Modified KdV hierarchy : Lax pair representation and bi-Hamiltonian structure

Mo died KdV hierar h y : Lax pair represen tation and bi-Hamiltonian struture Amita v a Choudh uri † , Beno y T alukdar ‡ and Umapada Das ∗ † ‡ Dep artment of Physis, Visva-Bhar ati University, Santiniketan 731235, India ∗ and ∗ A bhe dananda Mahavidyalaya, Sainthia 731234, India W e onsider equations in the mo died KdV (mKdV) hierar h y and mak e use of the Miura trans- formation to onstrut expressions for their Lax pair. W e deriv e a Lagrangian-based approa h to study the bi-Hamiltonian struture of the mKdV equations. W e also sho w that the omplex mo di- ed KdV (mKdV) equation follo ws from the ation priniple to ha v e a Lagrangian represen tation. This represen tation not only pro vides a basis to write the mKdV equation in the anonial form endo w ed with an appropriate P oisson struture but also help us onstrut a semianalytial solution of it. The solution obtained b y us ma y serv e as a useful guide for purely n umerial routines whi h are urren tly b eing used to solv e the mKdV eqution. P A CS n um b ers: 47.20.Ky , 42.81.Dp, 02.30.Jr Keyw ords: Real and omplex mo died KdV equations; Lax pair represen tation; Hamiltonian struture; Ritz optimization pro edure; Solitary-w a v e solution 1. In tro dution Nearly fort y y ears ago Lax [1℄ sho w ed that the K o- rtew eg de V ries (KdV) initial v alue problem for u = u ( x, t ) giv en b y u t = u 3 x − 6 uu x (1) with u ( x, 0) = V ( x ) (2) is but one of the innite family of equations that lea v e the eigen v alue of the S hrödinger equation with the p oten tial V ( x ) in v arian t in time. The subsripts of u in (1) denote dieren tiation with resp et to the asso iated indep enden t v ariables. The family of equations diso v ered b y Lax often go es b y the name KdV hierar h y and is generated b y making use of the reursion op erator [2℄ Λ = ∂ 2 x − 4 u − 2 u x ∂ − 1 x , ∂ x = ∂ ∂ x (3) in the dieren tial relation u t = Λ n u x , n = 0 , 1 , 2 , 3 , ... . (4) The KdV equation (1) is reognized as the solv abilit y ondition for the system Lψ = λψ (5 a ) and ∂ t ψ = Aψ , ∂ t = ∂ ∂ t (5 b ) with L = − ∂ 2 x + u , (6 a ) ∗ Eletroni address: bino y123bsnl.in, amita v a_ h26y aho o.om the so-alled S hrödinger op erator. Here A is a third- order linear op erator written as A = 4 ∂ 3 x − 3 u∂ x − 3 ∂ x .u . (6 b ) The existene of the solution ψ = ψ ( λ, x, t ) for ev ery onstan t λ is equiv alen t to ∂ t L = AL − LA = [ A, L ] . (7) The result in (7) is alled the Lax equation and the op- erators L and A are alled Lax pair [1℄. The Lax pair represen tation holds go o d for all equations in the KdV hierar h y . In the on text of Lax's metho d it is often said that L denes the original sp etral problem while A rep- resen ts an auxiliary sp etral problem. As one go es along the hierar h y , L remains un hanged but the dieren tial op erator asso iated with the auxiliary sp etral problems  hanges aording to A n = (4) n ∂ 2 n +1 x + n X j =1 { a j ∂ 2 j − 1 x + ∂ 2 j − 1 x a j } , n = 0 , 1 , 2 , 3 , ... . (8) The op erator A 0 = ∂ x and A 1 stands for A in (6 b ) . It is not alw a ys easy to obtain results for other A n 's whi h generate higher-order KdV equations. The o eien t a j dep ends on the solution u and deriv ativ es u n (= ∂ n u ∂ x n ) . F rom (8) it is lear that as j v aries, the dimension of a j  hanges. Th us a j should b e  hosen as a linear om bina- tion of p o w er and pro duts of u and u n 's su h that the terms in the urly bra k et ha v e the righ t dimension of ∂ 2 n +1 x . The onstruted expression for A n will then gen- erate the KdV hierar h y when used in the Lax equation [3℄. On the other hand, one an p ostulate that for an ev olution equation of the form u t = K [ u ] the terms in the F ré het deriv ativ e of K [ u ] on tribute additiv ely with unequal w eigh ts to form the op erator A n su h that L and A n via (7) repro dues the equations in the hierar h y 2 [4℄. Of ourse, there should not b e an y inonsisteny in determining the v alues of the w eigh t fators. Zakharo v and F addeev [5℄ dev elop ed the Hamiltonian approa h to in tegrabilit y of nonlinear ev olution equations in one spatial and one temp oral (1+1) dimensions and, in partiular, Gardner [6℄ in terpreted the KdV equation as a ompletely in tegrable Hamiltonian system with ∂ x as the relev an t Hamiltonian op erator. A signian t de- v elopmen t in the Hamiltonian theory is due to Magri [7℄, who realized that in tegrable Hamiltonian systems ha v e an additional struture. They are bi-Hamiltonian i.e. they are Hamiltonian with resp et to t w o dieren t om- patible Hamiltonian op erators ∂ x and  ∂ 3 x − 4 u∂ x − 2 u x  su h that u t = ∂ x  δ H n δ u  =  ∂ 3 x − 4 u∂ x − 2 u x   δ H n − 1 δ u  , n = 1 , 2 , 3 ... . (9) Here H n = R H n dx with H n , the onserv ed densities for the equations in the KdV hierar h y . These onserv ed densities generate o ws whi h omm ute with the KdV o w and as su h giv e rise to an appropriate hierar h y . T raditionally , the expression for H n is onstruted using a mathematial form ulation that do es not mak e expliit referene to the Lagrangians of the equations in the hi- erar h y . Ho w ev er, a Lagrangian-based approa h an b e used to iden tify H n as the Hamiltonian densit y of the n th hierar hial equation [8℄. The nonlinear transformation of Miura or the so-alled Miura transformation [9℄ u = v x + v 2 , v = v ( x, t ) (10) on v erts the KdV equation in to a mo died KdV (mKdV) equation v t = v 3 x − 6 v 2 v x . (11) This equation diers from the KdV equation only b e- ause of its ubi nonlinearit y . It has man y appliativ e relev ane. F or example, mKdV equation has b een used to desrib e aousti w a v es in anharmoni latties and Alfv én w a v es in ollisionless plasma. It is of in terest to note that the reursion op erator Λ for the mKdV equa- tion [10℄ Λ m = ∂ 2 x − 4 v 2 − 4 v x ∂ − 1 x .v (12) an b e iden tied from (4) and (10) . The equation of the mKdV hierar h y an b e generated b y using the relation v t = Λ n m v x , n = 0 , 1 , 2 , 3 , ... . (13) It is straigh tforw ard to obtain the equations in the mKdV hierar h y from those in the KdV hierar h y b y the use of Miura transformation. Ho w ev er, it is a non trivial prob- lem to deriv e the Lax represen tation and onstrut the bi-Hamiltonian struture of the equations in mKdV hi- erar h y starting from orresp onding results for the KdV equation [11℄. In this w ork w e shall deal with these prob- lems. T o deriv e the Magri struture w e shall mak e use of a Lagrangian-based approa h. In addition to the ab o v e another system of our in terest is the omplex mo died KdV (CMKdV) equation giv en b y v t = v 3 x − 6 | v | 2 v x , (14) This equation follo ws from the third-order nonlinear S hrö dinger equation via an appropriate v ariable trans- formation [12℄. W e shall pro vide a v ariational form ula- tion of (14) whi h, on the one hand, allo ws us to study its anonial struture and, on the other hand, serv es as a useful basis to onstrut an appro ximate analytial solution in term of a trial funtion. In this on text w e note that the n umerial routine for solving su h equa- tions is quite ompliated [13℄ and requires the use of Crank Niolson metho d for time in tegration and quin ti B-spline funtion for spae in tegration. W e b eliev e the solution presen ted b y us ma y serv e as an initial guide for the more am bitious programmes. In § 2 w e in tro due the equations in the mKdV hi- erar h y and deriv e their Lax pair represen tation. W e nd that the system of equations follo ws from the a- tion priniple and as su h an b e obtained from appropri- ate Lagrangian densities via the so-alled Euler-Lagrange equations. The orresp onding Hamiltonian densities on- stitute the in v olutiv e onserv ed densities of the mKdV equation. W e then study the bi-Hamiltonian struture of the mKdV equations. In § 3 w e on v ert (14) to a v ari- ational problem and th us obtain a Lagrangian represen- tation for the equation. As a useful appliation of the Lagrangian densit y so deriv ed w e w ork out the anoni- al form [5℄ of the mKdV equation and also onstrut a solution of it b y means of sech trial funtions and a Ritz optimization pro edure [14℄. Finally , in § 4 w e try to summarize our outlo ok on the presen t w ork. 2. mKdV hierar h y The equations of the mKdV hierar h y follo w from (13) for n = 0 , 1 , 2 , 3 ... . W e shall onstrut Lax pair represen tations of these equations b y taking reourse to the use of (10) in (6 a ) and (8) . F or these equations w e shall use a Lagrangian-based metho d to obtain the onserv ed densities whi h are in in v olution and generate the so-alled mKdV o w. W e shall then try to realize the bi-Hamiltonian struture b y an appropriate mo diation of (9) b y the use of Miura transform. (a) Lax pair represen tation F rom (6 a ) , (6 b ) and (10) w e write L = − ∂ 2 x + v 2 + v x (15 a ) 3 and A = 4 ∂ 3 x − 3( v 2 + v x ) ∂ x − 3 ∂ x . ( v 2 + v x ) . (15 b ) Using (15) in (7) w e get ( ∂ x + 2 v )  v t − v 3 x + 6 v 2 v x  = 0 . (16) As with (11) , (16) giv es the mKdV equation. In view of this w e shall denote the Lax pair in (15) b y L m and A m just to indiate that these refer to the mKdV equation. W e shall follo w this on v en tion for all op erators and fun- tions related to the mKdV equation. Consisten tly with the notation of (8) A m (= A ) stands for A m 1 . In lose anal- ogy with the ase of higher KdV equations the original sp etral problem for the mKdV equations  haraterized b y the op erator L m remains un hanged as w e go up the hierar h y but the dieren tial op erator A m n 's  hange with n . F rom (10) and the results giv en in refs 3 and 4 one an alulate the expressions for A m n , n = 2 , 3 , 4 , ... . In the follo wing w e presen t some of our results. A m 2 = 16 ∂ 5 x + (25 v 2 x + 30 v 2 v x + 10 v v 2 x + 15 v 4 + 5 v 3 x ) ∂ x − 20( v 2 + v x ) ∂ 3 x + ∂ x . (25 v 2 x + 30 v 2 v x + 10 v v 2 x + 15 v 4 + 5 v 3 x ) − 20 ∂ 3 x . ( v 2 + v x ) , (17 a ) A m 3 = 64 ∂ 7 x − 140( v 4 + 3 v 2 x + 2 v 2 v x + 2 v v 2 x + v 3 x ) ∂ 3 x + 112( v 2 + v x ) ∂ 5 x − 140 ∂ 3 x . ( v 4 + 3 v 2 x + 2 v 2 v x + 2 v v 2 x + v 3 x )+ 112 ∂ 5 x . ( v 2 + v x ) + (70 v 6 + 21 0 v 4 v x + 10 50 v 2 v 2 x + 21 0 v 3 x + 140 v 3 v 2 x + 840 v v x v 2 x + 721 v 2 2 x + 70 v 2 v 3 x ) ∂ x + (798 v x v 3 x + 182 v x v 4 + 91 v 5 x ) ∂ x + ∂ x . (70 v 6 + 210 v 4 v x + 1050 v 2 v 2 x + 210 v 3 x ) + ∂ x . (140 v 3 v 2 x + 840 v v x v 2 x + 721 v 2 2 x + 70 v 2 v 3 x + 798 v x v 3 x + 182 v x v 4 + 91 v 5 x ) (17 b ) and A m 4 = 25 6 ∂ 9 x − 255 v 8 x − 1794 v 7 x ∂ x − 510 v v 7 x − 1152 v x ∂ 7 x − 1152 v 2 ∂ 7 x − 5628 v 6 x ∂ 2 x +2100 v x v 6 x − 3588 v v 6 x ∂ x +5670 v 2 v 6 x − 4032 v 2 x ∂ 6 x − 8064 v v x ∂ 6 x − 10248 v 5 x ∂ 3 x − 7224 v 2 x v 5 x − 11256 v v 5 x ∂ 2 x − 1559 4 v x v 5 x ∂ x +1831 2 v v x v 5 x +5934 v 2 v 5 x ∂ x + 11340 v 3 v 5 x − 8736 v 3 x ∂ 5 x − 17472 v v 2 x ∂ 5 x − 15456 v 2 x ∂ 5 x + 4032 v 2 v x ∂ 5 x + 2016 v 4 ∂ 5 x − 11760 v 4 x ∂ 4 x − 11760 v 3 x v 4 x − 20496 v v 4 x ∂ 3 x − 39204 v 2 x v 4 x ∂ x + 19152 v v 2 x v 4 x − 43680 v x v 4 x ∂ 2 x +1260 0 v 2 v 4 x ∂ 2 x +6867 0 v 2 x v 4 x +4110 0 v v x v 4 x ∂ x + 70224 v 2 v x v 4 x + 11868 v 3 v 4 x ∂ x − 210 v 4 v 4 x − 23520 v v 3 x ∂ 4 x − 60240 v x v 2 x ∂ 4 x +1032 0 v 2 v 2 x ∂ 4 x +2064 0 v v 2 x ∂ 4 x +2064 0 v 3 v x ∂ 4 x − 25758 v 2 3 x ∂ x + 12180 v v 2 3 x − 66864 v x v 3 x ∂ 3 x + 15120 v 2 v 3 x ∂ 3 x − 87360 v 2 x v 3 x ∂ 2 x + 170268 v x v 2 x v 3 x + 69720 v x v 2 x v 3 x ∂ x + 13020 0 v 2 v 2 x v 3 x + 75600 v v x v 3 x ∂ 2 x + 25200 v 3 v 3 x ∂ 2 x + 81660 v 2 x v 3 x ∂ x + 64596 v v 2 x v 3 x + 93336 v 2 v x v 3 x ∂ x − 15960 v 3 v x v 3 x − 6300 v 4 v 3 x ∂ x − 420 v 5 v 3 x − 50568 v 2 2 x ∂ 3 x + 73920 v v x v 2 x ∂ 3 x +28 560 v 3 x ∂ 3 x +68 880 v 2 v 2 x ∂ 3 x − 50 40 v 4 v x ∂ 3 x − 1680 v 6 ∂ 3 x + 19026 v 2 2 x + 50400 v v 2 2 x ∂ 2 x + 114480 v x v 2 2 x ∂ x + 88452 v v x v 2 2 x +6727 2 v 2 v 2 2 x ∂ x − 1512 0 v 3 v 2 2 x +1184 40 v 2 x v 2 x ∂ 2 x + 16128 0 v 2 v x v 2 x ∂ 2 x − 7560 v 4 v 2 x ∂ 2 x + 208488 v v 2 x v 2 x ∂ x + 57960 v 3 x v 2 x − 66780 v 2 v 2 x v 2 x − 60480 v 3 v x v 2 x ∂ x − 27720 v 4 v x v 2 x − 12600 v 5 v 2 x ∂ x + 1260 v 6 v 2 x + 85680 v v 3 x ∂ 2 x − 3 0240 v 3 v 2 x ∂ 2 x − 1 5120 v 5 v x ∂ 2 x + 2 8518 v 4 x ∂ x − 27720 v v 4 x − 57960 v 2 v 3 x ∂ x − 37800 v 3 v 3 x − 44100 v 4 v 2 x ∂ x + 7560 v 5 v 2 x + 2520 v 6 v x ∂ x + 2520 v 7 v x + 630 v 8 ∂ x . (1 7 c ) Using (15 a ) and (17 a ) in (7) w e get ( ∂ x +2 v ) { v t − v 5 x +40 v v x v 2 x +10 v 2 v 3 x +10 v 3 x − 30 v 4 v x } = 0 . (18) 4 The expression inside the urly bra k et represen ts the equation obtained from (13) with n = 2 . Results similar to that in (18) hold go o d for an y pair lik e [ A m n , L m ] . This observ ation serv es as a useful  he k on our results for A m n with arbitrary v alues of n . (b) Bi-Hamiltonian struture Here w e shall demonstrate that the bi-Hamiltonian struture of (11) and all higher-order equations obtained from (13) with n = 2 , 3 , 4 ... . W e note that a single ev olution equation u t = P [ u ] , u ǫ R is nev er the Euler- Lagrange equation of a v ariational problem [10℄. One ommon tri k to put a single ev olution equation in to a v ariational form is to replae v b y a p oten tial funtion v = − w x , w = w ( x, t ) . (19) The funtion w is often alled the Casimir p oten tial. Our expressions for the Lagrangian densities will b e written in terms of w and its appropriate deriv ativ es. Hamil- tonian densities obtained b y use of Legendre map an, ho w ev er, b e expressed in terms of eld v ariable v ( x, t ) and its deriv ativ es. The linear equation obtained from (13) with n = 0 reads v t = v x . (20) F rom (19) and (20) w xt = w xx = P [ w x ] (say) . (21) Equiv alen tly , w t = w x . (22) In writing (22) w e ha v e used the b oundary ondition limit w ( x, t ) = 0 as x → ±∞ . The self adjoin tness of P [ w x ] ensures the existane of a Lagrangian for (21) and (22) . In this ase, the Lagrangian densit y an b e onstruted using the homotop y form ula [10℄ L [ ξ ] = Z 1 0 ξ P [ λξ ] dλ . (23) F rom (23) w e get L m 0 = 1 2 w t w x − 1 2 w 2 x , (24 a ) The subsript zero is self explanatory . The Hamiltonian densit y obtained from (24 a ) is giv en b y H m 0 = 1 2 w 2 x = 1 2 v 2 . (25 a ) The Lagrangian and Hamiltonian densities for the mKdV ( n = 1) and higher-order equations obtained from (13) for n = 2 , 3 and 4 are giv en b y L m 1 = 1 2 w t w x − 1 2 w x w 3 x + 1 2 w 4 x , (24 b ) H m 1 = 1 2 v v 2 x − 1 2 v 4 , (25 b ) L m 2 = 1 2 w t w x − 1 2 w 2 3 x − w 6 x − 5 w 2 x w 2 2 x , (24 c ) H m 2 = 1 2 v 2 2 x + v 6 + 5 v 2 v 2 x , (25 c ) L m 3 = 1 2 w t w x − 1 2 w x w 7 x + 7 2 w 3 x w 5 x + 14 w 2 x w 2 x w 4 x + 21 2 w 2 x w 2 3 x + 35 2 w x w 2 2 x w 3 x − 35 3 w 5 x w 3 x − 70 3 w 4 x w 2 2 x + 5 2 w 8 x , (24 d ) H m 3 = 1 2 v v 6 x − 7 2 v 3 v 4 x − 14 v 2 v x v 3 x − 21 2 v 2 v 2 2 x − 35 2 v v 2 x v 2 x + 35 3 v 5 v 2 x − 70 3 v 4 v 2 x − 5 2 v 8 (25 d ) and L m 4 = 1 2 w t w x − 1 2 w x w 9 x + 9 2 w 3 x w 7 x + 27 w 2 x w 2 x w 6 x + 57 w 2 x w 3 x w 5 x + 105 2 w x w 2 2 x w 5 x − 21 w 5 x w 5 x + 69 2 w 2 x w 2 4 x + 189 w x w 2 x w 3 x w 4 x − 16 8 w 4 x w 2 x w 4 x + 91 2 w x w 3 3 x − 12 6 w 4 x w 2 3 x − 518 w 3 x w 2 2 x w 3 x + 105 2 w 7 x w 3 x − 133 w 2 x w 4 2 x + 315 2 w 6 x w 2 2 x − 7 w 10 x (24 e ) H m 4 = 1 2 v v 8 x − 9 2 v 3 v 6 x − 27 v 2 v x v 5 x − 57 v 2 v 2 x v 4 x − 105 2 v v 2 x v 4 x +21 v 5 v 4 x − 69 2 v 2 v 2 3 x − 189 v v x v 2 x v 3 x +168 v 4 v x v 3 x − 91 2 v v 3 2 x +126 v 4 v 2 2 x + 518 v 3 v 2 x v 2 x − 105 2 v 7 v 2 x + 133 v 2 v 4 x − 315 2 v 6 v 2 x + 7 v 10 . (25 e ) Results of H m n 's for still higher v alues of n an b e ob- tained in a similar manner. As a useful  he k on our expressions, one an v erify that these results are in ex- at agreemen t with those obtained b y the appliation of Miura transformation on the w ell kno wn onserv ed den- sities of the KdV equations. 5 The bi-Hamiltonian struture of equations in the mKdV hierar h y an easily b e v eried b y using our Hamiltonian funtionals in v t = ∂ x  δ H m n δ v  = E  δ H m n − 1 δ v  , n = 1 , 2 , 3 ... , (26) where E =  ∂ 3 x − 4 v 2 ∂ x − 4 v x ∂ − 1 x .v ∂ x  . (27) The rst Hamiltonian op erator ∂ x in (26) is the same as that in (9) while the seond has b een obtained from [10℄ E = Λ m ∂ x . (28 ) The op erators ∂ x and E are sk ew symmetri and satisfy the Jaobi-iden tit y . Th us they onstitute t w o ompatible Hamiltonian op erators su h that all equations obtained from (13) are in tegrable in the Liouville's sense [7℄. 3. mKdV equation The omplex mKdV equation in (14) an b e restated as a v ariational problem giv en b y δ Z Z L ( v , v ∗ , v x , v ∗ x , v 3 x , v ∗ 3 x , v t , v ∗ t , x, t ) dx dt = 0 (29) with the Lagrangian densit y written as L = 1 2 ( v ∗ v t − v v ∗ t ) − 1 2 ( v ∗ v 3 x − v v ∗ 3 x )+ 3 2 v v ∗ ( v ∗ v x − v v ∗ x ) . (30) The Euler-Lagrange equations orresp onding to (29) are d dt  ∂ L ∂ v t  − δ L δ v = 0 (31 a ) and d dt  ∂ L ∂ v ∗ t  − δ L δ v ∗ = 0 (31 b ) with the v ariational deriv ativ e δ δ ψ = 3 X n ≥ 0 ( − ∂ x ) n ∂ ∂ ψ n . (32) Here ψ n = ( ∂ x ) n ψ . (33) It is easy to v erify that (30) and (31 b ) giv e the mKdV equation while (30) and (31 a ) giv e the orresp onding omplex onjugate equation. The Hamiltonian orre- sp onding to the Lagrangian densit y (30) is giv en b y H = Z H dx (34) with the Hamiltonian densit y H = 1 2 ( v ∗ v 3 x − v v ∗ 3 x ) − 3 2 v v ∗ ( v ∗ v x − v v ∗ x ) . (35) In order to sho w that (14) is a Hamiltonian system w e will ha v e to write it and its omplex onjugate in t w o dieren t forms v t = { v ∗ ( x ) , H ( y ) } (36) and v ∗ t = −{ v ( x ) , H ( y ) } . (37) W e ha v e already found an expression for the Hamilto- nian. Th us our task is to lo ok for fundamen tal P oisson bra k et relation for the eld v ariables that redue (36) to the mKdV equation and (37) to the omplex onju- gate one. One an easily  he k that the required P oisson bra k et relations are giv en b y { v ( x ) , v ( y ) } = { v ∗ ( x ) , v ∗ ( y ) } = δ ( x − y ) . (38) The relations (36) and (37) an b e written in the sym- pleti form η t = J δ H δ η , η =  v v ∗  (39) with J =  0 1 − 1 0  , a sk ew-adjoin t matrix as the Hamiltonian op erator. Equation (14) arises in a n um b er of appliativ e on- texts inluding the nonlinear ev olution of plasma w a v es [15℄. T o our kno wledge there is no w ell-dened sp e- tral problem that an easily b e used to solv e the mKdV equation in terms of kno wn transenden tal funtions. But a n um b er of w orks has b een en visaged to obtain the solitory-w a v es and/or soliton solutions of this equation. See, for example, [13℄ and referenes therein. Here w e are in terested to pro vide an aurate appro ximation solution of (14) b y supplemen ting the Lagrangian densit y in (30) with sech trial funtions and a Ritz optimization pro e- dure. W e ha v e  hosen to w ork with the trial funtion written as v ( x, t ) = a ( t ) sech [( x − y ( t ) ) / w ( t )] × e [ i ( q ( t )+ r ( t )( x − y ( t ))+ b ( t ) 2 w ( t ) ( x − y ( t )) 2 )] . (40) Here the parameters a , y and w are related to the three lo w est-order momen ts of the v en v elop e and represen t, re- sp etiv ely , its amplitude, en tral p osition and width. The other parameters q , r and b stand for the phase, v elo it y (en ter of mass) and frequeny  hirp. Understandably , these parameters will all v ary with time t . Using (40) in (30) w e get L s = 3 X i =1 L ( i ) s , ( 4 1) 6 where L (1) s = 1 2  x − y w  2 a 2 bw sech 2  x − y w  + a 2 r ˙ y sech 2  x − y w  − a 2 ˙ q se ch 2  x − y w  − 1 2  x − y w  2 a 2 ˙ bw sech 2  x − y w  , (42 a ) L (2) s = 3 a 2 r w 2 sech 2  x − y w  tanh 2  x − y w  − 3 a 2 r w 2 sech 4  x − y w  − a 2 r 3 sech 2  x − y w  − 3  x − y w  2 a 2 b 2 r sech 2  x − y w  (42 b ) and L (3) s = − 3 a 4 r se ch 4  x − y w  . (42 c ) Here the dots stand for deriv ativ e with resp et to t . The subsript s on L merely indiates that w e ha v e inserted the sech ansatz for v ( x, t ) in to the Lagrangian densit y . In terms of (41) the v ariational priniple (29) leads to δ Z < L > dt = 0 , (43) with the a v eraged eetiv e Lagrangian < L > = Z ∞ ∞ L s dx . (44) The result for < L > is giv en b y < L > = − 2 w a 2 r 3 − 4 wa 4 r − π 2 2 a 2 b 2 rw − 2 a 2 r w − π 2 12 w 2 a 2 ˙ b − 2 a 2 w ˙ q + 2 a 2 wr ˙ y + π 2 12 a 2 bw ˙ w . (45) The redued v ariational priniple expressed b y (43) re- sults in a set of oupled ordinary dieren tial equations for the parameters of our trial funtion. F rom the v anishing ondition of the v ariationals δ < L > δ q , δ < L > δ a , δ < L > δ y , δ < L > δ w , δ < L > δ r and δ < L > δ b w e obtain d dt  2 a 2 w  = 0 , (46 a ) − 4 aw r 3 − 16 wa 3 r − π 2 ab 2 rw − 4 ar w − π 2 6 w 2 a ˙ b − 4 aw ˙ q + 4 aw r ˙ y + π 2 6 abw ˙ w = 0 , (46 b ) − d dt  2 a 2 wr  = 0 , (46 c ) − 2 a 2 r 3 − 4 a 4 r − π 2 2 a 2 b 2 r + 2 a 2 r w 2 − π 2 6 wa 2 ˙ b − 2 a 2 ˙ q + 2 a 2 r ˙ y + π 2 12 a 2 b ˙ w − d dt  π 2 12 a 2 bw  = 0 , (46 d ) − 6 w a 2 r 2 − 4 wa 4 − π 2 2 a 2 b 2 w − 2 a 2 w + 2 a 2 w ˙ y = 0 (46 e ) and − π 2 a 2 brw + π 2 12 a 2 w + d dt  π 2 12 w 2 a 2  = 0 . (46 f ) Equations in (46) an b e used to write a 2 w = consta n t = E 0 , (47 a ) r = c onstant , (47 b ) da dt = − 3 abr w , (47 c ) dy dt = 3 r 2 + 2 a 2 + π 2 4 b 2 + 1 w 2 , (47 d ) dw dt = 6 b r (47 e ) and db dt = 24 r π 2 a 2 w + 24 r π 2 1 w 3 . (47 f ) Equation (47 a ) expresses the v ariational v ersion of the energy onserv ation la w [16℄ while (47 b ) states that the en ter of mass of the solution of (14) mo v es with a on- stan t v elo it y . F or a giv en v alues of r , the set of oupled ordinary dieren tial equations (47 c ) - (47 f ) an easily b e solv ed n umerially . Note that kno wledge of a ( t ) , y ( t ) and w ( t ) an b e used to study the | v ( x, t ) | as funtions of x 7 r = 0.1 - 5 0 5 10 15 20 0.0 0.2 0.4 0.6 0.8 1.0 t È v H x,t LÈ FIG. 1: | v ( x, t ) | as a funtion of t for three dieren t v alues of x . Here r = 0 . 1 . r = 0.001 - 5 0 5 10 15 0.0 0.2 0.4 0.6 0.8 1.0 t È v H x,t LÈ FIG. 2: Same as that in FIG. 1 but with r = 0 . 001 and t . W e w ork ed with the initial onditions a (0) = 1 , b (0) = 0 , y (0) = 0 , w (0) = 1 and solv ed these equations using the fourth-order Runge-Kutta metho d [17℄. First w e tak e r = 0 . 1 and plot in FIG. 1 , | v ( x, t ) | as a funtion of t for three dieren t v alues of x , namely , x = 0 (solid urv e), x = 10 (dotted urv e) and x = 20 (dashed urv e). F rom these urv es it is lear that as x inreases | v ( x, t ) | dereases rapidly . This implies that for our  hosen v alue of the v elo it y w e ha v e dea ying solitory w a v e solution. W e ha v e v eried that for still higher v alues of r the solu- tions dea y more rapidly . In FIG. 2 w e presen t a similar plot of | v ( x, t ) | for r = 0 . 00 1 . In terestingly , | v ( x, t ) | re- mains un hanged as x inreases. Th us one an infer that the solutions of (14) for small v alues of r b eha v es lik e solitons. 4. Conlusion The nonlinear transformation of Miura or the so-alled Miura transformation is an aid to obtain the mo died KdV (mKdV) equation from the KdV equation. W e nd that this transformation also pro vides an eetiv e w a y to onstrut expressions for Lax pairs of all equations in the mKdV hierar h y . As with the KdV equations the bi- hamiltonian struture of the mKdV equations are tradi- tionally studied using in v olutiv e set of onserv ed Hamil- tonian densities without expliit referene to their La- grangians. W e deriv e a Lagrangian-based approa h to realize the bi-Hamiltonian struture. In lose analogy with the mKdV equation, the mKdV equation in (14) also follo ws from Hamilton's v ariational priniple pro vided the ation funtional is made to v anish for sim ultaneous v ariations of b oth v and v ∗ . In this ase the Lagrangian densit y is a funtion of v , v ∗ and their appropriate time and spae deriv ativ es. W e ould use the Hamiltonian orresp onding to this Lagrangian densit y to write the mKdV equation in the anonial form [5℄ with an appropriate P oisson struture. As an added realizm w e demonstrate that the Lagrangian densit y onstitutes a basis to deriv e a semianalytial solution of (14) . W e a hiev e this b y taking reourse to the use of sech trial funtions to dene a redued v ariational problem whi h in onjuntion with the Ritz optimization pro edure ould yield an unompliated solution of the mKdV equation. There exist some sophistiated n umerial routines [13,15℄ to solv e the equation. Ho w ev er, w e feel that the v ariational approa h sough t b y us will serv e as a omplemen tary to ol to w ards understanding the prop erties of solitory w a v e- and/or soliton-solutions of the omplex mo died KdV equation. A  kno wledgemen ts This w ork is supp orted b y the Univ ersit y Gran ts Commission, Go v ernmen t of India, through gran t No. F.32-39/2006(SR). Referenes [1℄ P . D. Lax, Comm un. Pure Appl. Math., 21 , 467 (1968). [2℄ F. Calogero and A. 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A. 27 , 3135 (1983). [15℄ G. M. Muslu and H. A. Erba y , Computers and Mathematis with Appliations 45 , 503 (2003). [16℄ B. A. Malomed, Progr. Optis 43 , 69 (2002). [17℄ J. B. Sarb orough, Numerial Mathematial Anal- ysis, Oxford and IBH Publishing C., New Delhi, 1971.