A coupled Volterra system and its exact solutions
aa r X i v : . [ n li n . S I] N ov A coupled Volterra system and its exact solutions
S. Y. Lou , , Bin Tong , Man Jia and Jin-hua Li Department of Physics, Shanghai Jiao Tong University, Shanghai, 200030, China Department of Physics, Ningbo University, Ningbo, 315211, China
Abstract
A coupled Volterra system is proposed. The model can be considered as one of the integrablediscrete form of the coupled integrable KdV system which is a significant physical model. Manytypes of cnoidal waves, positons, negatons (solitons) and complexitons of the model are obtainedby a simple rational expansion method of the Jacobi elliptic functions, trigonometric functions andhyperbolic functions. . INTRODUCTION. The Volterra system [1] a nt − a n ( a n − − a n +1 ) = 0 , (1)is one of the famous integrable differential-difference systems which has been applied invarious physical systems such as the network, statistical physics and biology [2]. Some typesof exact solutions of the model have been studied by many authors (say [3]).It is also interesting that the Volterra system is one of the simplest discrete form of theKdV equation. In fact, if we write a n = 1 + δ u (cid:18) ( n − t ) δ, δ t (cid:19) + O ( δ ) (2) ≡ δ u ( x, τ ) + O ( δ ) , (3) a n ± = 1 + δ u ( x ± δ, τ ) + O ( δ ) , (4)then (1) becomes the well known KdV equation13 ( u τ + 6 uu x + u xxx ) δ + O ( δ ) = 0 . (5)Recently, some types of integrable coupled KdV system have been derived from somedifferent physical fields including the atmospheric dynamics [4], Bose-Einstein condensation[5] and two-wave modes in a shallow stratified liquid [6]. A common special interesting caseof [4]–[6] has the following form u t + 6 αvv x + 6 uu x + u xxx = 0 , (6) v t + 6 vu x + 6 uv x + v xxx = 0 . (7)Some kinds of analytical negatons, positons and complexitons of the coupled KdV system(6)–(7) for α = − α > α < II. A COUPLED VOLTERRA SYSTEM
It is interesting that the following coupled Volterra system a nt − a n ( a n − − a n +1 ) − αb n ( b n − − b n +1 ) = 0 , (8) b nt − a n ( b n − − b n +1 ) − b n ( a n − − a n +1 ) = 0 , (9)is an integrable extension of the usual Volterra system (1). It is obvious that both b n = 0 and b n = √ α a ( n, t ) reduce the coupled Volterra system (8)–(9) to the usual Volterra equation(1).The integrability of the coupled Volterra system (8)–(9) is guaranteed by the followingtheorem. Theorem.
The coupled Volterra system (8) – (9) possesses the following Lax pair, Lψ n = Λ ψ n , (10) ψ nt = M ψ n , (11) where L ≡ a n T + + T − αb n T + b n T + a n T + + T − , Λ = λ αλ λ λ , ψ n ≡ ψ n ψ n , (12) M ≡ − a n a n +1 + αb n b n +1 α ( a n b n +1 + b n a n +1 ) a n b n +1 + b n a n +1 a n a n +1 + αb n b n +1 T , (13) and T + and T − are shift operators defined by T + f n ≡ f n +1 , T − f n ≡ f n − (14) for an arbitrary function f n . Proof.
By the direct calculations, one can prove that the matrices M and Λ are commutable3nd then the compatibility condition of (12) and (13) is just the zero curvature condition L t + LM − M L = 0 . (15)Substituting the definition equations of L and M into (15) just leads to the coupled Volterrasystem (8)–(9). The theorem is proved.Another interesting fact is that the coupled Volterra system (8)–(9) is really a discreteform of the coupled KdV system (6)–(7). Applying the following continuous limiting proce-dure a n = 1 + δ u (cid:18) ( n − t ) δ, δ t (cid:19) + O ( δ ) ≡ δ u ( x, τ ) + O ( δ ) , (16) b n = δ v (cid:18) ( n − t ) δ, δ t (cid:19) + O ( δ ) ≡ δ v ( x, τ ) + O ( δ ) , (17) a n ± = 1 + δ u ( x ± δ, τ ) + O ( δ ) , (18) b n ± = δ v ( x ± δ, τ ) + O ( δ ) (19)to (8)–(9), we have 13 ( u τ + 6 αvv x + 6 uu x + u xxx ) δ + O ( δ ) = 0 , (20)13 ( v τ + 6 vu x + 6 uv x + v xxx ) δ + O ( δ ) = 0 (21)which is just the special coupled KdV system (6)–(7) with t → τ . III. CNOIDAL WAVES, SOLITONS AND POSITONS OF THE COUPLEDVOLTERRA SYSTEM FOR α > In the continuous case, the rational expansion of the elliptic and hyperbolic functions isone of the simplest methods to find travelling periodic and solitary wave solutions. Fortu-nately, this method is valid also in discrete cases.To find some types of cnoidal wave solutions of the coupled Volterra system (8)–(9), onemay take some types of the elliptic function expansion ansatzs, say, a ( n, t ) = P ( f, g, h ) R ( f, g, h ) , (22) b ( n, t ) = Q ( f, g, h ) R ( f, g, h ) (23)4here { P ( f, g, h ) , Q ( f, g, h ) , R ( f, g, h ) } are polynomial functions of { f, g, h } and f ≡ sn( kn + ct, m ), g ≡ cn( kn + ct, m ) and h ≡ dn( kn + ct, m ) are Jacobi elliptic functions withthree constant parameters, the wave number k , the angular frequency ω and the modulus m . Because of the computational difficulty, here, we just give some special examples of(22)–(23). Case 1.
The first simple expansion ansatz reads a ( n, t ) = a + c cn ( kn + ct, m ) a + c cn ( kn + ct, m ) , (24) b ( n, t ) = A [1 + a cn ( kn + ct, m )] a + c cn ( kn + ct, m ) (25)where a , c , a , c , A, k, m and c are constants that should be determined later.Substituting (24)–(25) into (8)–(9) and vanishing the coefficients of the different powers ofthe Jacobi elliptic function cn ( kn + ct ), we can obtain a complicated determining equationsystem for the undetermined constants. Fortunately, there exist a unique general solutionfor the determining equation system. The result reads ( s ≡ sn( kδ, m ) , d ≡ dn( kδ, m ) , C ≡ cn( kδ, m )) s = 4 a c ( a + c )[ m ( a + c ) − c ][ m ( a + c ) − c ] , (26) A = a c c ( a + c ) c [ m ( a + c ) − c ] α ( a a − c ) , (27) a = cm s ( a + c )( a − c + 2 a a )[( a + c ) m s − c ]4 dCc ( a a − c ) − ca dCs + cc s (2 a a − c )(2 a + 2 c − c s )4 dC ( a + c )( a a − c ) , (28) c = a ( a c + 2 c − a a ) a + 2 a a − c + c ( a + c − c s ) ( c − a a )2 dCs ( a + c )( a + 2 a a − c ) , (29)while the constants a , c , c, a and m remain to be free parameters.From (26) and (27), it is known that the real condition of the periodic solution requires α > . Fig. 1 shows the structure of the periodic solution (24) with the parameter selections α = c = a = c = 1 , a = 2 , δ = 1 , m = 0 .
999 (30)at time t = 0. 5 ig.1–4 –2 0 2 4n–10 –5 0 5 10t0.811.2a(n,t) FIG. 1: The structure of the periodic wave expressed by (24) with the parameters (30) at time t = 0. It is remarkable that for the α >
Case 2. a ( n, t ) = a + c cn( kn + ct, m ) a + c cn( kn + ct, m ) , (31) b ( n, t ) = A + A cn( kn + ct, m )] a + c cn( kn + ct, m ) (32)where the constants a , c , a , c , A , A , k, c and m satisfy the following conditions: a = sc a d ( m a + 1 − m ) , (33) A A = 2 a (cid:2) c + 2 c sd − cm s (1 − a ) (cid:3) cs (cid:2) m (1 − a ) − (cid:3) + 2 a ( c + 2 c sd ) , (34) A = ± cs (cid:2) m (1 − a ) − (cid:3) + 2 a ( c + 2 c sd )4 sda √ α (35) C = 1 + s a (cid:2) m (1 − a ) − (cid:3) , c = 1 . (36) Case 3. a ( n, t ) = a + c dn( kn + ct, m ) a + c dn( kn + ct, m ) , (37) b ( n, t ) = A + A dn( kn + ct, m )] a + c dn( kn + ct, m ) (38)6ith the constant constraints a = sc a C ( m + a −
1) + c a , (39) A A = 2 a (cid:2) c + 2 c sC − cs (1 − a ) (cid:3) cs (cid:2) (1 − a ) − m (cid:3) + 2 a ( c + 2 c sC ) , (40) A = ± cs (cid:2) (1 − a ) − m (cid:3) + 2 a ( c + 2 c sC )4 sCa √ α (41) d = 1 + s a (cid:2) (1 − a ) − m (cid:3) , c = 1 . (42) Case 4. a ( n, t ) = a + c sn( kn + ct, m ) a + c sn( kn + ct, m ) , (43) b ( n, t ) = A + A sn( kn + ct, m )] a + c sn( kn + ct, m ) (44)with the constant constraints c = a a − c ( Cd − s (1 + m a ) ( m a − , (45) A A = (1 + m a )(2 sa + cdCa ) a (cid:2) m a ( ca + 2 sa ) + 2 sa + 2 cdCa − ca (cid:3) , (46) A = ± m a ( ca + 2 sa ) + 2 sa + 2 cdCa − ca a p α (1 − dC )(1 + a m ) (47) s = 2 a (1 − dC )1 + m a , c = 1 . (48) Case 5. a ( n, t ) = a cn( kn + ct, m ) + c sn( kn + ct, m ) a cn( kn + ct, m ) + c sn( kn + ct, m ) , (49) b ( n, t ) = A cn( kn + ct, m ) + A sn( kn + ct, m ) a cn( kn + ct, m ) + c sn( kn + ct, m ) (50)with the constant constraints c = a a − c ( d − m a − a + 1)2 sC [ m a − (1 + a ) ] , (51) A A = [(1 + a ) − m a ]( cda + 2 sCa ) a (cid:2) m a ( ca + 2 sa C ) − (1 + a )(2 sCa + 2 cda − ca + ca + 2 sCa a ) (cid:3) , (52) A = ± m a ( ca + 2 sa C ) − (1 + a )(2 sCa + 2 cda − ca + ca + 2 sCa a )4 a C (1 − d ) √ α (53) s = 2 a (1 − d )(1 + a ) − m a , c = 1 . (54)7 ase 6. a ( n, t ) = a dn( kn + ct, m ) + c sn( kn + ct, m ) a dn( kn + ct, m ) + c sn( kn + ct, m ) , (55) b ( n, t ) = A dn( kn + ct, m ) + A sn( kn + ct, m ) a dn( kn + ct, m ) + c sn( kn + ct, m ) (56)with the constant constraints c = a a − c ( C − m a − − m a )2 sd [(1 + m a ) − m a ] , (57) A A = [(1 + m a ) − m a ]( cCa + 2 sda ) a (cid:2) sda [(1 + m a ) − m a ] + ca [ m a + m a (2 C − a ) + 2 C − (cid:3) , (58) A = ± s (cid:2) sda [(1 + m a ) − m a ] + ca [ m a + m a (2 C − a ) + 2 C − (cid:3) a d (1 − C ) √ α , (59) s = 2 a (1 − C )(1 + a m ) − m a , c = 1 . (60) Case 7. a ( n, t ) = a dn( kn + ct, m ) + c cn( kn + ct, m ) a dn( kn + ct, m ) + c cn( kn + ct, m ) , (61) b ( n, t ) = A dn( kn + ct, m ) + A cn( kn + ct, m ) a dn( kn + ct, m ) + c cn( kn + ct, m ) (62)with the constant constraints c = a a − c ( Cd − m a − s (1 + m a ) , (63) A A = (1 + m a )( cCda + 2 sa ) a (cid:2) m a ( ca + 2 sa ) + 2 sa + 2 cda C − ca (cid:3) , (64) A = ± m a ( ca + 2 sa ) + 2 sa + 2 cda C − ca a p α (1 − dC )(1 + m a ) , (65) s = 2 a (1 − dC )1 + a m , c = 1 . (66)Especially, when m →
1, the previous periodic wave solutions become negaton (soliton)solutions. For instance, (24) and (25) become the soliton solution a ( n, t ) = a + c sech ( kn + ct ) a + c sech ( kn + ct ) , (67) b ( n, t ) = A [1 + a sech ( kn + ct )] a + c sech ( kn + ct ) (68)8 ig.2 –4 –2 0 2 4n–10 –5 0 5 10t0.81.2a(n,t) FIG. 2: The structure of the solitary wave expressed by (67) which is the limit case of the figure1 for m = 1. with S = 4 c ( a + c )( a + 2 c ) , A = c a c ( a + c ) α ( a a − c ) , (69) a = 2 ca c ( a + c ) S ( a a − c )( a + 2 c ) − a c S ( a a − c ) , (70) c = 2 cc ( a + c ) S ( a a − c )( a + 2 c ) − c c (5 a + 12 a c + 8 c )2 S ( a a − c ) , (71)Fig. 2 shows the structure of the soliton solution (24) with the same parameter selectionsas (30) except for m = 1.In [7], for the α < m = 0. For instance, the first type of the positon solutions can beobtained from the periodic solution (24) and (25) by taking m = 0, which has the form of a ( n, t ) = 2[ a + c cos ( kn + ct )]cos(2 kn + 2 ct ) ± cos( kδ ) , (72) b ( n, t ) = 2 A [1 + a cos ( kn + ct )] √ α [cos(2 kn + 2 ct ) ± cos( kδ )] , (73)9 ig.3–4 –2 0 2 4n –2 –1 0 1 2t–1000100a(n,t) FIG. 3: A typical positon structure expressed by (72)–(76) with the parameter selections (77). where a = c (cid:2) a (cid:0) kδ ) + cos(4 kδ ) (cid:1) ∓ (1 + a ) cos( kδ ) cos(2 kδ ) (cid:3) kδ )[2 + a ∓ a cos( kδ )] , (74) c = − c [2 + a ∓ a cos( kδ ) cos(2 kδ )]sin(2 kδ )[2 + a ∓ a cos( kδ )] , (75) A = sin( kδ ) [2 + a ∓ a cos( kδ )] , (76)and c, a and k are arbitrary constants.Obviously, the positon solution (72)–(73) is always singular. Fig. 3 shows a specialstructure of (72) and (73) with the parameter selections c = α = δ = 1 , a = 2 , k = 0 . . (77)Actually, all the positon solutions obtained from Cases 1 to 7 by taking m = 0 are singularexcept for the trivial constant solution in Case 3.For a coupled nonlinear system, there may be different types of solitons and positons.The first type of soliton solutions given by (67) and (68) possesses the property that thefields a ( n, t ) and b ( n, t ) both have the ring or bell shape. The coupled Volterra system canhave other kinds of soliton solutions. For instance, by substituting the following solutionansatz a ( n, t ) = a + a cosh( kn + ct ) + a cosh(2 kn + 2 ct ) b + b cosh( kn + ct ) + b cosh(2 kn + 2 ct ) , (78) b ( n, t ) = d sinh( kn + ct ) b + b cosh( kn + ct ) + b cosh(2 kn + 2 ct ) (79)10 ig.4a–8 –4 0 4 8n–4 –2 0 2 4t–1–0.8–0.6–0.4a(n,t) Fig.4b–8 –4 0 4 8n–4 –2 0 2 4t–0.200.2b(n,t) FIG. 4: Second type of soliton structure expressed by (78)–(85) with the parameter selections (86). into the coupled Volterra system (8) and (9), one can find that d = ∓ c sin( c ) (cid:20) cos (cid:18) kδ (cid:19) − cos (cid:18) kδ (cid:19)(cid:21) , (80) b = √ α sin( kδ )(1 + cos(2 c ) + cos( kδ )) , (81) b = ± √ α sin( kδ ) cos( c ) cos (cid:18) kδ (cid:19) , (82) a = − c (cid:20) cos(2 c ) + 2 cos (cid:18) kδ (cid:19) cos (cid:18) kδ (cid:19)(cid:21) , (83) a = ∓ c cos( c ) (cid:20) cos (cid:18) kδ (cid:19) + cos (cid:18) kδ (cid:19)(cid:21) , (84) b = √ α sin( kδ ) , a = − c , (85)where c, c and k are arbitrary constants.From (80)–(85), we know that this kind of soliton solution is also valid only for α > δ = c = α = 1 , c = k = 2 , (86)for the upper sign. It is clear that in this case the soliton structures are different for thefields a ( n, t ) and b ( n, t ). The soliton structure for a ( n, t ) is still a bell or ring shape whilethat for b ( n, t ) becomes staggered.It is noted that because of the arbitrariness of the constants k, c and c , if we take k → √− k, c → √− c, c → √− c (87)11hen the soliton solution (78)–(85) is transformed to another type of positon solutions a ( n, t ) = a + a cos( kn + ct ) + a cos(2 kn + 2 ct ) b + b cos( kn + ct ) + b cos(2 kn + 2 ct ) , (88) b ( n, t ) = d sin( kn + ct ) b + b cos( kn + ct ) + b cos(2 kn + 2 ct ) (89)with d = ∓ c sinh( c ) (cid:20) cosh (cid:18) kδ (cid:19) − cosh (cid:18) kδ (cid:19)(cid:21) , (90) b = √ α sinh( kδ )[1 + cosh(2 c ) + cosh( kδ )] , (91) b = ± √ α sinh( kδ ) cosh( c ) cosh (cid:18) kδ (cid:19) , (92) a = − c (cid:20) cosh(2 c ) + 2 cosh (cid:18) kδ (cid:19) cosh (cid:18) kδ (cid:19)(cid:21) , (93) a = ∓ c cosh( c ) (cid:20) cosh (cid:18) kδ (cid:19) + cosh (cid:18) kδ (cid:19)(cid:21) , (94) b = √ α sinh( kδ ) , a = − c . (95)Fig. 5 displays the structure of the positon solution (88)–(95) for the upper sign with thespecial parameter selections δ = α = 1 , c = k = π , c = π . (96)Evidently, in this case the positon structure is still singular. Up to now, for the α > α > IV. SOLITONS, POSITONS AND COMPLEXITONS OF THE COUPLEDVOLTERRA SYSTEM FOR α < For the coupled real Volterra system (8) and (9) with α <
0, all the cnoidal wave solutionforms in the last section are not valid, however, the function expansion ansatz (78) and (79)is still applicable to obtain some soliton solutions.Similar to the last section, after substituting (78) and (79) into (8)–(9) for α < ig.5a–4 –2 0 2 4n –1 –0.50 0.5 1t–50050a(n,t)
Fig.5b–4 –2 0 2 4n –1 –0.5 0 0.5 1t–50050b(n,t)
FIG. 5: A typical positon structure expressed by (88)–(95) with the parameter selections (96). solving the determining equations of the parameters, one can find that d = 2 c sin( c ) sinh (cid:0) kδ (cid:1) √− α , (97) b = 1 + cos(2 c ) + cosh( kδ ) , (98) b = 4 cos( c ) cosh (cid:18) kδ (cid:19) , (99) a = c sinh( kδ ) [cos(2 c ) + cosh( kδ ) + cosh(2 kδ )] , (100) a = c cos( c )sinh( kδ ) (cid:20) cosh (cid:18) kδ (cid:19) + cosh (cid:18) kδ (cid:19)(cid:21) , (101) b = 1 , a = c kδ ) , (102)where c, c and k are arbitrary constants.Fig. 6 shows the structure of the soliton solution expressed by (78) and (79) with (97)–(102) and the parameter selections c = π , c = k = 15 , δ = 1 , α = − . (103)In the continuous case, it has been proved that the coupled KdV system (6) and (7) with α < k → √− k, c → √− c, c → √− c , ig.6a–20 –10 0 10 20n–4 –2 0 2 4t–1.02–1a(n,t) Fig.6b–20 –10 0 10 20n–4–2024 t–0.02–0.0100.010.02b(n,t)
FIG. 6: The structures of the soliton expressed by (a) (78) and (b) (79) with (97)–(102) and theparameter selections (103). to (78)–(79) with (97)–(102). The result still has the form (88)–(89) but with the constants d = 2 c sinh( c ) sin (cid:0) kδ (cid:1) √− α , (104) b = 1 + cosh(2 c ) + cos( kδ ) , (105) b = 4 cosh( c ) cos (cid:18) kδ (cid:19) , (106) a = − c kδ ) [cosh(2 c ) + cos( kδ ) + cos(2 kδ )] , (107) a = − c cosh( c )sin( kδ ) (cid:20) cos (cid:18) kδ (cid:19) + cos (cid:18) kδ (cid:19)(cid:21) , (108) b = 1 , a = − c kδ ) , (109)Fig. 7 reveals the structure of the analytical positon solution expressed by (78) and (79)with (104)–(109) and the parameter selections c = 1 , c = 1 , k = 2 , δ = 1 , α = − . (110)To get analytical complexiton solutions of the coupled Volterra system (8)-(9), we mayuse the following solution ansatz a ( n, t ) = a + a cosh( ξ ) cos( ξ ) + a sinh( ξ ) sin( ξ ) + a [cosh(2 ξ ) + cos(2 ξ )] b + b cosh( ξ ) cos( ξ ) + b sinh( ξ ) sin( ξ ) + b [cosh(2 ξ ) + cos(2 ξ )] , (111) b ( n, t ) = d + d cosh( ξ ) cos( ξ ) + d sinh( ξ ) sin( ξ ) + d [cosh(2 ξ ) + cos(2 ξ )] b + b cosh( ξ ) cos( ξ ) + b sinh( ξ ) sin( ξ ) + b [cosh(2 ξ ) + cos(2 ξ )] , (112)where ξ i = k i n + c i t + ξ i , i = 1 , . (113)14 ig.7a–10 –5 0 5 10n –4 –2 0 2 4t11.5a(n,t) Fig.7a–10 –5 0 5 10n –4 –2 0 2 4t–0.500.5b(n,t) FIG. 7: The structure of the positon expressed by (a) (78) and (b) (79) with (104)–(109) and theparameter selections (110).
After finishing tedious calculations, we find that (8)-(9) really possesses analytical complexi-ton solutions (an analytical complexiton is just a usual breather) if α < a j , b j and c j for j = 0 , , a = c [cos(4 c ) sinh(2 b ) − sinh(6 b ) − cos(2 c ) sinh(4 b )]+ c [sin(2 c ) cosh(4 b ) − sin(4 c ) cosh(2 b ) − sin(6 c )] , (114) a = ∓ { [sinh(5 b ) cos( c ) − cos(5 c ) sinh( b ) + cos( c ) sinh( b ) + sinh(3 b ) cos(3 c )] c +[sin( c ) cosh( b ) − sin( c ) cosh(5 b ) + sin(5 c ) cosh( b ) + sin(3 c ) cosh(3 b )] c } , (115) a = ∓ { [sin(3 c ) cosh(3 b ) − sin(5 c ) cosh( b ) + sin( c ) cosh( b ) + sin( c ) cosh(5 b )] c +[sinh(5 b ) cos( c ) − sinh(3 b ) cos(3 c ) − cos(5 c ) sinh( b ) − cos( c ) sinh( b )] c } , (116) a = − c cosh(2 b ) sin(2 c ) − c sinh(2 b ) cos(2 c ) , (117) b = 1 √− α [cos(2 c )(1 + 2 cosh(4 b )) + cosh(2 b )(1 − c )) + cosh(6 b ) − cos(6 c )] , (118) b = ∓ √− α { cosh( b ) cos(5 c ) + cosh( b ) cos(3 c ) − cos( c ) cosh(5 b ) − cos( c ) cosh(3 b ) } , (119) b = ± √− α {− sinh( b ) sin(5 c ) + sinh( b ) sin(3 c ) + sin( c ) sinh(5 b ) − sin( c ) sinh(3 b ) } , (120) b = 2 √− α [cosh(4 b ) − cos(4 c )] , (121)15 ig.8a–15 –10 –5 0 5 10 15n –10 –5 0 5 10t–15–10–50510a(n,t) Fig.8b–15 –10 –5 0 5 10 15n –10 –5 0 5 10t–30–20–10010a(n,t) FIG. 8: The structure of the complexiton expressed by (a) (111) and (b) (112) with (114)–(125)and the parameter selections (126). d = − c cos(4 c ) sinh(2 b ) − c sin(4 c ) cosh(2 b ) − c sin(6 c )+ c sinh(6 b ) + c sin(2 c ) cosh(4 b ) + c cos(2 c ) sinh(4 b ) , (122) d = ± { [sinh(5 b ) cos( c ) − cos(5 c ) sinh( b ) + cos( c ) sinh( b ) + sinh(3 b ) cos(3 c )] c +[sin( c ) cosh( b ) − sin( c ) cosh(5 b ) + sin(5 c ) cosh( b ) + sin(3 c ) cosh(3 b )] c } , (123) d = ± { [cos(5 c ) sinh( b ) + cos( c ) sinh( b ) − sinh(5 b ) cos( c ) + sinh(3 b ) cos(3 c )] c +[sin(3 c ) cosh(3 b ) − sin(5 c ) cosh( b ) + sin( c ) cosh( b ) + sin( c ) cosh(5 b )] c } , (124) d = 2 c cos(2 c ) sinh(2 b ) − c sin(2 c ) cosh(2 b ) , (125)with b ≡ k δ, c ≡ k δ and k , k , c , ξ and ξ being arbitrary constants.Fig. 8 displays the complexiton structure expressed by (111) and (112) with (114)–(125)and the special parameter selections k = ξ = 0 , k = 0 . , c = 0 . , c = 0 . , ξ = π , δ = 1 . (126) V. SUMMARY AND DISCUSSIONS
In summary, the special coupled integrable KdV system (6)–(7) is discreterized to anintegrable coupled Volterra system. The Lax integrability of the coupled Volterra systemis proved. By using a simple rational expansion method of the Jacobi elliptic functions,trigonometric functions and hyperbolic functions, various exact solutions are found.16or the coupled Vorterra system (8)–(9) with α > analytical negaton solutionwhen the modulus, m , of the model tends to 1. The soliton solution has a ring or bell shapefor both fields a ( n, t ) and b ( n, t ). On the other hand, whence the modulus of the cnoidalwave tends to 0, the wave tends to a positon solution. It is found that all the nontrivialpositons obtained from the cnoidal waves of section III are singular. This fact is similar tothe KdV and Toda system [11].In the α > a ( n, t ) and b ( n, t ). The new type of soliton solution is obtainedby taking a more complicated rational expansion of the hyperbolic functions. Because ofthe calculation difficulty, we have not yet found any cnoidal wave extension of this typeof negaton solution even utilizing computer algebras. The field a ( n, t ) for the second typeof soliton solution also possesses the ring or bell shape while the field b ( n, t ) possesses astaggered shape.For the coupled Vorterra system (8)–(9) with α <
0, though we have not yet found thecnoidal wave solutions, many other kinds of physically significant solutions, such as thesolitons, analytical positons and analytical complexitons are found. The structure of thesoliton solution in this case is similar to that of the second type of the solitons for α > α < α <
0, a more complicated rational expansion of both the hyperbolic functions and trigono-metric functions is used.Finally it is worth to indicated that the real solutions of the coupled Vorterra system (8)–(9) with α < ig.9a –30–20–100102030 n–6 –4 –2 0 2 4 6t–0.55–0.5–0.45a(n,t) Fig.9b –30–20–100102030 n–6 –4 –2 0 2 4 6t55.05b(n,t)
FIG. 9: The evolution of the two-soliton interaction solution expressed by (a) (111) and (b) (112)with the transformations (127) and the parameter selections (128). we just mention a further interesting example. If we apply the constant transformations k → √− k , c → √− c , ξ → √− ξ , (127)to (111)–(112), then the complexiton solution of α < α > k = ξ = 0 , k = 0 . , c = 0 . , c = 1 , ξ = π , δ = 1 . (128)Though we have obtained many types of exact solutions of the model via a simple functionexpansion method, various problems, such as the general multiple soliton solutions and τ function solutions are still open. As a discrete form of the significant physical model, themore about the model will be studied further. Acknowledgments
The authors are in debt to the helpful discussions with Drs. X. Y. Tang, P. Liu, Y. Gaoand X. Y. Jiao. The work was supported by the National Natural Science Foundations ofChina (Nos. 10475055, 10735030 and 90503006), the Scientific Research Fund of ZhejiangProvincial Education Department (No. 20040969) and National Basic Research Program of18hina (973 Program 2007CB814800). [1] O. Babelon, Commun. Math. Phys. 266 (2006) 819.[2] L. E. Reichl, A Modern Course in Statistical Physics, Edward Arnold (publishers) pp641-644(1980);P. A. Damianou and S. P. Kouzaris, Physica D 195 (2004) 50.[3] S. D. Zhu, Discrete Volterra equation via Exp-function Method, 2007 International Symposiumon Nonlinear Dynamics 27-30 Oct., 2007, Shanghai, China.[4] S. Y. Lou, B. Tong, H. C. Hu and X. Y. Tang,[5] V. A. Brazhnyi and V. V. Konotop, Phys. Rev. E 72 (2005) 026616.[6] J. A. Gear, Stud. Appl. Math. 72 (1985) 95.[7] H. C. Hu, B. Tong and S. Y. Lou, Phys. Lett. A 351 (2006) 403.[8] S. V. Maanakov, Sov. Phys. JETP 40 (1975) 269.[9] M. J. Ablowitz, SIAM Review 19 (1977) 663.[10] M. J. Ablowitz and J. F. Ladik, Stud. Appl. Math. 57 (1977) 1.[11] W. X. Ma, Phys. Lett. A 301 (2002) 35;W. X. Ma and K. Maruno, Physica A 343 (2004) 219.19