Self consistent sources and conservation laws for super broer kaup kupershmidt equation hierarchy

Mathematical Computation June 2013, Volume 2, Issue 2, PP.24-31

Self-Consistent Sources and Conservation Laws for Super Broer-Kaup-Kupershmidt Equation Hierarchy Sixing Tao 1# School of Mathematics and Information Science, Shangqiu Normal University, 476000, China #Email: taosixing@163.com

Abstract Based upon the basis of Lie super algebra B(0,1), the super Broer-Kaup-Kupershmidt equation hierarchy with self-consistent sources was presented. Furthermore, the infinite conservation laws of above hierarchy were given.

Keywords: Super Broer-Kaup-Kupershmidt Equation Hierarchy; Self-Consistent Sources; Conservation Laws; Lie Super Algebra

1 INTRODUCTION Soliton equations with self-consistent sources have received growing attention in recent years. Physically, the sources may result in solitary waves with a non-constant velocity and therefore leading to a variety of dynamics of physical models. For applications, these kinds of systems usually used to describe interactions between different solitary waves are relevant to some problems of hydrodynamics, solid state physics, plasma physics, etc. In Refs [1, 2], Ma, Strampp and Fuchssteiner systematically applied explicit symmetry constraint and binary nonlinearization of Lax pairs to generate the solution equation with sources. Furthermore, Ma presented the soliton solutions of the Schrรถdinger equation with self-consistent sources in Ref [3]. The discrete case of using variational derivatives in generating sources has been discussed in Ref [4].

It is known that conservation laws play an important role in discussing the integrability for soliton equations. Since the discovery of infinite conservation laws for KdV equation by MGK in Ref [5], lots of methods have been developed to detect them. This should be mainly due to the contribution of Wadati et al in Ref. [6]. Conservation laws also play an important part in mathematics as well.

With the development of soliton theory, super integrable systems associated with fermi variables have been receiving growing attention. Various methods have been developed to search for new super integrable systems, Lax pairs, soliton solutions, symmetries and conservation laws, etc in Refs [7-18]. In 1997, Hu proposed the supertrace identity and applied it to establish the super Hamiltonian structures of super-integrable systems in Ref [7]. Then Professor Ma gave a systematic proof of super trace identity and presented the super Hamiltonian structures of super AKNS hierarchy and super Dirac hierarchy for application in Ref [8]. The super Broer-Kaup-Kupershmidt hierarchy and its super-Hamiltonian structure were considered in Ref [9]. Recently, Yu et al considered the binary nonlinearization of the super AKNS hierarchy under an implicit symmetry constraint in Ref [10] and the Bargmann symmetry constraint and binary nonlinearization of the super Dirac systems in Ref [11]. Meanwhile, various systematic methods on classical integrable systems have been developed to obtain exact solutions of the super integrable such as the inverse transformations, the Backlund and Darboux transformations, the bilinear transformation of Hirota and other ones in Refs [19-21].

This paper is organized as follows. In section 2, the method to establish super integrable soliton hierarchy with selfconsistent sources by means of Lie super algebra B (0, 1) was presented. For application, the super Broer-KaupKupershmidt hierarchy with self-consistent sources was obtained in Section 3. In Section 4, the infinite conservation laws of the Broer-Kaup-Kupershmidt hierarchy were given.

- 24 www.ivypub.org/mc

2 A KIND OF SUPER INTEGRABLE EQUATION HIERARCHY WITH SELF-CONSISTENT SOURCES In the following, a basis of Lie super algebra B(0,1) was considered[8]

1 0 0 0 1 0 0 0 0 0 0 1 0 0 0           e1   0 1 0  , e2   0 0 0  , e3   1 0 0  , e4   0 0 0  , e5   0 0 1  . 0 0 0 0 0 0 0 0 0  0 1 0  1 0 0          

(1)

the loop algebra B(0,1) was introduced as follows

B(0,1)  {A | A  R( )  B(0,1)}. n where the loop algebra B(0,1) is defined by span { | n  0, A  B(0,1)}.

(2)

Consider the auxiliary linear problem

 1   1   1   1  5         (3) ui ei ( ),  2   V (u,  )  2  ,  2   U (u,  )  2  ,U (u,  )  e0 ( )   i 1          3 x  3  3  tn  3 T where u  (u1 , , us ) ,U (u,  )  u1e1   u p e p , ui  ui ( x, t )(i  1, 2, , p), i  i ( x, t ) are field variables defined on x  R, t  R, ei  ei ( )  B(0,1).

From the spectral problem (3), the compatibility condition gives rise to the well-known zero curvature equation

Utn  Vx  [U ,V ]  0, n  1, 2,

,.

(4)

The general scheme to search for the consistent V (n) and generate a hierarchy of nonlinear equations was proposed as follows[8]. We solve the equation  Am Bm m     m m  (5) Vx  [U ,V ],V   Vm     Cm  Am  m  , m0 m0    0  m  m  (n) and search for n (u,  )  B(0,1), such that V can be constructed by n

V ( n )   Vm  n m   n (u,  ),

(6)

m0

and

  n1  n 2  n 4     n (u,  )    n3  n1  n5  ,    n5  n 4 0  where ni (1  i  5) are linear functions of Am , Bm,Cm , m ,  m .

(7)

We consider the super trace identity of super integrable systems[7,8]

   U    U    Str  V   Str  V .      u        u 

(8)

a scalar H  H (u,  ) was defined by the equation

  U  H  Str  V , H  H m (u,  )  m .      m0

(9)

The sets {H m } proves the conserved densities of (4). The Hamilltonian form with H n 1 can be written as

utn  J

 H n1 , n  1, 2, . u - 25 www.ivypub.org/mc

(10)

 Hn  H n1 L  u u

 Ln

 H0 , n  1, 2, . u

where L is a recursion operator and J is a symplectic operator, and

 u





  u1

(11)

,

, u p



T

.

According to (3) and (5), the auxiliary linear problem was in consideration. For N distinct following systems resulted from (1)

 j , j  1, , N , the

 1 j   1 j   1 j      5    2 j   U (u,  j )  2 j    ui ei ( j )  2 j  ,      i 1    3 j x  3j   3j 

(12a)

 1 j   1 j   1 j      n   (n) nm  2 j   V (u,  j )  2 j     Vm (u,  j ) j   n (u,  j )   2 j  .         m 0  3j   3j   3 j 

(12b)

tn

Based on the results in Ref.[21], it was showed that the following equations N  j  Hk   j  0, u u j 1 where  j are constants. Equation (13) determines a finite dimensional invariant set for the flows (11).

For (12a), it is known that

 j  U (u,  j )   Str  j   Str  j ei ( j )  , i  1, ,5. u u  

(13)

(14)

where Str denotes the super trace of a matrix and

 1 j2 j   j   22j  2 j3 j 

12j 1 j2 j 1 j3 j

1 j3 j   2 j3 j  , j  1, , N .

(15)

0 

According to (13), for a specific k0  n0 , it was demanded that N  N  H k0 j    Str  j ei ( j ) .  ui j 1  ui j 1

(16)

From (10) and (13), a kind of super integrable hierarchy with self-consistent sources can be presented as follows N  N  H H j j (17) ui ,tn  J n1  J   JLn 1  J  , n  1, 2, .  ui  u  u  u j 1 j 1 i i i

3 THE SUPER BROER-KAUP-KUPERSHMIDT EQUATION HIERARCHY WITH SELFCONSISTENT SOURCES The super Broer-Kaup-Kupershmidt spectral problem associated with Lie super algebra B(0,1)is given by [9] r s    r  1    s x  U ,U   1   r   , u    ,    2  ,        0      3  

where

 is a spectral parameter, r and s are even variables,  and  are odd variables[9].

Taking

A B    V  C A  ,    0    the co-adjoint equation associated with (18) Vx  [U ,V ] gives - 26 www.ivypub.org/mc

(18)

Ax   B  sC     , Bx  2 B  2rB  2sA  2 , Cx  2C  2rC  2 A  2 ,

 x   A   B  r     s ,  x   A   C    r   .

If we set

A   Ai  i , B   Bi  i , C   Ci  i ,    i i ,   i i , i 0

i 0

then (19) is equivalent to

i 0

i 0

(19)

(20)

i 0

Bi 1  sAi  12 Bi , x  rBi  i ,   Ci 1  Ai  rCi  12 Ci , x   i ,   i 1   Ai   Bi  r i  i , x  s i ,   i 1   Ai   Ci  i  r i   i , x ,   Ai 1, x   Bi 1  sCi 1  i 1   i 1 , i  0.

(21)

which results in the recurrence relations

 2 Ai 1 , Ci 1 , 2 i 1 , 2 i 1 T  L  2 Ai , Ci , 2 i , 2 i T ,  Ai   1 ( Bi  sCi  i   i ), i  0. 

(22)

where

 12    1r  1s  s  1  12   1  12     1 1  12   r 0 2 2  .

L    2 r   1   2s s   r      Upon choosing the initial conditions

(23)

B0  C0  0   0  0, A0  1,

all other Ai , Bi , Ci , i , i (i  1) can be worked out by the recurrence relations (22). The first few sets are as follows:

A1  0, B1  s, C1  1, 1   , 1   , A2   12 s   ,

B2  12 sx  rs, C2  r ,  2   x  r ,  2    x  r  , A3   14 sx  rs   x   x   2r , B3  14 sxx  12 rx s  rsx  12 s 2  s  r 2 s   x , C3  12 rx  12 s    r 2   x ,

3   xx  2r x  rx  12 s  12 sx   s x  r 2 ,  3   xx  rx   2r  x  12 s   x  r 2  .

Let us associate the problem (18) with the following auxiliary problem

t  V ( n ) , n

(24)

with

 Ai     Ci i 0   i n

V

(n)

Bi  Ai  i

i   Cn 1 0 0   n i    i    0 Cn 1 0  .  0 0  0 0  

The compatible conditions of the spectral problem (18) and the auxiliary problem (24) are

Utn  Vx( n )  [U ,V ( n) ]  0, - 27 www.ivypub.org/mc

(25)

which refer to the super Broer-Kaup-Kupershmidt soliton hierarchy utn  Kn   Cn1, x , 2 An1, x  2n1  2 n1 , n1   Cn1 ,  n1   Cn1  .

T

(26)

Here utn  K n in (26) is called the n-th Broer—Kaup-Kupershmidt flow of this hierarchy.

Using the super trace identity

   U    U    Str  V   Str  V .     u         u  where Str means the super trace[7,8], we have

(27)

 2 Ai 1    (28)  Ci 1    H , H   2 Ai2 dx, i  0.  2 i 1   u i i  i 1    2 i 1  Therefore, the super coupled Broer-Kaup-Kupershmidt soliton hierarchy (26) can be written as the following super Hamiltonian form:

utn  J

 Hn , u

(29)

where  0   J   0   0



0

0





0



 12

0     .  12   0 

is a super symplectic operator, and H n is given by (28).

The first non-trivial nonlinear of super Broer-Kaup-Kupershmidt hierarchy is given by its second flow

 rt2   12 rxx  12 sx  ( ) x  2rrx   xx ,  st2  12 sxx  2(rs) x  2 x  2s x ,   3 1  t2   xx  2r x  2 rx  2 sx   s x   x ,  t2    xx  12 rx   2r  x   x .  (2) which possesses a Lax pair of U defined in (18) and V defined by

  2  12 rx  r 2   x s  12 sx  rs    x  r    (2) 2 1 2 V   r   2 rx  r   x    x  r   .     x  r      x  r 0  

(30)

(31)

Next the super Broer-Kaup-Kupershmidt hierarchy with self-consistent sources will be established. Consider the linear system

For the system (32), the following is obtained

H u

 1 j   1 j     r s    1 j          2 j   U  2 j    1   r    2 j  ,        0   3 j   3 j x  3j  

(32a)

 1 j   1 j   A B    1 j          2 j   V  2 j    C  A    2 j  .         0      3 j   3 j tn  3j  

(32b)

  j 1  uj has been taken into account in the Lie super algebra B(0,1) and the N



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 Str( j   j  Str( j   u  Str( j   Str( j

where i  (i1 ,

U r U s U  U 

)   2 1 ,  2   )   2 , 2  )   2  2 ,  3   )   2 1 ,  3

  .   

(33)

, iN )T (i  1, 2, 3).

According to the results in (17), the super coupled Burgers hierarchy with self-consistent sources is presented as follows:  2 1 ,  2 r  0       2 , 2 s n 1   utn   JL J  2  2 ,  3    2         tn  2   2 1 , 3

  .   

(34)

The first nontrivial integrable super coupled Burgers hierarchy with self-consistent sources is its second flow  rt2   12 rxx  12 sx  ( ) x  2rrx   xx    2 ,  2 ,  1  st2  2 sxx  2(rs) x  2 x  2s x  2 1 ,  2  2  2 ,  3  2 1 ,  3 ,  3 1 1   t2   xx  2 r x  2 rx  2 sx   s x   x    2 ,  2  1 ,  3 ,  t2    xx  12 rx   2r  x   x    2 ,  2   2 ,  3 . 

(35)

when     0, and it is the well known nonlinear Broer-Kaup-Kupershmidt equation with self-consistent sources.

So system (34) is a novel super integrable equation hierarchy.

4 CONSERVATION LAWS FOR THE SUPER BROER-KAUP-KUPERSHMIDT EQUATION HIERARCHY Then, conservation laws of the super coupled Burgers equation will be constructed and the variables are introduced:   (36) K  2 ,G  3 , 1 1 where p( K )  0, p(G)  1. From (12), we have

 K x  1  2(  r ) K   G  sK 2   KG,  2  Gx     K  qG  (  r ) KG   G .

We expand K , G in powers of

(37)

 1 as follows 



j 1

j 1

K   k j   j , G  g j   j , where p(k j )  0, p( g j )  1. Substituting (38) into (37) and comparing the coefficients of the same powers of we obtain

k1  12 , g1   , k2   12 r , g 2   12    x  r  , k3  14 rx  12 r 2  12  x  18 s, g3  12  x   xx  rx   2r  x  r  r 2  . and a recursion formula for k n and g n ,

n 1   1  n1  1 1 1  k   k  rk   g  s k k  n n l n l   2    kl g n l  ,  n1 2 nx 2 2    l 1   l 1   n n  g   g   k  rg  s  k g     g g  , n  2. nx n n   l n l    l n l   n1  l 0   l 0 

Because of

 1, x  1,t  , t 1 x 1

we derive the conservation laws of (30) - 29 www.ivypub.org/mc

(38)

, (39)

(40)

(41)

  (  r  sK   G)  ( A  BK  G), t x

(42)

where

A  c0 2  c1  c1r  c0r 2  12 c0rx  c0  x , B  c0 s  c1s  12 c0 sx  c0rs,   c0  c0  c0 x  c1r .

Assume that     r  sK   G,  A  BK  G, then (42) can be written as  t   x , which is the right form of conservation laws. We expand  and  as series in powers of  according with the coefficients which are called conserved densities and currents respectively 



j 0

j 0

     j   j ,  c0 2  c1   j   j ,

(43)

where c0 , c1 are constants of integration. Then the first two conserved densities and currents are

 1  12 s   ,  2   12 rs   x   x  , 1  c0 (rs  14 sx   x   x   2r )  c1 ( 12 s   ),

1  c0 ( 14 rx s  12 s x  81 s 2  14 rsx   x   xx  rx  3r x  2r 2   x  x  r x  )+ c1 ( 12 rs   x   x  ).

The recursion relations for

(44)

 n and  n are

 n  skn   gn ,   1 n  c0 (skn1  2 sx kn  rskn   g n1 + x g n  r g n )  c1 (skn   g n ).

(45)

where k n and g n can be calculated based on (40). The infinitely conservations laws of (40) can be easily obtained from (36)-(45) respectively.

ACKNOWLEDGMENT This work was supported by the National Natural Science Foundation of China (No.61072147), the Natural Science Foundation of Henan Province (No.132300410202), the Science and Technology Key Research Foundation of the Education Department of Henan Province (No.12A110017) and the Youth Research Foundation of Shangqiu Normal University (No. 2011QN12).

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AUTHORS Sixing Tao was born in March 1981, in Qingzhou, Shandong Province of China. He received the bachelor degree in Mathematics and Applied Mathematics and the master degree in Basic Mathematics both from Qufu Normal University in 2003 and 2006, respectively, then earned the doctor degree in Basic Mathematics from Shanghai University in 2011. Currently, he is a lecturer in School of Mathematics and Information Science, Shangqiu Normal University. His research interests are in soliton theory and integrable systems and published more than 10 articles in recent years.

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