δ u n = 1 2 Tr ( ψ j U ( u n , λ j ) u n ) , ψ j = ( φ 1 j φ 1 j φ 1 j φ 2 j φ 1 j φ 3 j φ 2 j φ 1 j φ 2 j φ 2 j φ 2 j φ 3 j φ 3 j φ 1 j φ 3 j φ 2 j φ 3 j φ 3 j ) , j = 1,2 , , N . Through a direct computation, we obtain the lattice hierarchy with self-consistent sources as follows: (45) ( p n r n w n ) t m = J ( δ H ~ n ( m ) δ p n + j = 1 N δ λ j δ p n δ H ~ n ( m ) δ r n + j = 1 N δ λ j δ r n δ H ~ n ( m ) δ w n + j = 1 N δ λ j δ w n ) = J ( δ H ~ n ( m ) δ p n δ H ~ n ( m ) δ r n δ H ~ n ( m ) δ w n ) + J ( 1 2 j = 1 N φ 1 j φ 3 j 1 2 j = 1 N φ 2 j φ 3 j 1 2 j = 1 N φ 3 j φ 3 j ) , m 1 .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by the Project Global Change and Air-Sea Interaction (GASI-03-01-01-02), Nature Science Foundation of Shandong Province of China (no. ZR2013AQ017), Science and Technology Plan Project of Qingdao (no. 14-2-4-77-jch), Open Fund of the Key Laboratory of Ocean Circulation and Waves, Chinese Academy of Science (no. KLOCAW1401), and Open Fund of the Key Laboratory of Data Analysis and Application, State Oceanic Administration (no. LDAA-2013-04).

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