Two-Componet Coupled KdV Equations and its Connection with the Generalized Harry Dym Equations

Transcription

1 Two-Componet Coupled KdV Equations and its Connection with the Generalized Harr Dm Equations Ziemowit Popowicz arxiv: v1 [nlin.si] 22 Oct 2012 Ma 5, 2014 Institute of Theoretical Phsics, Universit of Wrocław, Wrocław pl. M. Borna 9, Wrocław Poland, Abstract It is shown that, three different Lax operators in the Dm hierarch, produce three generalized coupled Harr Dm equations. These equations transform, via the reciprocal link, to the coupled two-component KdV sstem. The first equation gives us known integrable two-component KdV sstem while the second reduces to the known smmetrical two-component KdV equation. The last one reduces to the Drienfeld-Sokolov equation. This approach gives us new Lax representation for these equations. 1

2 1 Introduction There are man different methods of the classification of integrable equations. The most popular is the utilizations of the conservation laws and the generalized smmetr. These methods led to the discover of man new integrable sstems [1, 2], both S-integrable and C-integrable in Calogero s terminolog [3]. On the other side these methods have been used as well as to the classifications of the two-component coupled KdV tpe equations. For example Foursov in 2003 [4, 5] tested the integrabilit considering the following sstem of equations u t = F(u,v, v t = G(u,v (1 where F(u,v = F(u,v,u x,v x,u xx... denotes a differentail polnomial function of u and v. As the result Foursov presented five non-smmetrical [4] and 12 smmetrical [5] two-component coupled KdV sstems which possesses several higher-order smmetries and conserved quantities. The smmetrical sstem is such where G(u, v = F(v,u. The first three non-smmetrical sstems in this classification are known to be integrable equations u t = u xxx +6uu x 12vv x, v t = 2v xxx 6uv x (2 u t = u xxx +3uu x +3vv x, v t = u x v +uv x (3 u t = u xxx +2vu x +uv x, v t = uu x (4 The first equation is the Hirota - Satsuma sstem [6], second is the Ito sstem [7], third is the rescaled Drinfeld - Sokolov equation [8]. The fourth sstem u t = u xxx +v xxx +2vu x +2uv x (5 v t = v xxx 9uu x +6vu x +3uv x +2vv x possesses several generalized smmetries and several higher order conserved densities and therefore Foursov conjectured that it is integrable and should possesses infinitel man generalized smmetries. The last sstem in this classification u t = 4u xxx +3v xxx +4uu x +vu x +2uv x (6 v t = 3u xxx +v xxx 4vu x 2uv x 2vv x is integrable and possesses the Lax pair representations L = ( xxx u x u x( xx 1 3 v, L t = 5[L,L3/5 + ] (7 and has been first considered man ears ago b Drinfeld and Sokolov [8] and rediscovered b S. Sakovich [9] In this paper we will discuss the problem how the coupled two-component KdV sstems are connected with the generalized two component Harr Dm sstems. It is 2

3 ver well known that, Korteweg de Vries equation is connected with the Harr Dm equation [10, 11]. Strictl speaking the Harr Dm equation is reciprocal linked with the Korteweg de Vries equation. However the problem how it is possible to transforms five of the mentioned two-component coupled KdV sstems to some new generalization of the Harr Dm equations is an open problem. Such equivalence exists onl for the Hirota-Satsuma equation [12, 13]. The one possible manner to find such connections is to appl the inverse reciprocal link to the coupled KdV sstems. However in this procedure we have to assume the most general ansatz on the coupled modified KdV sstem. Next from these modified KdV functions we have to construct new functions, assuming once more the most general ansats, in order to ields the generalized Harr Dm equation. This approach lead us to the algebraic sstem which is impossible to solve. The next manner is to first find some generalization of the coupled Harr Dm equations and next construct the reciprocal link. If we progress it we expect to obtain the coupled KdV tpe sstems. At the moment we known two different two-component generalization of the Harr Dm equation. The first considered in [12, 13] is connected with the Hirota-Satsuma equation. The second, defined in [14], does not lead us to the coupled two component KdV sstem as we checked using the reciprocal link. In this paper we use the second manner and present three Lax representations which produce new generalizations of the Harr Dm equation. Appling the reciprocal link to these equations we obtained the equations 6, the Drinfeld-Sokolv and one known smmetrical coupled KdV equation listed in [5]. The idea of the construction of the Lax representation for the generalized Harr Dm sstem follows from the observation that the Lax operator 7 which, generates the sstem 6, is exactl the product of two operators. The first is the Lax operator, which produces the Kupershmidt equation, while the second is the KdV Lax operator. It is known that analogon of the Lax operator for the Kupershmidt equation, in the Dm hierarch, is third order operator while analogon of the Lax operator for the KdV equation is the Harr Dm Lax operator. If we consider the product of these two operators we expect to obtain some new generalization of the Harr Dm equation and in the next, b application of the reciprocal link, new interacted two-component KdV equations. In principle we should consider three different operators because the Harr Dm equation possesses two different Lax operators the standard [15] and recentl discovered the nonstandard [16]. We show that the Lax operator constructed as the product of third order and standard Harr Dm operators leads us to the equation 6. The operator constructed as the product of third order operator and non-standard Harr operator gives us known smmetrical two-component KdV equation. The Lax operator constructed as the product of standard and nonstandard Lax operator of Harr Dm leads us to the Drinfeld-Sokolov equation. The paper is oragnized as follows. In the second section we present three Lax operators which generate three generalization of the Harr Dm equation. The third section describes the reciprocal link from generalized Harr Dm equations to the coupled two-component KdV equations. In the fourth section we compare the Lax representation obtained using our approach with the known Lax representation for the coupled two-component KdV equations. 3

4 integrabilit properties of new coupled KdV equations. The last section contains concluding remarks. 2 New generalization of the Harr Dm equation We meet three equivalent expression on the Harr Dm (HD equation in the literature [11, 18] w t = (w 1/2 xxx, u t = ((u xx 1/2 x, v t = v 3 v xxx (8 where v = 2 1/3 w 1/2 and u xx = w respectivel. This equation have been discovered b H. Dm and M. Kruskal in 1975 [17] and share man of the properties tpical of the soliton equations as for example it has a Bi-Hamiltonian structure δh 1 w t = D 1 δw = D δh 2 2 (9 δw where D 1 = 3, D 2 = w +w (10 H 1 = 2 dx w 1/2, H 2 = 8 dx w 5/2 wx 2 and an infinite number of conservation laws and infinitel man smmetries [18]. This equation is connected with the Korteweg de Vries (KdV equation via the reciprocal transformation [11, 10]. The HD equation follows from the following Lax representation. L s = 1 w 2, L s,t = 2[L s,(l 3/2 s 2 ] (11 where subscript 2 denotes the projection to the part with the powers greater or equal to 2. On the other side there exists the second Lax operator for the Harr Dm equation, the nonstandard one, recentl discovered in [16], L ns = w 1/4 1 w 1/4 2, L ns,t = [L ns,(l 3 ns 2]/2 (12 Now taking into account that the sstem of equation 6 has the Lax representation 7 in which the Lax operator is factorized as a product of the Lax operator of the Kupershmidt equation and the Lax operator of the Korteweg de Vries equation let us consider four Lax operators L s,l sn,l HS,L DS L = w 3 xxx +k 0 w 2 w x x (13 L s = Lv 2 2, L SD = Lv 1 v 2 L HS = w 2 u 2, L DS = w 1/2 1 w 1/2 21 u 2 The operator L HS has been considered in [12] and leads us to the Hirota-Satsuma equation. From that reasosn we will not consider it in this paper. 4

5 Let us mention that the operator L generates b L t = [L 5/3 2,L] (14 the fifth order equation which, could be transformed b reciprocal link, to the Kupershmidt or Sawada-Kotera equation for k 0 = 3 or k = 3 2 respectivel. The time evolution of the ˆL s, ˆL DS and L DS L SD,t = [(L 3/5 SD 2,L SD, L s,t = [(L 3/5 s 2,L s ], L DS,t = [(L 3 DS 2,L DS ] (15 produces the consistent solution onl for k 0 = 3 2 For L SD operator we obtained while for L s w t = 2 3 w5/2 ( w 3/2 (v 6/5 w 6/5 x w t = 2 3 w5/2 ( w 3/2 (v 3/2 w 3/4 x For the L DS operator we obtained u t = 1 2 ( 1 w 3 ReciprocaL link xxx xx xx, v t = 1 4 v3 (w 9/5 v 4/5 xxx (16, v t = v 3 (w 3/4 v xx v 3/4 x (17 ( w( 1, w t = 2 u w xx x Introducing the parametriaztion of the function w,v for the L SD operator as and for the L s operator as and for the L DS operator as we obtained that the L SD operator generates (18 w = ae b, v = ae 3b/2 (19 w = ae b, v = ae 3b/2 (20 u = 1 a eb w = 1 a 2e b (21 a t = 1 10 ( a xxxa 3 +9b xxx a 3 +27b xx (a x a 3 3b x a 3 (22 81b 2 x a xa 3 9b x (a xx a 3 a 2 x a2, b t = 1 10 (a xxxa 2 +11b xxx a 3 +b xx (33a x a 2 9b x a b 3 x a3 9b 2 x a xa 2 +b x (21a xx a 2 +6a 2 x a 5

6 while the L s operator gives us a t = 1 4 (a xxxa 3 9b xxx a 4 27b xx (a x a 3 b x a 4 +27b 2 x a xa 3 (23 9b x (a xx a 3 +a 2 xa 2 b t = 1 4 ( a xxxa 2 +b xxx a 3 +3b xx (a x a 2 +3b x a 3 18b 3 x a3 + and for the L DS operator 9b 2 xa x a 2 3b x (a xx a 2 a 2 xa a t = a2 ( 4axx a+b 2 x 2 a2 +2(b x a 2 x (24 x b t = 1 ( 2a 3 2 x b 3 x a3 2b x a 2 (b x a x +a xx (2a 2 x a+b2 x a3 +2b x a x a 2 x We use the reciprocal transformation where now x,a and b are defined as x = p(,t, a(x,t = p(,t, b(x,t = q(,t (25 in order to find the time evolution of p,q. For L SD case we obtained p t = 1 20 ( 2p +3 p2 +p (18q 81q p 2 (26 q t = 1 ( 2 p 4 p (8 p 6 p2 +22q 20 p p 2 +q (9q +18 p 27 p p p p 2 while for L s p t = 1 8 (2p 3 p2 p p (18q 27q 2 (27 q t = 1 8 ( 2p 4 p +p (8 p p 2 and for the L DS operator 6 p +2q p 3 q (9q +6 p 9 p2 p p 2 p t = 1 4 (4p 4 p2 +2q p +q 2 p p +2q p (28 q t = 1 ( 8p p p 2 +8p 3 4 p 3 2q q q 3 4p 1 (q p +q 2 p +q p To verif this one can use the identities Next we appl the transformation x = 1 a, a t = p,t p p t p, b t = q t q p t p (29 f = p p, g = q (30 6

7 from which we conclude that for the L SD operator we obtained f t = 1 ( 2f +f 3 +18g +18g f 162gg 81g 2 f 20 g t = 1 (2f 2f f +22g +9g 3 +18gf 9gf 2 20 (31 while for the L s operator f t = 1 8 (2f f 3 18g 2 18g f +54g g +27g 2 f (32 g t = 1 8 ( 2f +f f +2g 9g 3 6gf +3gf 2 and for the L DS operator f t = 1 4 g t = 1 4 ((4f +2g +2f 2 +g 2 +2gf +2g f +g 2 f +2gf 2 ((4f 2 +g 2 +4gf +g 3 +4g 2 f +4gf 2 (33 If we appl the Miura-tpe transformation for the L SD case u = 1 8 (8f +3f 2 +27g 2 6fg (34 v = 1 24 (24g f 2 9g 2 +18fg then the sstem of equation 33 reduces to u t = 1 20 ( 2u +18v +9uu 9uv 405vv 27vu (35 v t = 1 20 (2u +22v 5uu 27uv +9vv 9vu If we appl the linear transformation of the variables u,v and appl the scale of the time t t 4, u v u, v u+v ( then the sstem 35 reduces to the Drinfeld-Sokolov equation. Appling different Miura-tpe transformation for the L s case we obtained u = 1 2 (2f f 2, v = 1 2 (2g 3g 2 (37 u t = 1 4 (u +3uu 9v 18uv 9vu (38 v t = 1 4 ( u +v 3uv 6vu +9vv 7

8 Appling the scaling u 2u,v 2 v the sstem of equation 38 reduces to the 3 9 smmetrical form u t = 2( 1 2 u uu v +2v u+vu. (39 It is one of the rescaled smmetrical coupled two-component KdV sstem considered b Foursov [5] 1. If we appl the Miura-tpe transformation s = f 1 2 f2, z = g +f g2 +fg (40 for the L DS case, then the sstem of equation 33 reduces to s t = 1 2 ( (2s +z +3s 2 +sz x +z x s (41 z t = 3 4 (4s2 +2zs+z 2. (42 If we further shift the function s s z/2 we obtain s t = 1 2 (2s xxx z x s 2zs x (43 z t = 6s s. (44 It is exactl the first Drinfeld - Sokolov equation DS1 if we appl the scaling s s/2,z z, t t /2. 4 The Lax representations of the coupled two-component KdV tpe sstems The known Lax representation of the Drienfeld-Sokolov [8] is L = ( 3 +(s+z (s+z x( 3 +(s z (s z x, L t = [L 1/2 0,L] (45 while for the sstem 6 is 7. We show that our approach produces quite different Lax represenation for these equations. First let us consider the L DS operator for which we appl the gauge transformation ˆL DS = e b/2 L DS e b/2 (46 and next the reciprocal transformation and once more the gauge transformation ˆL DS 1 p ˆLDS p (47 1 the equation 4.9 in [4] where α = 1. 8

9 Finall if we appl the Miura-tpe transformation 40 we obtain ˆL DS = 3 +(3s+2z + 1( s z 1 4 z2 1 4 zs (48 3 ( 1 2 (f +g2 + 1 (f +g( 2 (s+ 1 2 z(f +g s 1 2 z Strictl speaking we obtained the Lax operator which generates the time evolution of the function f, g and implicitl the Drinfeld-Sokolv equation. Interestingl this operator produces also the conserved quantities for this equation. It can be easil seen for the first quantities H 0 = Res(L = d s z2 +zs. (49 For the L s operator making the similar transformations we obtined ˆL s = 4x +(u+3v xx (u+3v x (u+3v xx (u 3v (u 3v x 1 (u 3v For the L DS operator making the similar transformations we obtined ˆL DS = e d((f+3g/2 L DS e d((f+3g/2 = ( 3 g (2f f 2 +6g +9g 2 ( +3g (2f f 2 +6g +9g 2. (50 The time evolution of ˆL DS ˆLDS,t = [(ˆL 3/5 DS +, ˆL DS ] (51 generates the coupled mkdv equation 31 and in the implicit form also the sstem of 6. However ˆL s can not be rewritten, after applications of the Miura transformation 34, purel in terms of the local KdV variables u,v and its derivatives. As we checked the residuum formula of the ˆL generates the conserved quantit for the MKdV equation as well as for the coupled KdV sstem 35. Indeed the first two nontrivial conserved laws obtained from ˆL operator are G 2 = res(ˆl 1/5 = (f 2 +9g 2 d (52 G 4 = res(ˆl 3/5 = (4f f f 4 72g 2 f 396g g 36g f g gf 81g g 2 f 2 d Here the lower index in G,H denotes the KdV weight of the function e.g [u] = [v] = 2,[f] = [g] = 1,[ ] = 1. In order to obtain the conserved quatities for the equation 6 from the ˆL s operator let us appl the Miura transformation 40 and rewrite these quantities as G 2 = (f 2 +9g 2 +3f +3g d = 3 (u+vd = H 2 (53 G 4 = d(u 2 99v 2 18uv. In the similar wa it is possible to obtain the higher order conserved densities. 9

10 5 Concluding remarks In this paper we presented three different generalizations of the Harr Dm equation. These equations have been obtained using different Lax operators in the Harr Dm hierarch. Using the reciprocal link we showed that our equations are reduced to the coupled two-component KdV equations. However how remaining coupled twocomponent KdV equations listed in [4, 5] are connected with the generalized Harr Dm equations is still an open problem. References [1] A. Mikhailov, A. Shabat, V. Sokolov The smmetr approach to classificatiobn of integrable equations in What is Integrabilit edited b. V. Zakharov (Springer- Verlag [2] A. Mikhailov, A. Shabat, R. Yamilov The smmetr approach to the classification of nonlinear equations. Complete lists of integrable sstems Russ. Math. Surves 42 ( [3] F. Calogero Wh are certain nonlinear PDEs both widel applicabe and integrable in What is Integrabilit edited b. V. Zakharov (Springer-Verlag [4] M. Foursov Towards the complete classification of homogenous two-component integrable equations J. Math. Phs. 44 ( [5] M. Foursov On integrable coupled KdV-tpe sstems Inverse Problems 16 (2000, [6] R. Hirota, J. Satsuma Soliton solutions of a coupled Korteweg-de Vries equation Phs. Lett 85A [7] M. Ito, Smmetries and conservation laws of a coupled nonlinear wave equation, Phs. Lett. A, 1982, V.91, [8] V. Drinfeld, Sokolov V, New evolutionar equations possessing an (L,A-pair, Trud Sem. S.L. Soboleva (1981, no (in Russian. [9] S.Sakovich Coupled KdV Equations of Hirota-Satsuma Tpe Journal of Nonlinear Mathematical Phsics 1999, V.6 N , ibid Addendum to: Coupled KdV Equations of Hirota-Satsuma Tpe Journal of Nonlinear Mathematical Phsics Volume 6, Number 2 (2001, [10] N.H Ibragimov Sur l équivalence des équations d évolution, qui admettent une algébre de Lie-Bäcklund infinie C. R. Acad. Sci., Paris ( [11] W. Hereman, P. P. Banerjee and M. R. Chatterjee, On the Nonlocal Equations and Nonlocal Charges Associated with the Harr Dm Hierarch Korteweg-de Vries equation J. Phs. A 22 (

11 [12] Z. Popowicz The Generalized Harr Dm Equation Phs. Lett. A (2003. [13] S. Sakovich Transformation of a generalized HD equation into the Hirota- Satsuma sstem Phs. Lett. A321 ( [14] M. Antonowicz, A. Ford Coupled Harr Dm equations with multi-hamiltonian structures J. Phs. A: Math. Gen. 21 (1988 L269-L275. [15] B. Konopelchenko, W.Oevel; An r-matrix Approach to Nonstandard Classes of Integrable Equations Publ. Rims Koto Unive. 29 ( [16] K. Tian, Z. Popowicz, Q. Liu A non-standard Lax formulation of the Harr Dm hierarch and its supersmmetric extension J. Phs A:Math.Theor. 45 ( (8pp. [17] M.D. Kruskal Lecture Notes in Phsics vol.38, Springer Berlin 1975, p [18] J.C. Brunelli, G.A.T.F. da Costa On the Nonlocal Equations and Nonlocal Charges Associated with the Harr Dm Hierarch J.Math. Phs. 43 (