On the supersymmetric nonlinear evolution equations

On the supersymmetric nonlinear ev olution equations Amita v a Choudhuri a , B . T a lukdar a and S. Ghosh b a Dep artment of Physics, V isva-Bhar ati U niversity, Santiniketan 7312 35, India b Patha Bhavana, Visva-Bhar ati Uni versity, Santiniketan 731 235, India e-mail : bi noy123@bsnl.in Abstract. Sup ersymmetrization o f a nonlinear ev olutio n equation in whic h the b o sonic equation is indep enden t of the fermionic v a r ia ble and the system is linear in fermionic field go es by the name B-sup ersymmetrization. This sp ecial t yp e of sup ersymmetrization pla ys a role in superstring theory . W e pro vide B-sup ersymmetric extension of a n umber o f quasilinear and fully non- linear ev olution equations a nd find that the sup ersymmetric system follow s from the usual action principle while the b o sonic and fermionic equations are individually non Lagrangian in the field v ariable. W e p oint out that B-sup ersymmetrization can also b e realized using a generalized No etherian symmetry suc h tha t the resulting set of Lagra ng ian symmetries coincides with symmetries of the b osonic field equations. This observ ation pro vides a basis to associat e t he b osonic and fermionic fields with the terms of brigh t and dark solitons. The in terpretation sough t b y us has its origin in the classic w ork of Bateman who in tro duced a rev erse-time system with negativ e friction to bring the linear dissipativ e systems within the framew ork of v a r iational principle. P A CS n umbers : 05.45.Yv, 52.3 5.Mw, 45.20 .-d, 45.20.Jj Key w o rds : Nonlinear ev olution equation, B-sup ersymmetrization, su- p erpartners and action principle, b osonic and f ermionic fields, phy sical real- ization 1. In tro du ction In the theories of elemen tary particles sup ersymmetry aims a t a unified description of fermions and b osons i.e. of matter and in teraction [1]. Under- standably , construction of supersymmetric theories in this case will in v olv e consideration of particles of differen t spin a nd of differen t statistics. As op- p osed to this, sup ersymmetrization of in tegrable nonlinear ev olution equa- 1 tions pro ceeds by considering only the space sup ersymmetric inv ariance [2]. More sp ecifically , in a sup erspace approac h to (1 + 1) dimensional systems the commuting space v ariable x is extended t o a doublet ( x, θ ), where θ is an an ticomm uting v a r ia ble of Grassmann type suc h that θ 2 = 0. A sup erfield F rega rded a s a function of x and θ has a v ery simple T a ylor expansion in terms of θ suc h tha t F ( x, θ ) = v ( x ) + θu ( x ) . (1) Here u ( x ) and v ( x ) are the comp onen t fields of F ( x, θ ) with v ( x ), t he sup er- partner of u ( x ) and the con vers ly . The function F ( x, θ ) is fermionic (b osonic) if u ( x ) is b osonic (fermionic). A sup ersymmetric system for a g iv en ev olution equation is constructe d in a wa y that will make it inv arian t under the tra nsfor ma t io ns x → x − η θ and θ → θ + η with η , an an ticomm uting parameter. Since the sup ersymmetric transformation do es not depend on time w e hav e suppresse d t dependence in writing (1). F or any function Φ of the do ublet ( x, θ ) it is easy to see that δ Φ = η ( ∂ θ − θ∂ x )Φ . (2) Th us Q = ∂ θ − θ∂ x (3) is the g enerator of the sup ersymmetric transformation. Using Φ = F ( x, θ ) from (1 ) in (2) w e get the transformations for the comp o nen t fields as δ v = η u and δ u = η v x . (4) The results in (4) giv e the so-called sup ersymmetry transforma t io n since the first equation relates the b osonic field to a fermionic field while the second one do es the opp osite. In other w or ds, the sup ersymmetry transformations regarded as transformations in sup erspace exhibit the fermi-b ose symmetry . Tw o successiv e supersymmetry transformatio ns lead to δ 2 v = η 2 v x and δ 2 u = η 2 u x . (5) This sho ws that a supersymmetry transformat io n is a sort of square ro ot of an ordina r y translation. The deriv ativ e written a s D = θ∂ x + ∂ θ (6) an ticomm ute with Q and is called the co v arian t superderiv ative presumably b ecause expressions written in terms of D and sup erfield are manifestly in- v arian t under the supersymmetry transformat ion (4). It is of intere st to note that D 2 = ∂ x . (7) 2 The Kortew eg-de V ries (KdV) equation [3] u t = − u 3 x + 6 uu x . (8) represen ts the first nonlinear e v olution equation that could a ccoun t for s oliton formation [4]. The KdV equation can b e extended to a supersymmetric sys - tem by rewriting the KdV equation in terms of the sup erfield and co v a r ian t deriv ativ e. This is achie v ed b y m ultiplying (8) with θ and then sup ersym- metrizing t he result. Th us w e ha v e F t = − D 6 F + 6 D F D 2 F (9) F rom (1) and (9) w e obta in equations for the b osonic and fermionic fields as u t = − u 3 x + 6 uu x (8) and v t = − v 3 x + 6 uv x . (8 ′ ) W e note tha t θ (6 uu x ) can b e regarded as a fermionic part of eithe r 3 D 2 ( F D F ) or 6 D F D 2 F . Therefore, in order to construct the most general p ossible su- p ersymmetric extension of the KdV equation, one must consider a linear com bination of these tw o terms. In writing θ (6 uu x ) w e hav e not tak en this p oint in to consideration but chos en to w ork with only the term 6 D F D 2 F . This resulted in a set of simple equations giv en in (8) and (8 ′ ). Clearly , the b osonic equation do es not dep end on the fermionic v ariable and moreov er, the system is linear in the fermionic field. This kind o f sup ersymmetric ex- tension w as originally discarded as b eing a ‘trivial’ sup ersymmetrization [2]. Ho w ev er, it generated a lot of interes t af t er it w as realized tha t equations like (8 ′ ) arise in the study of sup erstring theory [5 ]. A sup ersymmetric extension in whic h b osonic equation do not c hange in the presenc e of fermions has now b een giv en the name B-sup ersymmetrization. The aim of the presen t w ork is to provide a Lag rangian realization for the pair of equations whic h arise from B-supersymmetrization of nonlinear ev o lution equations. Our equations of in terest are t he mo dified KdV (mKdV) [6], Hun ter-Saxton (HS) [7] and Camassa-Holm (CH) [8] equations. These equations ar e related to the celebrated KdV equation in somew ay or others. In this contex t w e note that the KdV and mKdV equation are quasilinear in the sens e t ha t their dis p ersiv e terms are linear while the HS and CH equations are f ully nonlinear b ecause their disp ersiv e terms are nonlinear. Besides Hun ter-Saxton and Camassa-Holm equations there exist another class of fully nonlinear ev o lution equations (FNE) in tro duced b y Rosenau a nd Hyman (RH) [9]. These equation is apparen tly not related to the KdV equation. 3 In section 2 w e out line how the mKdV, Hun ter-Saxton and Camassa-Holm equations are r elat ed to the KdV equation and study their sup ersymmetric structure. W e also tr y to prese n t a similar supersymmetric generalization of the third-order RH equation. In section 3 we con template deriving all these sup ersymmetric equations fro m the action principle. A single ev o lutio n equation is nev er an Euler-Lagrange expression. W e need to couple a g iv en equation with a n asso ciated one to construct a ph ys- ically complete system in the sense of the v ariational principle. As an inter- esting curiosit y we note that the asso ciated equation is the fermionic partner of the orig inal b osonic equation. The tec hanique of in tro ducing an auxil- iary equation fo r v ariational formulation o f ph ysical problems has a n old ro ot in the classical mec hanics literature. F or example, in a celebrated work Bateman [10] noted that a dissipativ e sys tem is ph ysically incomplete and one needs an additional equation when one attempts to deriv e the defining equations fro m an a ctio n principle. In section 4 w e make some concluding remarks. In particular, we judiciously exploit the similarit y b et w een sup er- symmetric a nd Bateman’s dual systems to prov ide a ph ysical realization for the b o sonic and fermionic fields in terms o f bright and dark solitons. 2. Ev olution equ ations and their sup ersymmetric part- ners In tro duction In t his section w e collect a n umber of w ell- known in tegrable nonlinear ev o lution equations whic h are related to t he KdV equation and study their sup ersymmetric structure. W e also presen t a similar treatmen t fo r the non in tegrable third-order RH equation. (i) Mo dified K dV equation: The nonlinear transformation of Miura or the so-called Miura tra nsformation [11] u = ξ x + ξ 2 , ξ = ξ ( x, t ) (10) con v erts the KdV equation in (8) in to a mo dified KdV ( mKdV) equation ξ t = − ξ 3 x + 6 ξ 2 ξ x F or breviet y , intro ducing ξ = u w e write the mKdV equation in the form u t = − u 3 x + 6 u 2 u x . (11) This equation differs f rom the KdV equation only because of its cubic non- linearit y . It has man y applicativ e relev ance. F or example, mKdV equation 4 has been used to describe acoustic w av es in anharmonic lattices and Alfv´ en w a v es in collisionless plasma. (ii)Hun ter-Saxton equation: Consider the hereditary recursion op erator R = c∂ 2 + λ ( ∂ u∂ − 1 + u ) , (12) for the KdV hierarc h y . Here c and λ are an a r bit r a ry cons tants . This op erato r generates a hierarc hy of in tegrable equations in whic h the first member is ∂ u ∂ t = Ru x = cu xxx + 3 λuu x . (13) F or c = − 1 and λ = 2 w e ha v e the fa mous KdV equation. A second recursion op erator can b e extracted from (12) b y shifting the function u to u + γ , where γ is a constan t. W e th us hav e R ( u + γ ) = ( c 1 ∂ 2 + λ ( ∂ u∂ − 1 + u )) + ( c 2 ∂ 2 + 2 λγ ) = R 1 + R 2 (sa y) . (14) In writing (1 4) w e hav e used c = c 1 + c 2 . I t app ears that t he recursion op erator R = R 1 R − 1 2 generates new hierarc hy of in t egr a ble equations u t = ( R 1 R − 1 2 ) n u x . (15) Assuming that u = R 2 ξ = c 2 ξ xx + 2 λγ ξ the first mem b er of the hierarc hy is 2 λγ ξ t + c 2 ξ xxt = c 1 ξ 3 x + λc 2 ξ ξ 3 x + 2 λc 2 ξ x ξ 2 x + 6 λ 2 γ ξ ξ x . (16) F or γ = 0 , c 1 = 0 , λ = − 1 and ξ = u (16) giv es the Hun ter-Saxton equation [7] u xxt + 2 u x u 2 x + uu 3 x = 0 . (17) (iii) Camassa-Holm equation: The choice c 1 = 0 , c 2 = 1 , γ = 1 2 , λ = − 1 and ξ = u (16) leads to the w ell known Camassa-Holm equation [8] u t − u xxt + 3 uu x − 2 u x u 2 x − uu 3 x = 0 . (18) The CH equation describ es the unidirectional propagation of shallo w w ater w a v es o v er a flat b ottom. It can also represen t the geo desic flo w on the Bott- Virasoro g roup. 5 (iv) Rosenau-Hyman equation: The KdV and mKdV equations are quasi- linear. Here the disp ersion pro duced is comp ensated b y nonlinear effects resulting in the formation o f exp o nen tially lo calized solitons. As opp osed to KdV and mKdV equations, the HS and CH equations are fully nonlinear. These equations do not ha v e exp onentially lo calized solito n solutions. In- stead they suppor t p eak ed- and cusp-lik e solutions often called p eak on a nd cuspo n. Bo th HS and CH are in tegr a ble. Unlike HS and CH equations, the family of FNE equations prop osed by Rosenau and Hyman [9] are nonin te- grable but hav e solito r y w av e solutions with compact supp ort. That is, they v anish iden tically outside the finite ra nge. These solutions were giv en the name compactons. The compacto ns are robust within their ra nge of exis- tence. In con tra st with the in teraction of solito ns suppor ted by KdV and mKdV equations, the p oin t at whic h tw o compactons collide is mark ed by the birth of a low-amplitude compacton-a nticompacton pair. The third- order RH equation is giv en b y u t + 3 u 2 u x + 6 u x u 2 x + 2 uu 3 x = 0 . (19) Although (11), (17), (18) and (19) appear to b e structurally differen t from the KdV equation, eac h of them can b e sup ersymmetrized in a rather straigh t- forw ard manner. In T able 1 w e presen t results for equations of the superfield and supersymmetric partners resulting from B-sup ersymmetrization of these equations. F o r completenes s w e also include results for the KdV equation. T able 1 : Nonlinear ev olution equations and their sup ersymmetric partners Ev olution equations Equations for S u p erfi eld Sup ersymmetric partners KdV: u t + u 3 x − 6 uu x = 0 F t + D 6 F − 6 DF D 2 F = 0 v t + v 3 x − 6 uv x = 0 mKdV: u t + u 3 x − 6 u 2 u x = 0 F t + D 6 F − 6( DF ) 2 D 2 F = 0 v t + v 3 x − 6 u 2 v x = 0 Hun ter-Saxton: u xxt + uu 3 x + 2 u x u 2 x = 0 D 4 F t + D F D 6 F + D 3 F D 4 F v xxt + uv 3 x + u x v 2 x + D 2 F D 5 F = 0 + u 2 x v x = 0 Camassa-Holm: u t − u xxt + 3 uu x − uu 3 x F t − D 4 F t + 3 D F D 2 F − D F D 6 F v t − v xxt + 3 uv x − uv 3 x − 2 u x u 2 x = 0 − D 3 F D 4 F − D 2 F D 5 F = 0 − u x v 2 x − u 2 x v x = 0 FNE: u t + 3 u 2 u x + 6 u x u 2 x + F t + 3( D F ) 2 D 2 F + 3 D 3 F D 4 F v t + 3 u 2 v x + 3 u x v 2 x 2 uu 3 x = 0 +3 D 2 F D 5 F + 2 D F D 6 F = 0 +3 u 2 x v x + 2 uv 3 x = 0 Lo oking closely in to this table w e see that the sup erfield equations of KdV, mKdV and FNE equations con tain only F t as the time deriv ative parts 6 of these equations. In con trast to this, the corresp onding time deriv ativ e parts of HS and CH equations a r e D 4 F t and F t − D 4 F t . Understandably , the presence of D 4 F t is asso ciated with the mixed deriv at ive term u xxt presen t in these equations. Note that, as expected, all fermionic fields satisfy linear equations. 3. F ermionic equations from an action p rinciple Our curren t understanding of all protot ypical ph ysical theories is la r gely based on the action principle as enunciated by Hamilton during mid 1930’s. In the con text of field theory , classical or quantum, the dynamical equations are obtained using the machine ries of the Hamilton’s v ariational principle. Th us it is exp ected that the b osonic and fermionic equations a s given in T able 1 will follo w from judicious use of the action principle. T o achie v e this w e pro ceed b y noting the fo llo wing. A single ev olution equation do es not follow from an action principle [12]. When written in terms of the Casimir p otential an evolution equation can either b e Lagrangian or nonLagrangian. The example of a nonLagrangian ev o lution equation ev en in p o ten tial space is prov ided by Rosenau-Hyman equation [13]. Most of nonlinear ev olution equation ha ve at least one con- serv ed densit y suc h t hat w e can write these equations as u t + ∂ ρ [ u ] ∂ x = 0 . (20) Clearly , the KdV, mKdV and FNE equations are of the form (20). The HS and CH equations can b e recast in a similar form b y in t ro ducing n = u xx and m = u − u xx . W e can mak e use of an elemen tary lemma to get a Lagrangia n represen - tation of (20). Lemma 1. There exists a pro longation of (20) into ano t her equation v t + δ δ u ( ρ [ u ] v x ) = 0 (21) with t he v aria t io nal deriv ative δ δ u = n X k =0 ( − 1) k ∂ k ∂ x k ∂ ∂ u k x , u k x = ∂ k u ∂ x k (22) suc h that the system of equations follows fro m the action principle δ Z L d xdt = 0 . (23) 7 Here L stands for the Lagrangian densit y . Pro of. F or a direct pro of of the lemma let us in tro duce L in the form L = 1 2 ( v u t − uv t ) − ρ [ u ] v x . (24) F rom (23) and (2 4) w e obtain the Euler-Lag range equations d dt ( ∂ L ∂ v t ) − δ L δ v = 0 (25) and d dt ( ∂ L ∂ u t ) − δ L δ u = 0 . (26) Using ( 24) in (25 ) and (26) w e obtain (20) and (21) resp ectiv ely . Lemma 1 giv es the fermionic equations for the KdV a nd mKdV equation in a rather straig h tforw ard manner. Since the HS a nd CH equations inv olve mixed deriv ativ e lik e u xxt , one needs t o mak e use of a simple v arian t of Lemma 1 to construct the corresp onding fermionic equations. T o ac hiev e this we write the CH equation as m t + ∂ ρ [ u ] ∂ x = 0 (27) with t he asso ciated equation η t + δ δ u ( ρ [ u ] v x ) = 0 (28) where η = v − v xx . The system of equations will follo w from a Lagrangia n densit y L = 1 2 ( v m t − mv t ) − ρ [ u ] v x (29) via the Euler-Lagrange equation d dt d 2 dx 2 ( ∂ L ∂ φ xxt ) + d dt ( ∂ L ∂ φ t ) − δ L δ φ = 0 (30) obtained from (23) and (29). Here φ is either u or v . A similar treatmen t also applies for the HS equation. Application of Lemma 1 t o the F NE equation giv es the sup erpart ner v t + 3 u 2 v x + 2 uv 3 x = 0 . (31) 8 This equation do es not agree with the result giv en in T able 1 presumably b ecause (19) is nonLagrangian ev en in the p otential represen tation. In the recen t past one of us (BT) [13] found that the Helmholtz solutio n of the in v erse problem [14] can b e judiciously exploited to construct Lagra ngian systems f rom (19), whic h suppo r t compacton solutions. F or example, the equation u t + 3 u 2 u x + 8 u x u 2 x + 4 uu 3 x = 0 (32) when written in terms of Casimir p oten tial w ( x, t ) = R ∞ x u ( y , t ) dy w as fo und to result fro m an action principle and supp ort compacton solution. The fermionic equation corresp onding to (32) can b e obtained as v t + 3 u 2 v x + 4 u x v 2 x + 4 u 2 x v x + 4 uv 3 x = 0 . (33) It is easy to v erify that (33) can also b e obtained b y the use of Lemma 1 . Th us w e infer that all fermionic equations resulting from B-sup ersymmetrization of Lagrangian system of nonlinear ev o lution equations can b e o bta ined b y using our Lemma 1. Ab out a decade ago Kaup and Malomed [15] sough t an a pplication of the v ariational pr inciple to nonlinear field equations in volving dissipativ e terms. These aut ho rs demanded that the Lag rangian densit y L = v ( x, t ) × (equation for u ( x, t )) . (34) should b e minimize the action functional. An imp ortan t consequen ce of writing (34) is that in the presence o f the a uxiliary field v ( x, t ) the resulting set up Lagra ng ian symmetries coincides with t he symm etries of the ev olution equation for u ( x, t ). In p o in t mec hanics this t yp e of generalization for t he traditional No etherian symmetry w as in tro duced b y Ho j ma n [16]. F or the KdV equation ( 3 4) reads L = v ( u t + 6 uu x + u 3 x ) . (35) Clearly , for L in (35 ) the Euler-Lagrange equation in v ( x, t ) give s the KdV equation and more significan tly w e get the corresp onding fermionic equation from the Euler-Lagrang e equation for u ( x, t ). Starting fro m (3 4) w e can v erify the results giv en in T able 1 for the mKdV, HS and CH equations. Consisten tly with our previous analysis the use of (34) do es no t repro duced the tabulated result for t he R H equation. Ra ther w e find that (33) is the sup ersymmetric partner of ( 32) whic h is Lagr angian in the p otential space. Th us there m ust exist a relation b et w een the Lag rangian densities in (24) and (34), F or the KdV field the g auge terms d dt  1 2 uv  + d dx (3 u 2 v ) + d dx ( u 2 x v ) can b e a dded to (24) to get (35). A similar conclusion a lso holds go o d for 9 other Lagrangian equ ations in T able 1 implying that the Lagrangians in (24) and (34 ) are gauge equiv alen t . 4. Conclusion Nonlinear evolution equations do not admit Lagrangian represen tation in terms of the field v ariables b ecause the F r ´ ec het deriv ative D P of a n y equation written as u t = P [ u ] is non self-adjo int suc h that the Helmholtz theorem [14] b ecomes inapplicable. Similar nonLag r a ngian systems also o ccur in p oint mec hanics. F or example, it is w ell kno wn that there is no direct metho d of applying v aria tional principles to nonconserv ativ e system s, whic h a re c har- acterized b y friction or other diss ipativ e pro cesses. In fact, one can not write a time-independen t Lagra ng ia n ev en for a simple syste m like the damp ed Harmonic oscillator represen ted by ¨ q 1 + λ ˙ q 1 + ω 2 q 1 = 0 (36) since it is non self-adjoin t. Here λ is the frictional co efficien t and ω , the natural f r equency of the oscillator. Bateman [10] sho w ed tha t while ( 3 6) is non Lagrangian, the v ariatio nal principle can be a pplied to a dual system consisting of (36) and the equation ¨ q 2 − λ ˙ q 2 + ω 2 q 2 = 0 . (37) Equations (3 6 ) and (37) can b e obtained by using t he Lagrangian L = q 2 ( ¨ q 1 + λ ˙ q 1 + ω 2 q 1 ) (38) in the Euler-Lagrange equations ∂ L ∂ q 2 = 0 (39) and ∂ L ∂ q 1 − d dt ∂ L ∂ ˙ q 1 ! + d 2 dt 2 ∂ L ∂ ¨ q 1 ! = 0 . (40) Note that L in (38) can b e con v erted to the well-kno wn Morse-F esh bac h [1 7] Lagrangian by adding appro pr ia te gaug e terms. Equation (36) represen ts the motion of a simple p endulum em b edded in a fluid whic h opp oses the motion through frictional forces prop ortional to the v elo cit y . As a result the energy is drained out o f t he system. The complemen- tary equation (37) represen ts a second ph ysical system whic h absorbs energy 10 dissipated in the first suc h that (36) and (37) taken tog ether can b e view ed as a conserv ativ e system to hav e an explicitly time independent Lagrangian represen tatio n. Thus t he phys ical in terpretatio n of Bateman dual system is fairly simple and straightforw ord. This is, how ev er, not the case with the pair of equations obtained b y B-sup ersymmetrization and finally sho wn to form a Lagrangian sys tem. No nintegrabilit y of these equations [2,5] migh t b e o ne the reasons f or t hat. The single soliton solution of the KdV equation in (8) is giv en b y u = − 2 k 2 sech 2 k ( x − 4 k 2 t ). The solution has amplitude − 2 k 2 and mov es to the right with sp eed 4 k 2 . F or this v alue of u w e hav e found v in (8 ′ ) as v = tanh 2 k ( x − 4 k 2 t ). The v a v e represen ted by v is of constan t amplitude and mov es with same v elo cit y as that of u . Moreov er, | u | with a p eak while | v | is a not ch. Lo oking from this p oint o f view u and v may b e regarded as bright and dark soliton resp ective ly . The b osonic and fermionic fields of (32) and (33) are associated with compaction and an ticompacton expressed in terms of appropriate Ja cobi elliptic function. This kind of realization for bo sonic and fermionic fields marks a p oin t of departure from our under- standing of quan tum field theory in that the particles and a n tiparticles are not sup erpar t ner. Ac kno wledgemen t s This w ork is supp orted b y the Unive rsit y Grants Commiss ion, Gov ernmen t of India, through grant No. F.3 2-39/2 006(SR). References [1] W ess J and Zumino B 19 74 Nucl. Phys. 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