The Maxwell equations as a Bäcklund transformation

Transcription

1 ADVANCED ELECTROMAGNETICS, VOL. 4, NO. 1, JULY 15 The Mawell equaons as a Bäklund ransformaon C. J. Papahrsou Deparmen of Physal Senes, Naval Aademy of Greee, Praeus, Greee papahrsou@snd.edu.gr Absra Bäklund ransformaons (BTs) are a useful ool for negrang nonlnear paral dfferenal equaons (PDEs). However, he sgnfane of BTs n lnear problems should no be gnored. In fa, an mporan lnear sysem of PDEs n Physs, namely, he Mawell equaons of eleromagnesm, may be vewed as a BT relang he wave equaons for he eler and he magne feld, hese equaons represenng negrably ondons for soluon of he Mawell sysem. We eamne he BT propery of hs sysem n deal, boh for he vauum ase and for he ase of a lnear ondung medum. 1. Inroduon Bäklund ransformaons (BTs) are an effeve ool for negrang paral dfferenal equaons (PDEs). They are parularly useful for obanng soluons of nonlnear PDEs, gven ha hese equaons are ofen noorously hard o solve by dre mehods (see [1] and he referenes heren). Generally speakng, gven wo PDEs say (a) and (b) for he unknown funons u and v, respevely, a BT relang hese PDEs s a sysem of aulary PDEs onanng boh u and v, suh ha he onsseny (negrably) of hs sysem requres ha he orgnal PDEs (a) and (b) be separaely sasfed. Then, f a soluon of PDE (a) s known, a soluon of PDE (b) s found smply by negrang he BT, whou havng o negrae he PDE (b) drely (whh, presumably, s a muh harder ask). In addon o beng a soluon-generang mehansm, BTs may also serve as reurson operaors for obanng nfne herarhes of (generally nonloal) symmeres and onservaon laws of a PDE [1 7]. I s by hs mehod ha he full symmery Le algebra of he self-dual Yang-Mlls equaon was found [3,6]. In hs arle, he naure of whh s mosly pedagogal, we adop a somewha dfferen (n a sense, nverse) vew of a BT, suable for he reamen of lnear problems. Suppose we are gven a sysem of PDEs for he unknown funons u and v. Suppose, furher, ha he onsseny of hs sysem requres ha wo PDEs, one for u and one for v, be separaely sasfed (hus, he gven sysem s a BT onneng hese PDEs). The PDEs are assumed o possess known soluons for u and v, eah soluon dependng on a number of parameers. If, by a proper hoe of he parameers, hese funons are made o sasfy he orgnal dfferenal sysem, hen a soluon o hs sysem has been found. In oher words, we are seekng soluons of he gven sysem by usng known, parameer-dependen soluons of he ndvdual PDEs epressng he negrably ondons of hs sysem. Pars of funons (u,v) sasfyng he sysem wll be sad o represen BT-onjugae soluons. Ths modfed vew of he onep of a BT has an mporan applaon n eleromagnesm ha serves as a paradgm for he sgnfane of BTs n lnear problems. As dsussed n hs paper, he Mawell equaons for a lnear medum ealy f hs BT sheme. Indeed, as s well known, he onsseny of he Mawell sysem requres ha he eler and he magne feld sasfy separae wave equaons. These equaons have known, parameerdependen soluons, namely, monohroma plane waves wh arbrary ampludes, wave veors, frequenes, e. (he parameers of he problem). By nserng hese soluons no he Mawell sysem, one may fnd he neessary ondons on he parameers n order ha he plane waves for he wo felds represen BT-onjugae soluons of Mawell s equaons. The paper s organzed as follows: Seon revews he lassal onep of a BT. The soluon-generang proess by usng a BT s demonsraed n a number of eamples. In Se. 3 he onep of paramer, BT-onjugae soluons s nrodued. A smple eample llusraes he dea. In Se. 4 he Mawell equaons n empy spae are shown o onsue a BT n he sense desrbed n Se. 3. For ompleeness of presenaon (and for he benef of he suden) he proess of onsrung BT-onjugae planewave soluons s presened n deal. Fnally, n Se. 5 he Mawell sysem for a lnear ondung medum s smlarly eamned. The resuls of Ses. 4 and 5 are, of ourse, well known from lassal eleromagne heory. I s mahemaally neresng, however, o revs he problem of onsrung soluons of Mawell s equaons from a novel pon of vew by usng he onep of a BT and by reang he eler and he magne omponen of a plane e/m wave as BTonjugae soluons.. Bäklund ransformaons: defnon and eamples The general dea of a Bäklund ransformaon (BT) was eplaned n [1] (see also he referenes heren). Le us revew he man pons:

2 We onsder wo PDEs P[u]= and Q[v]=, where he epressons P[u] and Q[v] may onan he unknown funons u and v, respevely, as well as some of her paral dervaves wh respe o he ndependen varables. For smply, we assume ha u and v are funons of only wo varables,. Paral dervaves wh respe o hese varables wll be denoed by usng subsrps, e.g., u, u, u, u, u, e. We also onsder a sysem of oupled PDEs for u and v, B [ u, v] =, = 1, (1) where he epressons B [u,v] may onan u, v and eran of her paral dervaves wh respe o and. The sysem (1) s assumed o be negrable for v (he wo equaons are ompable wh eah oher for soluon for v) when u sasfes he PDE P[u]=. The soluon v, hen, sasfes he PDE Q[v]=. Conversely, he sysem (1) s negrable for u f v sasfes he PDE Q[v]=, he soluon u hen sasfyng P[u]=. If he above assumpons are vald, we say ha he sysem (1) onsues a BT onneng soluons of P[u]= wh soluons of Q[v]=. In he speal ase where P Q,.e., when u and v sasfy he same PDE, he sysem (1) s alled an auo-bäklund ransformaon (auo-bt). Suppose now ha we seek soluons of he PDE P[u]=. Also, assume ha we possess a BT onneng soluons u of hs equaon wh soluons v of he PDE Q[v]= (f P Q he auo-bt onnes soluons u and v of he same PDE). Le v=v (,) be a known soluon of Q[v]=. The BT s hen a sysem of equaons for he unknown u: B [ u, v ] =, = 1, () Gven ha Q[v ]=, he sysem () s negrable for u and s soluon sasfes he PDE P[u]=. We may hus fnd a soluon u(,) of P[u]= whou solvng he equaon self, smply by negrang he BT () wh respe o u. Of ourse, he use of hs mehod s meanngful provded ha we know a soluon v (,) of Q[v]= beforehand, as well as ha negrang he sysem () for u s smpler han negrang he PDE P[u]= drely. If he ransformaon () s an auo-bt, hen, sarng wh a known soluon v (,) of P[u]= and negrang he sysem (), we fnd anoher soluon u(,) of he same equaon. Le us see some eamples of usng a BT o generae soluons of a PDE: 1. The Cauhy-Remann relaons of omple analyss, u = v ( a) u = v ( b) (3) y y (here, he varable has been renamed y) onsue an auo- BT for he (lnear) Laplae equaon, P [ w] w + w = (4) yy Indeed, dfferenang (3a) wh respe o y and (3b) wh respe o, and demandng ha he negrably ondon (u ) y =(u y ) be sasfed, we elmnae he varable u o fnd he onsseny ondon ha mus be obeyed by v(,y) n order ha he sysem (3) be negrable for u: P [ v] v + v =. yy Conversely, elmnang v from he sysem (3) by usng he negrably ondon (v ) y =(v y ), we fnd he neessary ondon for u n order for he sysem o be negrable for v: P [ u] u + u =. yy Now, le v (,y) be a known soluon of he Laplae equaon (4). Subsung v=v n he sysem (3), we an negrae he laer wh respe o u o fnd anoher soluon of he Laplae equaon. For eample, by hoosng v (,y)=y we fnd he soluon u(,y)= ( y )/ +C.. The Louvlle equaon s wren u P[ u] u e = u = e (5) Solvng he PDE (5) drely s a dfful ask n vew of hs equaon s nonlneary. A soluon an be found, however, by usng a BT. We hus onsder an aulary funon v(,) and an assoaed lnear PDE, Q[ v] v = (6) We also onsder he sysem of frs-order PDEs, u + v = e u v = e ( u v) / ( u+ v) / I an be shown ha he self-onsseny of he sysem (7) requres ha u and v ndependenly sasfy he PDEs (5) and (6), respevely. Thus, hs sysem onsues a BT onneng soluons of (5) and (6). Sarng wh he rval soluon v= of (6) and negrang he sysem u / u / u = e, u = e, we fnd a soluon of (5): + u (, ) = ln C. 3. The sne-gordon equaon has applaons n varous areas of Physs, suh as n he sudy of rysallne solds, n he ransmsson of elas waves, n magnesm, n elemenary-parle models, e. The equaon (whose name u (7) 53

3 s a pun on he relaed lnear Klen-Gordon equaon) s wren u = sn u (8) As an be proven, he dfferenal sysem 1 u v ( u+ v ) = a sn 1 1 u+ v ( u v) = sn a [where a ( ) s an arbrary real onsan] s a paramer auo-bt for he PDE (8). Sarng wh he rval soluon v= of v = sn v, and negrang he sysem u u u = a sn, u = sn, a we oban a new soluon of (8): { } u (, ) = 4 aran C ep a + a. 3. BT-onjugae soluons Consder a sysem of oupled PDEs for he funons u and v of wo ndependen varables, y: (9) B [ u, v] =, = 1, (1) Assume ha he negrably of hs sysem for boh u and v requres ha he followng PDEs be ndependenly sasfed: P[ u] = ( a) Q[ v] = ( b) (11) Tha s, he sysem (1) represens a BT onneng he PDEs (11). Assume, furher, ha he PDEs (11) possess parameer-dependen soluons of he form u = f (, y ; α, β, γ, ), v = g(, y ; κ, λ, µ, ) (1) where α, β, κ, λ, e., are (real or omple) parameers. If values of hese parameers an be deermned for whh u and v sasfy he sysem (1), we say ha he soluons u and v of he PDEs (11a) and (11b), respevely, are onjugae hrough he BT (1) (or BT-onjugae, for shor). Le us see an eample: Gong bak o he Cauhy- Remann relaons (3), we ry he followng paramer soluons of he Laplae equaon (4): u (, y) = α ( y ) + β + γ y, v (, y) = κ y+ λ + µ y. Subsung hese no he BT (3), we fnd ha κ=α, µ=β and λ= γ. Therefore, he soluons u (, y) = α ( y ) + β + γ y, v (, y) = α y γ + β y of he Laplae equaon are BT-onjugae hrough he Cauhy-Remann relaons. As a ouner-eample, le us ry a dfferen ombnaon: u (, y) = α y, v (, y) = β y. Inserng hese no he sysem (3) and akng no aoun he ndependene of and y, we fnd ha he only possble values of he parameers α and β are α=β=, so ha u(,y)= v(,y)=. Thus, no non-rval BT-onjugae soluons es n hs ase. 4. Applaon o he Mawell equaons n empy spae As s well known, aordng o he Mawell heory all eleromagne (e/m) dsurbanes propagae n spae as waves runnng a he speed of lgh. I s neresng from he mahemaal pon of vew ha he vauum wave equaons for he eler and he magne feld are onneed o eah oher hrough he Mawell sysem of equaons n muh he same way wo PDEs are onneed va a Bäklund ransformaon. In fa, eran parameer-dependen soluons of he wo wave equaons are BT-onjugae hrough he Mawell sysem. In empy spae, where no harges or urrens (wheher free or bound) es, he Mawell equaons are wren n S.I. uns [8]: B ( a) E = ( ) E = ( b) B = ( d) B = ε µ E (13) where E and B are he eler and he magne feld, respevely. In order ha hs sysem of PDEs be selfonssen (hus negrable for he wo felds), eran onsseny ondons (or negrably ondons) mus be sasfed. Four are sasfed auomaally: ( E) =, ( B) =, ( E) = E, ( B) = B. 54

4 Two ohers read: ( ) = ( ) E E E ( ) = ( ) B B B (14) (15) Takng he ro of (13) and usng (14), (13a) and (13d), we fnd: E ε µ E = (16) Smlarly, akng he ro of (13d) and usng (15), (13b) and (13), we ge: B ε µ B = (17) No new nformaon s furnshed by he remanng wo negrably ondons, ( E) = E, ( B) = B. Pung 1 1 ε µ = (18) ε µ we rewre Eqs. (16) and (17) n wave-equaon form: 1 E = E (19) ω = where k = k k (1) The smples suh soluons are monohroma plane waves of angular frequeny ω, propagang n he dreon of he wave veor k : E ( r, ) = E ep{ ( k r ω )} ( a) () B ( r, ) = B ep{ ( k r ω )} ( b) where he E and B represen onsan omple ampludes. Sne all onsans appearng n equaons () (ha s, ampludes, frequeny and wave veor) an be arbrarly hosen, hey an be regarded as parameers on whh he soluons () of he wave equaons depend. Clearly, alhough every par of felds ( E, B) ha sasfes he Mawell equaons (13) also sasfes he respeve wave equaons (19) and (), he onverse s no rue. Ths means ha he soluons () of he wave equaon are no a pror soluons of he Mawell sysem of equaons (.e., do no represen e/m felds). Ths problem an be remeded, however, by approprae hoe of he parameers. To hs end, we subsue he general soluons () no he sysem (13) n order o fnd he era ondons hs sysem requres; ha s, n order o make he wo funons n () BT-onjugae soluons of he respeve wave equaons (19) and (). Subsung (a) and (b) no (13a) and (13b), respevely, and akng no aoun ha e = k e, k r k r we have: ω k r ( k r ω ( E e ) e = ( k E ) e =, ω k r ( k r ω ( B e ) e = ( k B ) e =, 1 B = B () so ha k E =, k B =. (3) The PDEs (19) and () are onsseny ondons ha mus be separaely sasfed by E and B n order ha he dfferenal sysem (13) be negrable for eher feld, gven he value of he oher feld. In oher words, he sysem (13) s a BT relang soluons of he wave equaons (19) and (). I should be noed arefully ha he BT (13) s no an auo-bt! Indeed, alhough he PDEs (19) and () look smlar, hey onern dfferen felds wh dfferen physal dmensons and physal properes. A rue auo-bt should onne smlar objes (suh as, e.g., dfferen mahemaal epressons for he eler feld). The above wave equaons adm plane-wave soluons of he form F ( k r ω ), wh Physally, hs means ha he monohroma plane e/m wave s a ransverse wave. Ne, subsung (a) and (b) no (13) and (13d), we fnd: ω k r ( k r ω e ( e ) E = ω B e ( k r ω ( k r ω ( k E ) e = ω B e, ω k r ( k r ω e ( e ) B = ωε µ E e ω ( k r ω ( k r ω ( k B ) e = E e, 55

5 so ha ω k E = ω B, k B = E (4) Ths means ha he felds E and B are normal o eah oher as well as beng normal o he dreon of propagaon. I an be seen ha he wo veor equaons n (4) are no ndependen of eah oher; ndeed, rossmulplyng he frs relaon by k we ge he seond one. Inrodung a un veor ˆ τ n he dreon of he wave veor k, ˆ τ = k / k ( k = k = ω / ), we rewre he frs of Eqs. (4) as k 1 B = ( ˆ τ E ) = ( ˆ τ E ). ω The BT-onjugae soluons n () are now wren: E ( r, ) = E ep{ ( k r ω )}, 1 B ( r, = ( ˆ τ E ) ep{ ( k r ω } 1 = ˆ τ E (5) As onsrued, he omple veor felds n (5) sasfy he Mawell sysem (13), whh s a homogeneous lnear sysem wh real oeffens. Evdenly, he real pars of hese felds also sasfy hs sysem. To fnd he epressons for he real soluons (whh, afer all, arry he physs of he suaon) we ake he smples ase of a lnearly polarzed e/m wave and wre: E = E e α, R (6) where he veor E and he number α are real. The real,r versons of he felds (5), hen, read: E = E os ( k r ω + α ),, R 1 B = ( ˆ τ E, R ) os ( k r ω + α ) 1 = ˆ τ E (7) We noe, n parular, ha he felds E and B osllae n phase. Our resuls for he Mawell equaons n vauum an be eended o he ase of a lnear non-ondung medum upon replaemen of ε and µ wh ε and µ, respevely. The speed of propagaon of he e/m wave s, n hs ase, ω 1 υ = =. k εµ 5. The Mawell sysem for a lnear ondung medum In a lnear ondung medum of onduvy σ, n whh Ohm s law s sasfed, J =σ E f (where J s he free f urren densy), he Mawell equaons read [8]: B ( a) E = ( ) E = E ( b) B = ( d) B = µσ E+ εµ By he negrably ondons = = ( E) ( E) E, ( B) ( B) B, we ge he modfed wave equaons E E E εµ µσ = B B B εµ µσ = (8) (9) No new nformaon s furnshed by he remanng negrably ondons (f. Se. 4). We observe ha he lnear dfferenal sysem (8) s a BT relang soluons of he wave equaons (9) (as eplaned n he prevous seon, hs BT s no an auo-bt). As n he vauum ase, we seek BT-onjugae suh soluons. As an be verfed by dre subsuon no Eqs. (9), hese PDEs adm paramer plane-wave soluons of he form E ( r, ) = E ep{ s ˆ τ r + ( k r ω)} s { } = E ep k r ep ( ω ), k B ( r, ) = B ep{ s ˆ τ r + ( k r ω )} s { } = B ep k r ep ( ω ) k (3) 56

6 where ˆ τ s he un veor n he dreon of he wave veor k, ˆ τ = k / k ( k = k = ω / υ ) (υ s he speed of propagaon of he wave nsde he ondung medum) and where, for gven physal haraerss ε, µ, σ of he medum, he parameers s, k and ω sasfy he algebra sysem s k + εµω =, µσω sk = (31) Up o hs pon he omple ampludes E and B n relaons (3) are arbrary and he veor felds (3) are no a pror soluons of he Mawell equaons (8), hus are no ye BT-onjugae soluons of he respeve wave equaons n (9). To fnd he resrons hese ampludes mus sasfy, we nser Eqs. (3) no he sysem (8). Wh he ad of he relaon s s ( ) s ( ) k r k r k k e = k e k s no hard o show ha (8a) and (8b) mpose he ondons k E =, k B =, (3) Agan, hs means ha he e/m wave s a ransverse wave. Subsung (3) no (8) and (8d), we fnd wo more ondons: ( k+ s) ˆ τ E = ωb ( k+ s) ˆ τ B = ( εµω + µσ ) E (33) (34) However, (34) s no an ndependen equaon sne an be reprodued by ross-mulplaon of (33) by ˆ τ and use of relaons (31). The BT-onjugae soluons of he wave equaons (9) are now wren: s ˆ τ r ( k r ω E( r, ) = E e e, k+ s B( r, ) = ( ˆ τ E ) e e ω s ˆ τ r ( k r ω (35) To fnd he orrespondng real soluons, we assume lnear polarzaon of he e/m wave and se, as before, E = E e α, R. We also se k+ s = k+ s e = k + s e an ϕ = s / k. ϕ ϕ ; Takng he real pars of Eqs. (35), we fnally have: s ˆ τ r E( r, ) = E e os ( k r ω + α ),, R k + s s ˆ τ r B( r, ) = ( ˆ τ E ) e os ( k r ω + α + ϕ)., R ω 6. Summary and onludng remarks Bäklund ransformaons (BTs) were orgnally devsed as a ool for fndng soluons of nonlnear paral dfferenal equaons (PDEs). They were laer also proven useful as nonloal reurson operaors for onsrung nfne sequenes of symmeres and onservaon laws of eran PDEs [ 7]. Generally speakng, a BT s a sysem of PDEs onneng wo felds ha are requred o ndependenly sasfy wo respeve PDEs n order for he sysem o be negrable for eher feld. If a soluon of eher PDE s known, hen a soluon of he oher PDE s obaned by negrang he BT, whou havng o aually solve he laer PDE eplly (whh, presumably, would be a muh harder ask). In he ase where he wo PDEs are denal, an auo-bt produes new soluons of a PDE from old ones. As desrbed above, a BT s an aulary ool for fndng soluons of a gven (usually nonlnear) PDE, usng known soluons of he same or anoher PDE. In hs arle, however, we approahed he BT onep dfferenly by aually nverng he problem. Aordng o hs sheme, s he soluons of he BT self ha we are afer, havng parameer-dependen soluons of he PDEs ha epress he negrably ondons a hand. By a proper hoe of he parameers, a par of soluons of hese PDEs may possbly be found ha sasfes he gven BT. These soluons are hen sad o be onjugae wh respe o he BT. A pedagogal paradgm for demonsrang hs parular approah o he onep of a BT s offered by he Mawell sysem of equaons of eleromagnesm. We showed ha hs sysem an be hough of as a BT whose negrably ondons are he wave equaons for he eler and he magne feld. These wave equaons have known, parameer-dependen soluons (monohroma plane waves) wh arbrary ampludes, frequenes, wave veors, e. By subsung hese soluons no he BT, one may deermne he requred relaons among he parameers n order ha he plane waves also represen eleromagne felds,.e., are BT-onjugae soluons of he Mawell sysem. The resuls arrved a by hs mehod are, of ourse, well known n advaned elerodynams. The proess of dervng hem, however, s seen here n a new lgh by employng he onep of a BT. 57

7 We remark ha he physal suaon was eamned from he pon of vew of a fed neral observer. Thus, sne no spaeme ransformaons were nvolved, we used he lassal form of he Mawell equaons (wh E and B reanng her ndvdual haraers) raher han he manfesly ovaran form of hese equaons. An neresng onluson s ha he onep of a Bäklund ransformaon, whh has been proven eremely useful for fndng soluons of nonlnear PDEs, an n eran ases also prove useful for negrang lnear sysems of PDEs. Suh sysems appear ofen n Physs and Eleral Engneerng (see, e.g., [9]) and would eranly be of neres o eplore he possbly of usng BT mehods for her negraon. Aknowledgmen I hank Arsds N. Magoulas for many fruful dsussons. Referenes [1] C. J. Papahrsou, Symmery and negrably of lassal feld equaons, hp://arv.org/abs/ [] C. J. Papahrsou, Poenal symmeres for selfdual gauge felds, Phys. Le. A 145 (199) 5. [3] C. J. Papahrsou, Reurson operaor and urren algebras for he poenal SL(N,C) self-dual Yang- Mlls equaon, Phys. Le. A 154 (1991) 9. [4] C. J. Papahrsou, La par, hdden symmeres, and nfne sequenes of onserved urrens for self-dual Yang-Mlls felds, J. Phys. A 4 (1991) L 151. [5] C. J. Papahrsou, Symmery, onserved harges, and La represenaons of nonlnear feld equaons: A unfed approah, Eleron. J. Theor. Phys. 7, No. 3 (1) 1. [6] C. J. Papahrsou, B. K. Harrson, Bäklundransformaon-relaed reurson operaors: Applaon o he self-dual Yang-Mlls equaon, J. Nonln. Mah. Phys., Vol. 17, No. 1 (1) 35. [7] C. J. Papahrsou, Symmery and negrably of a redued, 3-dmensonal self-dual gauge feld model, Eleron. J. Theor. Phys. 9, No. 6 (1) 119. [8] D. J. Grffhs, Inroduon o Elerodynams, 3rd Edon (Prene-Hall, 1999). [9] E. C. Zahmanoglou, D. W. Thoe, Inroduon o Paral Dfferenal Equaons wh Applaons (Dover, 1986). 58