Commun. Theor. Phys. Beijing, China 54 21 pp. 1 6 c Chinese Physical Society and IOP Publishing Ltd Vol. 54, No. 1, July 15, 21 A New Integrable Couplings of Classical-Boussinesq Hierarchy with Self-Consistent Sources GE Jian-Ya 1 and XIA Tie-Cheng 2, 1 Jinhua College of Profession and Technology, Jinhua 32, China 2 Department of Mathematics, Shanghai University, Shanghai 2444, China Received September 25, 29; revised manuscript received January 25, 21 Abstract A kind of integrable couplings of soliton equations hierarchy with self-consistent sources associated with sl4 is presented by Yu. Based on this method, we construct a new integrable couplings of the classical-boussinesq hierarchy with self-consistent sources by using of loop algebra sl4. In this paper, we also point out that there exist some errors in Yu s paper and have corrected these errors and set up new formula. The method can be generalized other soliton hierarchy with self-consistent sources. PACS numbers: 2.3.Ik, 2.3.Jr, 2.2.Sv Key words: the classical-boussinesq hierarchy, self-consistent sources, integrable couplings, loop algebra 1 Introduction In the study of nonlinear science, soliton theory plays a very important role and has been applied in almost all the natural sciences especially in the physics branches, such as condensed matter physics, field theory, fluid dynamics, etc. Recently, the study of integrable couplings of soliton equations has attracted much attention. The study of integrable couplings of soliton equations originated from the investigations into the symmetry problems associated centerless Virassro algebras. [1 3] It is an approach to obtain new larger integrable hierarchies. Some integrable couplings of the well-known integrable systems have been obtained. [4 7] In addition, the Hamiltonian structures of the integrable systems play very great important role in soliton theory. [8 9] Soliton equations with self-consistent sources SESCS [1 11] may also have attracted considerable attention. Physically, the sources may result in solitary waves with a non-constant velocity and therefore lead to a variety of dynamics of physical models. For applications, these kinds of systems are usually used to describe interactions between different solitary waves and are relevant to some problems of hydrodynamics, solid state physics, plasma physics, etc. Very recently, the integrable couplings of the SESCS have also been receiving growing attention. Yu [12 14] has derived some integrable couplings of integrable hierarchy with self-consistent sources such as the Dirac soliton with self-consistent sources, the Yang soliton hierarchy with self-consistent sources and the C-KdV soliton hierarchy with self-consistent by taking use of the loop algebra sl4. But there are some errors in Refs. [12 14]. In this paper, we correct these errors and construct a new integrable coupling with self-consistent sources by using of the loop algebra sl4 and new idea. This paper is organized as follows. In Sec. 2, we will lead to a kind of integrable couplings of soliton equations with self-consistent sources associated with sl4. In Sec. 3, an integrable couplings of the classical-boussinesq hierarchy with self-consistent sources is derived by using of the loop algebra sl4. Finally, some conclusions are given. 2 A New Integrable Couplings of Soliton Equation Hierarchy with Self-Consistent Sources Associated with sl4 In the following, we consider a set of matrix Lie algebra sl4 1 1 1 e 1 = 1, e 1 2 = 1, 1 1 1 1 1 e 3 = 1, e 1 4 =, 1 1 1 1 e 5 =, e 1 6 =, 1 which was presented by Zhang and Guo in Ref. [15]. Supported by the Natural Science Foundation of Shanghai under Grant No. 9ZR1418, the Science Foundation of Key Laboratory of Mathematics Mechanization under Grant No. KLMM86, the Shanghai Leading Academic Discipline Project under Grant No. J511 and by Key Disciplines of Shanghai Municipality S314 E-mail: xiatc@yahoo.com.cn
2 GE Jian-Ya and XIA Tie-Cheng Vol. 54 Let sl4 = span{e 1, e 2, e 3, e 4, e 5, e 6 }, sl4 1 = span{e 1, e 2, e 3 }, sl4 2 = span{e 4, e 5, e 6 }. Therefore we construct three Lie algebra, and satisfy sl4 = sl4 1 sl4 2, [sl4 1, sl4 2 ] = sl4 1 sl4 2 sl4 2 sl4 1 sl4 2. In what follow, we introduce the loop algebra sl4 sl4 = {A A R[λ] sl4}, sl41 = {A A R[λ] sl4 1 }, sl42 = {A A R[λ] sl4 2 }, 2 where the loop algebra sl4 is defined by span {λ n A n Z, A sl4}. Consider the auxiliary linear problem x = Uu, λ, Uu, λ = ẽ 1λ + 6 u i ẽ i λ, i=2 t n = V n u, λ, 3 where u = u 1,..., u s T, u i = u i x, t i = 1, 2,..., s, φ j = φ j x, t j = 1, 2, 3, 4 are field variables defining on x R, t R, ẽ i λ sl4, sl4 denotes a finite-dimensional Lie algebra over R. The general scheme of searching for the consistent V n and generating a hierarchy of zero curvature equations was proposed as follows. We solve the equation a m b m + c m d m e m + f m V x = UV V U, V = V m uλ m b m c m a m e m f m d m = a m b m + c m λ m. 4 b m c m a m We search for n sl4 such that V n can be constructed by n1 n2 + n3 n4 n5 + n6 n V n = V m uλ n m n2 n3 n1 n5 n6 n4 + n u, λ, n u, λ = n1 n2 + n3, 5 n2 n3 n1 where ni are linear functions of a n, b n, c n, d n, e n, and f n. Under certain conditions, there is a constant γ such that the so-called bi-trace identity holds. [16] i Tr V U = λ γ λ γ Tr V U, 6 where Tr denotes the trace of a matrix. Define a scalar H = Hu, λ by the equation λ γ λ γ H = Tr V U, H = H m uλ m. 7 From the trace identity, we yield that H m λ m = Tr V U = TrV m e i λ m. 8 i j j x t n = ẽ 1 λ + 6 u i ẽ i λ i=2 The set {H m } proves the conserved densities of 6. In [17], Ma proposed the generalized Tu scheme to find a Hamiltonian function, a recursion operator L, and symplectic operator J of the hierarchy based on zero curvature equation. The Hamiltonian form with H n+1 can be written as u tn = J H n+1, n = 1, 2,... 9 H n = LH n 1 = = L nh n, n = 1, 2,..., 1 where n is the minimum integer such that H n / and H k / = for k < n, / = / 1,...,/ s T. According to the condition 8, we obtain a new auxiliary linear problem. For N distinct λ j, j = 1,..., N, the following systems result from 3 j n = V m uλ n m + n u, λ, j = 1,...,N, 11a j, j = 1,..., N. 11b
No. 1 A New Integrable Couplings of Classical-Boussinesq Hierarchy with Self-Consistent Sources 3 The following equation is presented in Refs. [18 19] H N k + α j =, 12 where α j are constants, H k / determines a finite dimensional invariant set for the flow 12. For 11a, we define that Uu, λ j Ψ j = Tr Ψ j e i λ j, i = 1, 2, 13 i = Tr which is first presented by Tu, [2] where 1j j 3j 2j 4j j Ψ j = 1j, 2j j = 1,...,N. For i = 3, 4, we have = Tr i where U1 U U = U 1 U u, λ j Ψ ja, 14, Ψ ja = 4 4 φ3j 3j 4j j. According to Eqs. 13 and 14, we obtain a kind of the integrable couplings of Hamiltonian soliton equation hierarchy with self-consistent sources as follows u tn = J H n+n + J = JL n H n + J N, n = 1, 2,... 15 Remarks Here we point out that there are some errors in Refs. [12 14]. For example, Eqs. 17 18 are not true. We see that the left side of Eq. 17 and Eq. 18 are numbers entries. But the right side of Eq. 17 and Eq. 18 is a 4 4 matrix. And Eq. 16 in Refs. [12 14] is also inaccurate! 3 Integrable Couplings of Classical- Boussinesq Hierarchy and Its Hamiltonian Structures In Ref. [21], we have obtained integrable couplings of the classical-boussinesq hierarchy. In order to set up integrable couplings of the classical-boussinesq hierarchy with self-consistent sources, we consider the following matrix spectral problem λ 1 4 u 1 u 2 u 3 u 3 + u 4 1 λ + 1 4 φ x = Uu, λφ, Uu, λ = u 1 u 3 + u 4 u 3 λ 1 4 u 1 u 2 = U1 U, 16 U 1 1 λ + 1 4 u 1 where λ is a spectral parameter and U 1 satisfies ϕ x = U 1 ϕ, which is matrix spectral problem of the classical-boussinesq soliton hierarchy. Therefore, by using the zero curvature equation U t V n x + [U, V n ] =, 17 we have the following integrable couplings of the classical-boussinesq soliton hierarchy: u 1 4c n+1, x 4c n+1, x u 2 2u 2 c n+1 2b n+1 u t = = a n+1, x = 2g n+1, x a n+1, x u 3 where = u 4 t 2u 4 c n+1 + 2u 2 g n+1 2f n+1 1 2 a n+1 + d n+1 c n+1 + g n+1 2a n+1 c n+1 4 4 4u 2 4u 2 2d n+1, x + 4u 2 g n+1 1 2 a n+1 + d n+1 = J c n+1 + g n+1 2a n+1 = J L n+1 c n+1 1 M 1 4 1 u 2 1 4 u 2 M 2 M 3 L 2 1 2 = u1 4 1 2u 3 1 4 1 u 1 1 u 2 + u 2 1 2 M 1 = 1 u 2 1 2 1 4 1 u 1 1 u 4, 1 2 u1 4, 1 2 α 2α, 18 M 2 = 1 4 1 u 3 1 4 1 u 2 1 4 1 u 4, M 3 = 1 2 1 u 2 1 4 1 u 4 1 4 u 4 + 1 2 1 u 2 u 3.
4 GE Jian-Ya and XIA Tie-Cheng Vol. 54 To set up the Hamiltonian structure of the integrable couplings of c-b hierarchy, we need the following main idea in Ref. [9] for a = a 1, a 2,...,a 6 T, b = b 1, b 2,..., b 6 T R 6, define a commutator 2b 2 2b 3 2b 5 2b 6 b 3 2b 1 b 6 2b 4 [a, b] T b 2 2b 1 b 5 2b 4 = a 1, a 2,..., a 6 a T Rb. 19 2b 2 2b 3 b 3 2b 1 b 2 2b 1 In order to apply the quadratic-form identity, we need to construct the symmetric matrix F 2 2 1 1 1 1 F =. 2 2 1 1 And constant matrix F = f ij s s is determined by F = F T, RbF = RbF T. For a, b R 6, we define a function {a, b} = a T Fb = 2a 1 b 1 + a 2 b 3 + a 3 b 2 + 2a 4 b 1 + a 5 b 3 + a 6 b 2 + 2a 1 b 4 + a 2 b 6 + a 3 b 5. 21 A direct calculation by 15 gives { V, U } = 1 a + d, u 1 2 { V, U } { = a + g, V, U } = 2a, u 2 u 3 { V, U } { = c, V, U } = 2a + d, u 4 where a = m a m λ m, b = m b m λ m,... Substituting the above computing results into the quadratic-form identity [9] { } V, U λ = λ γ i λγ{ V, U }, i = 1, 2, 3, 4, 22 we get 2a + d = λ γ λγ 1 2 a + d c + g 2a c. 23 Comparing the coefficients of λ n 2 in 23 gives 1 2 a n + d n 2a c n+1 + g n+1 n+2+d n+2 = n+γ 1 2a n+1.24 c n+1 Taking n =, gives γ =. Therefore, we obtain the Hamiltonian structure of the Lax integrable hierarchy 12 u t = J H n, 25 where H n = 2a n+2 + d n+2 /n + 1 are Hamiltonian functions. Hence, the system 12 can be written as u 1 1 2 u 2 u t = = J L n 1 u 3 2α = J H n. 26 u 4 t Taking u 1 = q, u 2 = r, u 3 = u 4 =, the system 19 reduces to the c-boussinesq hierarchy 4. 4 Integrable Couplings of Classical- Boussinesq Hierarchy with Self-Consistent Sources According to Eqs. 13 14, we have the following results by direct computations: 1 Φ 1, Φ 2 λ N j = 2 Φ 1, Φ 2 Φ 2, Φ 2 = λ j 2 Φ 3, Φ 4 + Φ 3, Φ 3 + Φ 4, Φ 4. 3 Φ 4, Φ 4 Φ 3, Φ 3 4 Therefore we have a novel integrable coupling system of the classical-boussinesq equations hierarchy with self-
No. 1 A New Integrable Couplings of Classical-Boussinesq Hierarchy with Self-Consistent Sources 5 consistent sources as follows: 1 2 α u t = J L n+1 2α + J Φ 1, Φ 2 Φ 1, Φ 2 Φ 2, Φ 2 2 Φ 3, Φ 4 + Φ 3, Φ 3 + Φ 4, Φ 4 Φ 4, Φ 4 Φ 3, Φ 3, 27 where Φ i = φ i1,...,φ in T, i = 1, 2, 3, 4., is the standard inner product in R N and J and L satisfy Eq. 18. When n = 1, we have u 1t = u 1x + 8 x 8 j jx, u 2t = u 2x jx + j x 2 j jx 2 x, u 3t = 2αu 3x + x + x + 2 x + 2 j jx x, u 4t = 8αu 2 u 3 + 4 + 2 N x + x 4u 2 + 2j N jx + j x + 2jx j + 2x + 4u 2 3j φ2 4j, x = λ j 1 4 u 1 + u 2 + u 3 j + u 3 + u 4, x = + λ j + 1 4 u 1 + u 4 u 3 j u 3, jx = λ j 1 4 u 1 j + +u 2, x = j + λ j + 1 4 u 1, j = 1,...,N, 28 which are nonlinear evolution equations with self-consistent sources. When n = 2, we have u 1t = α 2 u 1xx u 2x u 1 u 1x + 8 x 8 j jx, u 2t = α 4 u 1xu 2 + u 1 u 2x + 3 8 αu 2u 2 1 + jx + j x 2 j jx 2 x, u 3t = α u 3x + 3 4 u 1u 3 x + x + x + 2 x + 2 j jx x, 1 u 4t = α 2 u 1xu 4 + 1 4 u 1u 4x 1 2 u 4xx u 2 u 3x + 1 2 u 1u 2 u 3 3u 2 u 3 + 4 x + x N N 4u 2 + 2j + 2 jx + j x + 2jx j + 2x + 4u 2 3j 4j, x = λ j 1 4 u 1 + u 2 + u 3 j + u 3 + u 4, x = + λ j + 1 4 u 1 + u 4 u 3 j u 3, jx = λ j 1 4 u 1 j + +u 2, x = j + λ j + 1 4 u 1, j = 1,...,N, 29 which are also nonlinear evolution equations with self-consistent sources. 5 Conclusions In this paper, we have obtained a new integrable couplings of soliton equations hierarchy with self-consistent sources
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