Exact Solutions of Supersymmetric KdV-a System via Bosonization Approach

Commun. Theor. Phys. 58 1 617 6 Vol. 58, No. 5, November 15, 1 Exact Solutions of Supersymmetric KdV-a System via Bosonization Approach GAO Xiao-Nan Ô é, 1 YANG Xu-Dong Êü, and LOU Sen-Yue 1,, 1 Department of Physics, Shanghai Jiao Tong University, Shanghai 4, China Faculty of Science, Ningbo University, Ningbo 1511, China Received April 6, 1; revised manuscript received July 5, 1 Abstract Bosonization approach is applied in solving the most general N = 1 supersymmetric Korteweg de-vries equation with an arbitrary parameter a skdv-a equation. By introducing some fermionic parameters in the expansion of the superfield, the skdv-a equation is transformed to a new coupled bosonic system. The Lie point symmetries of this model are considered and similarity reductions of it are conducted. Several types of similarity reduction solutions of the coupled bosonic equations are simply obtained for all values of a. Some kinds of exact solutions of the skdv-a equation are discussed which was not considered integrable previously. PACS numbers:..ik,..sv,..jr, 11.1.Lm, 5.45.Yv Key words: supersymmetric KdV-a equation, bosonization, symmetry reductions, exact solutions The supersymmetric integrable systems are very important in many physical fields especially in quantum field theory and cosmology such as superstring theory. [1 In quantum level, the bosonization approach is one of the powerful methods which simplifies the procedure to treat complex fermionic fields. [ However, it is difficult to find a proper bosonization procedure for classical supersymmetric integrable models though the supersymmetric quantum mechanical problems can be successfully bosonized. [4 It is significant if one can establish a proper bosonization procedure to treat the supersymmetric systems even if in the classical level. In Ref. [5, we have established a bosonization method for the usual N = 1 supersymmetric KdV skdv equation to find various new types of exact solutions. In this paper, we use the method to Mathieu s skdv-a equation, [6 which represents the most general nontrivial supersymmetric extension of the KdV equation. As the fermionic superfield extension of the KdV equation, the skdv-a equation can be shown as Φ t + Φ xxx + adφ x Φ + 6 adφφ x =, 1 with an arbitrary parameter a, where Φθ, x, t = ξx, t + θux, t, θ is a Grassmann variable, and D = θ +θ x is the covariant derivative. The component version of Eq. reads u t + u xxx aξξ xx + 6uu x =, ξ t + ξ xxx + au x ξ + 6 auξ x =, where u and ξ are bosonic component field and fermionic component field, respectively. Vanishing ξ in Eq., only the usual classical KdV equation remains. If a =, the skdv- model is clearly trivial integrable. For nontrivial cases, only the skdv- model is proven to possess an infinite number of conservation laws, a second Hamiltonian structure and a nontrivial Lax representation. [6 Thereafter, many remarkable properties of the skdv- model have been discovered, such as the Painlevé property, [7 the Darboux transformation, [8 the bilinear forms, [9 1 the Bäklund transformation BT, [11 the nonlocal conservation laws, [1 and the bosonizations. [5 However, study on integrability and exact solutions of the skdv-a system in the case of a and a are much rarer. The bosonization approach can effectively avoid difficulties caused by intractable fermionic fields that are anticommuting. This method has been used by Andrea et al. [1 to obtain new integrable bosonic system. In addition, the method is one of the powerful methods, which simplify the procedure to treat complex fermionic fields in quantum field theory. [14 17 In this paper, we will apply this powerful bosonization approach to the skdv-a equation under discussion for all values of the parameter a. Then this model can be changed to a system of equations with respect to bosonic fields, including the completely integrable KdV equation and several variable-coefficient linear equations, which are relatively easy to deal with. By applying the Lie point symmetry analysis, the coupled bosonic equations can be reduced to some systems of ordinary differential equations. As a particular case only two fermionic parameters are included in the solution of the skdv-a model. One can expand the component fields ξ and u in the form of ξx, t = pζ 1 + qζ, ux, t = u + u 1 ζ 1 ζ, Supported by the National Natural Science Foundation of China under Nos. 111759, 11751 and 1958, Scientific Research Fund of Zhejiang Provincial Education Department under Grant No. Y117148, and K.C. Wong Magna Fund in Ningbo University E-mail: lousenyue@nbu.edu.cn c 11 Chinese Physical Society and IOP Publishing Ltd http://www.iop.org/ej/journal/ctp http://ctp.itp.ac.cn

618 Communications in Theoretical Physics Vol. 58 where ζ 1 and ζ are two Grassmann parameters ζ 1 = ζ = ζ 1ζ + ζ ζ 1 =, while the coefficients p px, t, q qx, t, u u x, t, and u 1 u 1 x, t are four usual real or complex functions with respect to the spacetime variables x and t. Then the skdv-a system is changed to u t + u xxx + 6u u x =, p t + p xxx + 6 au p x + au x p =, q t + q xxx + 6 au q x + au x q =, 4a 4b 4c u 1t + u 1xxx + 6u u 1x + 6u x u 1 = apq xx qp xx. 4d That is just the bosonic-looking form of the skdv-a system in two fermionic parameter case. Apparently, the system 4a 4d has no fermionic variables. Moreover, Eq. 4a is exactly the usual KdV equation, which has been widely studied. Equations 4b 4b are linear homogeneous in p and q, respectively. Equation 4d is linear nonhomogeneous in u 1 and contains a source in terms of p and q. Thereby, in principle, these equations can be easily solved. This fact shows one of the main advantages of the bosonization approach. Now let us consider the Lie point symmetries σ u = Xu x + Tu t H u, σ u1 = Xu 1x + Tu 1t H u1, σ p = Xp x + Tp t H p, σ q = Xq x + Tq t H q, 5 and the symmetry reduction of the bosonic-looking equations 4, where X, T, H u, H u1, H p, H q are functions of variables x, t, u, u 1, p, q. The Lie point symmetries σ u, σ u1, σ p, and σ q of the form 5 satisfy the linearized equations of the model 4 σ u t + σ u x + 6u σ u x + 6 u x σ u =, σ p t + σ p x + 6 au σ p x + a u x σ p p +6 aσ u x + a σ u x p =, σ q t + σ q x + 6 au σ q x + a u x σ q 6a 6b q +6 aσ u x + a σ u x q =, 6c σ u1 + σ u1 σ u1 t x + 6u x + 6 u x σ u 1 + 6 x u 1σ u +a q σ p x x p σ q x + p x σ q q x σ p =, 6d which means the system 4 is form-invariant under the transformations u u + ǫσ u, u 1 u 1 + ǫσ u1, p p + ǫσ p, q q + ǫσ q, 7 where ǫ is an infinitesimal parameter. Substituting symmetries 5 into Eqs. 6, then eliminating the quantities u t, u 1t, p t, q t as well as their higher order derivatives via Eqs. 4, and then vanishing all the coefficients of the independent terms of the polynomials of u, u 1, p, q and their partial derivatives, the general form of the transformations X, T, H u, H u1, H p, H q is easily found as X = c 1 x + c 7, T = c 1 t + c, H u = c 1 u, H u1 = c 1 + c + c 6 u 1, H p = c p + c 4 q, H p + c 6 q, 8 with seven arbitrary constants c 1, c, c, c 4, c 5, c 6, c 7. Thus, the Lie point symmetries of the model 4 have the form σ u = c 1 x + c 7 u x + c 1 t + c u t + c 1 u, σ u1 = c 1 x + c 7 u 1x + c 1 t + c u 1t c 1 + c + c 6 u 1, σ p = c 1 x + c 7 p x + c 1 t + c p t c p c 4 q, σ q = c 1 x + c 7 q x + c 1 t + c q t c 5 p c 6 q. 9 Solving the characteristic equations, dx X = dt T = du = du 1 = dp = dq, 1 H u H u1 H p H q with X, T, H u, H u1, H p, H q given by Eq. 8, one can get the reduction variables and similarity reduction solutions. According to the different selections of constant parameters, there are some different types of similarity reduction solutions, in which similarity reduction functions satisfy the corresponding similarity reduction equations at the following concrete cases. Case 1 c 1 and c 4. In the case of c 1 and c 4, the independent reduction variable reads c 1 x + c 7 η =, 11 c 1 c 1 t + c 1/ and the similarity solutions possess the form U u = c 1 t + c, / p = t + c b+b1/6c1 P + t + c b b1/6c1 Q, c 1 c 1 q = b + b 1 t + c b+b1/6c1 P c 4 c 1 + b b 1 t + c b b1/6c1 Q, c 4 c 1 u 1 = c 1 t + c c1+b/c1 U 1, 1 where U, U 1, P, and Q are functions of η express as Eq. 11, and the redefined arbitrary constants b 1, b, b are described as follow b 1 = c c 6 + 4c 4 c 5, b = c + c 6, b = c 6 c. 1 The corresponding reduction equations can be simply obtained by substituting Eqs. 1 to the model 4 with the

No. 5 Communications in Theoretical Physics 619 final forms: U ηηη + 6U c 1 ηu η c 1 U =, P ηηη + [6 au c 1 ηp η + au η + b + b 1 Q ηηη + [6 au c 1 ηq η + au η + b b 1 U 1ηηη + 6U c 1 ηu 1η + 6U η + c 1 + b U 1 P =, Q =, = a c 4 c 1 b/c1 b 1 QP ηη PQ ηη. 14 Case c 1, c 4 =, and c 6 c. In this case we obtain the similarity solution U u = c 1 t + c, p = c 1t + c / c/c1 P, c c 6 c 1 t + c c/c1 P + c 1 t + c c6/c1 Q, u 1 = U 1 c 1 t + c c1+b/c1, 15 with the reduction equations U ηηη + 6U c 1 ηu η c 1 U =, P ηηη + [6 au c 1 ηp η + au η + c P =, Q ηηη + [6 au c 1 ηq η + au η + c 6 Q =, U 1ηηη + 6U c 1 ηu 1η + 6U η + c 1 + b U 1 = apq ηη QP ηη, 16 while the reduction variable η is the same as in Eq. 11. Case c 1, c 4 =, and c 6 = c. The third type of similarity solutions for c 1 can be obtained as follow U u = c 1 t + c, p = c 1t + c / c/c1 P, c 1 lnc 1 t + c c 1 t + c c/c1 P + c 1 t + c c/c1 Q, u 1 = c 1 t + c c1+c/c1 U 1, 17 with the same η in Eq. 11 and the reduction equations in the form U ηηη + 6U c 1 ηu η c 1 U =, P ηηη + [6 au c 1 ηp η + au η + c P =, Q ηηη + [6 au c 1 ηq η + au η + c Q + c 5 P =, U 1ηηη + 6U c 1 ηu 1η + 6U η + c 1 + c U 1 = apq ηη QP ηη. 18 Case 4 c 1 =, c, and c 4. In this case the reduced independent variable is a simply traveling wave variable η = c x b 4 t, b 4 = c 7, 19 c while the similarity solutions can be expressed as u = U, p = e b+b1t/c P + e b b1t/c Q, q = b + b 1 c 4 e b+b1t/c P + b b 1 c 4 e b b1t/c Q, u 1 = e bt/c U 1, where U, U 1, P, and Q are similarity reduction functions related to the variable η in Eq. 19. The corresponding reduction equations of the model 4 are U ηηη + 6U b 4 U η =, P ηηη + [6 au b 4 P η + au η + b + b 1 P =, Q ηηη + [6 au b 4 Q η + U 1ηηη + 6U b 4 U 1η + au η + b b 1 6U η + b U 1 Q =, = ac b 1 c 4 QP ηη PQ ηη, 1 where b 1, b, and b are shown as the relationship 1. Case 5 c 1 =, c, c 4 =, and c 6 c. In this case, we have the similarity solution u = U, p = e ct/c P, c c 6 e ct/c P + e c6t/c Q, u 1 = e bt/c U 1, and the related reduction equations for reduction functions U, P, Q, and U 1 U ηηη + 6U b 4 U η =, P ηηη + [6 au b 4 P η + au η + c P =, Q ηηη + [6 au b 4 Q η + U 1ηηη + 6U b 4 U 1η + au η + c 6 6U η + b U 1 Q =, = ac PQ ηη QP ηη, with the traveling wave variable η as same as in Eq. 19. Case 6 c 1 =, c, c 4 =, and c 6 = c. In this case, the similarity solutions can be obtained as follows u = U, p = e ct/c P, c e ct/c tp + e ct/c Q, u 1 = e ct/c U 1. 4 The corresponding reduction equations possess the forms U ηηη + 6U b 4 U η =, P ηηη + [6 au b 4 P η + Q ηηη + [6 au b 4 Q η + au η + c c Q + c 5 c P =, U 1ηηη + 6U b 4 U 1η + au η + c 6U η + c 5a P =, 5b U 1 5c = ac PQ ηη QP ηη, 5d with U, U 1, P, Q being functions of variable η like Eq. 19.

6 Communications in Theoretical Physics Vol. 58 On the basis of the above studying, once any system of the coupled ordinary differential equations ODEs 14, 16, 18, 1,, and 5 are solved, the related special solutions of the skdv-a system can be obtained via corresponding similarity solutions and Eqs.. For the first three cases, the reduction equation for U is equivalent to the Painlevé II equation. The remained equations for P, Q, and U 1 are linear ones with variable coefficients. For the last three reduction cases, the U equation can be directly integrated, and the result reads a U η = U + b 4U c 8U c 9, 6 c where c 8 and c 9 are two integral constants and a = ±1. Introducing the variable transformations Pη = U PU du, Qη = U QU du, U 1 η = U η Ũ 1 U, 7 into the reduction equations 5b, 5c, and 5d in the last case, the linear ODEs for P, Q, and Ũ1 with the particular conditions of c = and c 5 = can be constructed as U U b 4U + c 8U + c 9 d P du U U b 4U + c 8U + c 9 d Q du U b 4U + c 8U + c 9 d Ũ 1 = ac U P du QdU Q + 5U 4b 4U + c 8U + c 9 d P du + [a + 18U 1b 4U + 6c 8 P =, + 5U 4b 4U + c 8U + c 9 d Q du + [a + 18U 1b 4U + 6c 8 Q =, + U b 4U + c 8 dũ1 du P du A 1 U + b 4 U c 8U c 9, 8 where A 1 is an integral constant. Solving these linear ODEs and taking a = for example, one kind of dependent variable transformation of skdv- system can be constructed as c9 U U + b 4 U c {B 1 cos[ru + B sin [RU }, if c 9, 8U c 9 B H 1 + B 4 H P = U U U b 6, if c 9 = and c 8, where [ b 4 B 5 U arctanh 1 U + b 4 b 4 B 6 b 4 U b 4 U U, if c 9 = and c 8 =, b 4 U c9 U U + b 4 U c {C 1 cos[ru + C sin [RU }, if c 9, 8U c 9 C H 1 + C 4 H Q = U U U b 6, if c 9 = and c 8, [ b 4 C 5 arctanh 1 U + b 4, if c 9 = and c 8 =, b 4 U b 4 U b 4 U Ũ 1 = U C 6 b 4 U FU du + A U + b 4U c 8U c 9 / du + A, 9 c9 RU = U U + b 4 U c du, 8U c 9 FU = c U Q P du P QdU U + b 4U c 8U c 9 A 1 = c P dq du Q dp du U + b 4U c 8U c 9 A 1. Involved in above result 9, A i i = 1,,, B i i = 1,,...,6, and C i i = 1,,..., 6 are arbitrary integral constants, b5 H 1 H 1, b 6 4, 1,,, 1, U b5, H H, b 5,, b 6 b 6 4b 6, 1, 1, U b 6

No. 5 Communications in Theoretical Physics 61 are two special Heun functions, while constant transformations c 8 = b 5 b 6 and b 4 = b 5 +b 6 are taken into consideration. Substituting Eqs. 9 into transformation equations 7, four component fields, which describes superfield with the bosonization procedure follow the relationship {B 1 sin [RU + B cos[ru + B 7 }U, if c 9, [ B H 1 + B 4 H U U U b 6 du + B 8 U, if c 9 = and c 8, P = { [ B 5 1 U + arctanh 1 U U b 4 b 4 [ +B 6 1 U + arctanh 1 U + B 9 }U, if c 9 = and c 8 =, U b 4 b 4 {C 1 sin[ru + C cos[ru + C 7 }U, if c 9, [ C H 1 + C 4 H U, if c 9 = and c 8, Q = U 1 = U U U b 6 du + C 8 { [ C 5 1 U + arctanh 1 U U b 4 b [ 4 +C 6 U 1 U b 4 + arctanh 1 U b 4 + C 9 }U, if c 9 = and c 8 =, a c U + b 4U c 8U c 9 [ FU du + A U + b 4U c 8U c 9 / du + A, 1 where B i and C i i = 7, 8, 9 are also arbitrary integral constants. Thus, we have obtained one type of traveling wave solutions of the skdv- system with the traveling wave solution U of the KdV equation. Particularly, one traveling wave solution of KdV equation is known as u = sech c x 4c t, corresponding to c 8 =, c 9 =, b 4 = in Eq. 6. The solution 4 can be constructed as p = B 5[sinhc x 4c t + c x 4c tsechc x 4c t + B 6[tanhc x 4c t + c x 4c tsech c x 4c t + c B 9sech c x 8c t, q = C 5 [sinhc x 4c t + c x 4c tsechc x 4c t + C 6 [tanhc x 4c t + c x 4c tsech c x 4c t + C 9 sech c x 8c t, u 1 = 4c C 56c x 4c t sech c x 4c t[c x 4c ttanhc x 4c t + c C 56c x 4c ttanhc x 4c t[5sech c x 4c t 8 cosh c x 4c t 8 + c 6 C 56[1 cosh c x 4c t 15sech c x 4c t + 1, + 4c C 59 c x 4c tsech c x 4c t[c x 4c ttanhc x 4c t 4c C 59 tanhc x 4c t[ lnsechc x 4c t + cosh c x 4c t + 1 c C 69[7c x 4c tsech c x 4c ttanhc x 4c t + 7 tanh c x 4c t + sinh c x 4c t + 4A sech c x 4c ttanhc x 4c t A c 5 cosh c x 4c t + 5A A 1 16c 5 [ sinh c x 4c t + c x 4c tsech c x 4c ttanhc x 4c t + tanh c x 4c t, where C 56 = B 5 C 6 C 5 B 6, C 59 = B 5 C 9 C 5 B 9, and C 69 = B 6 C 9 C 6 B 9. Actually, this traveling wave solution is the soliton solution of the skdv- model. More generally, a new type of solution of skdv-a system can be obtained from Eqs. 4 directly, p = a 1 u, q = a u, u 1 = σu, 4 where a 1, a are two arbitrary constants, and σu represents arbitrary symmetry of the usual KdV equation 4a

6 Communications in Theoretical Physics Vol. 58 for all possible solution u of it. This means that if the KdV equation with u as a solution u is not restricted to traveling wave form, solution 4 can satisfy the skdv-a system 4 automatically. It should be emphasized that this type of solution is independent of the parameter a. In summary, with two fermionic parameters, the bosonization procedure of the supersymmetric systems has been successfully applied to the skdv-a equation in detail. Such a nonlinear supersymmetric system is simplified to the usual KdV equation together with several variable coefficient linear differential equations without fermionic variables. The bosonization procedure makes applying the theory of integrable system into the study on the supersymmetric system easier. The Lie point symmetry analysis has been applied on this bosonic-looking form of the skdv-a system, and the similarity solutions and corresponding reduction equations are obtained. In addition, a certain type of exact solution of the skdv-a system for all values of a has been discovered, says, solution 4. In other words, any kind of solutions of the usual KdV equation such as the N-soliton solutions, τ- function solutions can be extended to those of the skdva equation for all a that means the bosonized systems reduced from skdv-a may be integrable for all values of a. References [1 D.J. Gross and A. Migdal, Nucl. Phys. B 4 199 ; M. Douglas, Phys. Lett. B 8 199 176; R. Dijkgraaf and E. Witten, Nucl. Phys. B 4 199 486. [ P.P. Kulish and A.M. Zeitlin, Phys. Lett. B 597 4 9; Nucl. Phys. B 79 5 578; Nucl. Phys. B 7 5 89. [ K.B. Efetov, C. Pepin, and H. Meier, Phys. Rev. Lett. 1 9 1864; A.O. Gogolin, A.A. Nersesyan, and A.M. Tsvelik, Bosonization and Strongly Correlated Systems, Cambridge University Press, Cambridge 1998; T. Giamarchi, Quantum Physics in One Dimension, Oxford University Press, Oxford ; A. Luther, Phys. Rev. B 19 1979. [4 M.S. Plyushchay, Ann. Phys. 45 1996 9; F. Correa and M.S. Plyushchay, Ann. Phys. 7 49. [5 X.N. Gao and S.Y. Lou, Phys. Lett. B 77 1 9. [6 P. Mathieu, J. Math. Phys. 8 1988 499. [7 P. Mathieu, Phys. Lett. A 18 1988 169. [8 Q.P. Liu, Lett. Math. Phys. 5 1995 115. [9 A.S. Carstea, Nonlinearity 1 1645. [1 A.S. Carstea, A. Ramani, and B. Grammaticos, Nonlinearity 14 1 1419. [11 Q.P. Liu and Y.F. Xie, Phys. Lett. A 5 4 19. [1 S. Andrea, A. Restuccia, and A. Sotomayor, J. Math. Phys. 46 5 1517. [1 S. Andrea, A. Restuccia, and A. Sotomayor, J. Math. Phys. 4 1 65. [14 A. Luther, Phys. Rev. B 19 1979. [15 A.O. Gogolin, A.A. Nersesyan, and A.M. Tsvelik, Bosonization and Strongly Correlated Systems, Cambridge University Press, Cambridge 1998. [16 T. Giamarchi, Quantum Physics in One Dimension, Oxford University Press, Oxford. [17 K.B. Efetov, C. Pepin, and H. Meier, Phys. Rev. Lett. 1 9 1864.