Mathematical Formulation of Inverse Scattering and Korteweg- De Vries Equation

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1 Mahmaical Thor ad Modlig ISSN (Papr) ISSN 5-5 (Oli) Vol.3, No.3, 3 Mahmaical Formulaio of Ivrs Scarig ad Korwg- D Vris Equaio Absrac Bija Krisha Saha, S. M. Chapal Hossai & Md. Shafiul Alam Dparm of Mahmaics, Uivrsi of Barisal, Barisal-8, Bagladsh Dparm of Mahmaics, Jagaah Uivrsi, Dhaka-, Bagladsh Corrspodig Auhor: Tl: , bijadumah@gmail.com Ivrs scarig rfrs o h drmiaio of h soluios of a s of diffrial quaios basd o kow asmpoic soluios, ha is, h soluio of Marchko quaio. Marchko quaio was drivd usig igral quaio. Th poial fucio drivd from igvalus ad scarig daa sms o b h ivrs mhod of scarig problm. Th rflcio coffici wih o pol ad zro rflcio cofficis has b chos o solv ivrs scarig problm. Agai his papr dals wih h cocio bw ivrs scarig ad h Korwg-d Vris quaio ad dscribs vari of ampls wih Korwg-d Vris quaio: h sigl-solio soluio, h wo-solio soluio ad fiall h N-solio soluio. Throughou h work, h primar objciv is o sud som mahmaical chiqus applid i aalzig h bhavior of solio i h KdV quaios. Kwords: Marchko quaio, KdV quaio, Solios, Scarig, Ivrs Scarig, Caal.. Iroducio I h ara of scarig hor i phsics, h ivrs scarig problm drmis h characrisics of a objc (is shap, iral cosiuio, c.) from masurm daa of radiaio or paricls scard from h objc. I phsical rms h problm is ssiall o of fidig h shap (or prhaps mass disribuio) of a objc which is mchaicall vibrad, from a kowldg of all h souds ha maks, i.. from h rg or ampliud a ach frquc. Bu i mahmaics, ivrs scarig rfrs o h drmiaio of h soluios of a s of diffrial quaios basd o kow asmpoic soluios, ha is, h soluio of Marchko quaio []. Eampls of quaios ha hav b solvd b ivrs scarig ar h rflcio coffici wih pol ad zro rflcio coffici. I is h ivrs problm o h dirc scarig problm, which is drmiig h disribuio of scard radiaio o h characrisics of h scarr. Sic is arl sam for radio locaio, h problm has foud vas umbr of applicaios, such as cholocaio, gophsical surv, odsruciv sig, mdical imagig, quaum fild hor, o am jus a fw []. Th Korwg-d Vris quaio [3] dscribs h hor of war wavs i shallow Chals, such as caal. I is a o-liar quaio which hibis spcial soluios, kow as solios, which ar sabl ad do o disprs wih im. Furhrmor hr as soluios wih mor ha o solio which ca mov owards ach ohr, irac ad h mrg a h sam spd wih o chag i shap (bu wih a im lg or spd up ). Th solio phomo was firs dscribd b Joh Sco Russll (88-88) who obsrvd a soliar wav i h Uio Caal, rproducd h phomo i a wav ak, ad amd i h "Wav of Traslaio"[4]. Owig o is oliari, h KdV quaio rsisd aalsis for ma ars, ad i did o com udr sris scrui uil 965, wh Zabusk [5] ad Kruskal (s also [5]) obaid umrical soluios, whil ivsigaig h Frmi-Pasa-Ulam problm [6] of masss coupld b wakl oliar sprigs. From a gral iiial codiio, a soluio o KdV dvlops io a sris of soliar pulss of h varig ampliuds, which pass hrough o aohr wihou modificaio of shap or spd. Th ol ligrig rac of h srog oliar iracio bw hs so-calld solios (lik lcros, proos, barios ad ohr paricls dig wih os ) [] is a sligh forward phas shif for h largr, fasr wav ad sligh backward shif for h smallr, showr o. Solio soluios hav subsqul b dvlopd for a larg umbr of oliar quaios, icludig quaios of paricl phsics, ad mago hdrodamics. Havig ildig a sarlig w p of wav bhaviour, h KdV quaio soo simulad a furhr mahmaical dvlopm, wh i quaio soo simulad a furhr mahmaical dvlopm, wh i 967, Gard[], Gr, Kruskal ad Miura(s also []) iroducd h ivrs scarig rasform mhod for drmiig h solios ha aris from arbirar iiial codiios. This chiqu rprsd a major advac i h mahmaical hor of PDFs. As i mad i 8

2 Mahmaical Thor ad Modlig ISSN (Papr) ISSN 5-5 (Oli) Vol.3, No.3, 3 possibl obai closd-form soluios o oliar voluio quaios ha wr prviousl bod h rach of aalsis. This brakhrough iiiad a priod of rapid dvlopms boh i dscribig h propris of KdV solios ad i gralizig h approach o ohr oliar quaios icludig h Si-Gordo, o liar Schrödigr ad Boussisq quaio.. Formulaio of Ivrs Scarig Problm Th ivrs scarig chiqu is applicabl o h KdV, No liar Schrödigr ad Si-Gordo quaios.w ow dscrib how h ivrs scarig rasform ca b usd o cosruc h soluio o h iiial-valu problm for KdV quaio. I his scio w shall summariz h rsuls, ad w discuss som spcific ampls.w wish o solv h KdV quaio u 6 uu + u, >, - < <. () - wih u (,) f ( ). I is assumd ha f is a sufficil wll-bhavd fucio i ordr o sur h isc of a soluio of h KdV quaio ad also of h Sur-Liouvill quaio Y + ( l - u) Y, - < <. () Th firs sag o s u (,) f ( ) ad solv quaio (), a las o h of drmiig h discr spcrum, - k, h ormalizaio cosas, c (), ad h rflcio cofficis, (k;). of hs scarig daa is h giv b quaios. k Th fucio F, dfid b h quaio cos a ; c ( c ()p(4 k 3 (3) ad b ( k; b( k;)p(8 ik 3. (4) F( X ) N å c p( -k Now h Marchko quaio for h abov quaio is F( X; X ) + p ò - b( 3 3 c ()p(8 k- kx ) + b( k;)p(8 ik + p - N å ò ikx dk. ikx ) dk. b Th im voluio No ha F also dpds upo h paramr. Th Marchko quaio for K(, is hrfor K (, + F( + + ò K(, ; F( + d. Fiall h soluio of h KdV quaio ca b prssd as u(, K(, 3. Procdur - ad K (, K(, ;. W firs wri h KdV quaio i h covi form u 6 uu + u. (6) - Th o simpl wa of showig a cocio wih h Surm-Liouvill problm is o dfi a fucio u(, such ha u v + v (7) Equaio (7) is calld h Miura Trasformaio. Dirc subsiuio of (7) io quaio (6) h ilds vv + v - 6( v + v)(vv + v) + 6vv + vv + v which ca b rarragd o giv æ ö çv + ( v - 6v v+ v ) (8) è ø (5) 9

3 Mahmaical Thor ad Modlig ISSN (Papr) ISSN 5-5 (Oli) Vol.3, No.3, 3 Thus if v is a soluio of 6v v + v v (9) - Th abov quaio is calld h Modifid KdV quaio (mkdv). Th quaio (7) dfis a soluio of h KdV quaio (6). Now w rcogiz quaio (7) as a Riccai quaio for v which hrfor ma b liarisd b h subsiuio v () for som diffriabl fucio ( ; ¹. Th fac ha im ( occurs ol paramricall i quaio (7) is accommodad i our oaio for b h us of h smicolo. Equaio (7), upo h iroducio of () bcoms - u () which is almos h (im-dpd Surm-Liouvill quaio for. Th cocio is compld wh w obsrv ha h KdV quaio is Galila ivaria, ha is u(, + u( 6,, - < l <. l + l lavs quaio (6) uchagd for arbirar (ral) l. Sic h -dpdc is ualrd udr his rasformaio ( plas h rol of a paramr) w ma quall rplac u b u -l. Th quaio of ow bcoms + ( l-u) () which is h Surm-Liouvill quaio wih poial u ad igvalu l. Thus, if w abl o solv for, w ca h rcovr u from quaios () ad (7). Howvr, h procdur is far from sraighforward sic quaio () alrad ivolvs h fucio u which w wish o drmi. Th wa o avoid his dilmma is o irpr h problm i rms of scarig daa b h poial u. Ths daa ar dscribd b h bhaviors of h igfucio,, i h form -ik ïì + b( ( ;» í -ik ïî a( ik as as + - for l >, wih k l for h coiuous spcrum, ad ( ) c p( k ) as +» for l <, wih k (-l) for ach discr igvalu (,,..., N) W h show u( ) - K(, d whr (, d K, + F( + + ò K(, F( + d K is h soluio of h Marchko quaio ( (3) ad F is dfid b F( X ) åc p( -kx ) + ò N L u (, b h soluio of 6 uu + u - p - b( ik dk (4) u wih u (,) f ( ) giv: his dfis h iiial-valu problm for h KdV quaio. Furhr, l us iroduc h fucio which saisfis h quaio + ( l-u) for som l, ad b viru of h paramric dpdc o w mus allow l l(. Th soluio of h KdV quaio ca hrfor b dscribd as follows. A w ar giv u (,) f ( ) ad so (providd iss) w ma solv h scarig problm for his poial, ildig prssios for b (, k ad c (..., N). If h im voluio of hs scarig daa ca b drmid h w shall kow h scarig daa a a lar im. This iformaio hrfor allows us o solv h ivrs scarig problm ad so rcosruc u (, for >. Th procdur is rprsd schmaicall i h followig figur, whr S( dos h scarig daa, i.. b ( k,, k ( ad c ( (..., N).

4 Mahmaical Thor ad Modlig ISSN (Papr) ISSN 5-5 (Oli) Vol.3, No.3, 3 u (,) Scarig S () KdV Tim Evaluaio Rprsaio of h ivrs scarig rasform for h KdV quaio I is clar ha h succss or failur of his approach ow rss o whhr, or o, h im voluio of S ca b drmid. Furhrmor, i is o b hopd ha h voluio is fairl sraighforward so ha applicaio of his chiqu dos o prov oo difficul. W shall dmosra i h scio how S( ca b foud ad ls show ha i aks a surprisigl simpl form. Howvr, bfor w sar his, w o h paralll bw h schm rprsd i h abov figur ad h us of h Fourir rasform for h soluio of liar parial diffrial quaios. Cosidr h quaio u + u+ u which is o liarizaio of h KdV quaio. If u (,) f ( ) h w ca wri ik -ik f ( ) ò A( dk or A( f d - ò ( ) p - ad A ( is h aalogous o h scarig daa S(). Furhr, if ò - u A k i ( k w (, ) ( ) ) dk whr - 3 w w(, h w( k- k ad h rm i w prsss h im voluio of h scarig daa. 4. Rflciolss poials Th ivrs scarig rasform mhod is bs mplifid b choosig h iiial profil, u (,) o b a sch fucio ad i paricular o of hos which corrspods o a rflcio lss poial (i.. b ( for all k ).Alhough h soliar wav is alrad kow o b a ac soluio of h KdV quaio, i is possibl o obai his soluio b passig a suiabl iiial valu problm wihou akig h assumpio ha h soluio aks h form of a sad progrssig wav. This ampl h affords a simpl iroducio o h applicaio of h chiqu. 4 (a): Sigl-solio soluio of KdV quaio Th iiial profil is ak o b u(,) -sc h (5) ad so h Surm-Liouvill quaio, a bcoms + ( l+ sch ) (6) which is covil rasformd b h subsiuio T ah (so ha < < Thus Ivrs Scarig u (, S ( d d d º sc h (- T ) d dt dt d ü ( ) í ì d - T ( - T ) ý+ l + (- T ) dt î dt þ d ì dü æ l ö í( - T ) ý+ ç+ dt î dt þ è -T ø ad so { } or - T for - < < ).

5 Mahmaical Thor ad Modlig ISSN (Papr) ISSN 5-5 (Oli) Vol.3, No.3, 3 which is h associad Lgdr quaio. Th ol boudd soluio for -k ( < ) l occurs if k k ad h soluio is proporio o h associad Lgdr fucio P (ah ) i.. h corrspodig igfucio is ( ) µ P (ah ) -sch. ad sic ò sc - h d h ormalizd igfucio bcoms ( ) sc h. - (Th sig of is irrlva.) Th h asmpoic bhavior of his soluio ilds - 4 ( )» as + so ha c ( ), ad h quaio (.3) givs c (. This rasformaio is suffici o h rcosrucio of u (, sic w hav chos a iiial profil for which 8- b ( for all k. Now from quaio (.5) w obai F( X; which icorporas ol o rm form h summaio, h coribuio form h igral big zro. Th Marchko quaio is hrfor which implis 8-( + ò 8-( + K(, + + K(, ; d -z K(, L(, 8-8 form som fucio L such ha ò - L+ + L d This ca b solvd dircl o ild ad h 8 æ ç è+ - u(, L(, 8- ö ø sc h ( ( ) which is h soliar wav of ampliud - ad spd of propagaio 4. Codig ad Oupu: B usig programmig laguag MATHEMATICA u Fig Fig. Fig.3 4 (b): Two-solio soluio of KdV quaio

6 Mahmaical Thor ad Modlig ISSN (Papr) ISSN 5-5 (Oli) Vol.3, No.3, 3 W cosidr h problm for which h iiial profil is u(,) -6sc h (7) so ha h Surm-Liouvill quaio, a bcoms + ( l+ 6sc h ) (8) d ì dü æ l ö or í( - T ) ý+ ç6+ dt î dt þ è -T ø which is h associad Lgdr quaio. Whr T ah. This quaio has boudd soluios, for l -k ( < ) occurs if k or k of h form 3 3 ( ) ah sc h; ( ) sc h ; boh of which hav b mad o saisf h ormalizaio codiio. Th asmpoic bhaviors of hs soluios ar so ha () )» 6 ; ( )» 3 ; as + ( c ; c () 3, ad h quaio (3) givs 4 3 (, c ( 3. c As i h abov ampl, h choic of iiial profil surs ha b ( for all k ad so b ( k; for all. Th fucio F h bcoms F( X; (sic hr ar wo rms i h sris),ad h Marchko quaio is hrfor 8-( + 64-( + ò 8-( ( + K(, K(, ; {6 + } d I is clar ha h soluio for K mus ak h form K(, L z (, - + L (, Sic F is a sparabl fucio, collcig h cofficis of -z z - ad - z, w obai h pair of quaios L ( Lò d+ Lò d) L + + ( Lò d+ Lò d) for h fucios L ad L. Upo h valuaio of h dfii igrals hs wo quaios bcoms which ar solvd o ild whr L L + L L + + 4L + 3L L (, 6( - )/ D L (, -( + )/ D D(, Th soluio of h KdV quaio ca ow b prssd as 3

7 Mahmaical Thor ad Modlig ISSN (Papr) ISSN 5-5 (Oli) Vol.3, No.3, u(, ( L L ) {( + + )/ D} which ca b simplifid o giv + 4cosh(- 8 + cosh(4-64 u(, -. (9) {3cosh( cosh(3-36} which is h wo solio soluio. Codig ad Oupu: B usig programmig laguag MATHEMATICA 4

8 Mahmaical Thor ad Modlig ISSN (Papr) ISSN 5-5 (Oli) Vol.3, No.3, 3 Fig. Fig. 4 (c): N-solio soluio of KdV quaio W cosidr h problm for which h iiial profil is u(,) -N( N+ )sc h () h similarl h N -solio soluio is N å u(,»- sc h { ( - 4 ± } as ± Codig ad Oupu: B usig programmig laguag MATHEMATICA. (a) N u Fig.3 (b) N u Fig.4 5

9 Mahmaical Thor ad Modlig ISSN (Papr) ISSN 5-5 (Oli) Vol.3, No.3, 3 (c) N u Fig.5 (d) N Fig.6 - u Rsul ad discussio I is clar from h Fig. ha h wav movs forward as icrass wih dph ad spd 4 ad o chagig is shap. Ploig h soluio shows h caal propagaig o h righ. A coour plo ca also b usful i Fig.. To vrif ha h umrical soluio is h soluio, w plo boh for a paricular valu of (.5 hr) which illusrad i Fig.3. W s ha h wo plos agr vr wll. I fac hr is a whol famil of siglsolio soluios paramrizd b h dph of h chal. So h dpr h caal h fasr h solio movs ad h arrowr i is. W vrifid ha his dos saisf h KdV quaio. Sic h soluio is valid for posiiv ad gaiv, w ma ami h dvlopm of h profil spcifid a. Th wav profil, plod as a fucio of a si diffr ims, is show i Fig.4-Fig.9. Hr w hav chos o plo - u rahr ha u, his allows a dirc compariso o b mad wih h applicaio of h KdV quaio o war wavs. Th soluio shows wo wavs, which ar almos soliar whr h allr o cachs h shorr, mrgs o form a sigl wav a ad h rappars o h righ ad movs awa from h shorr o as icrass. Also w plod h soluio a im which showd i Fig. for a caal of dph 8 ad a caal of dph. Thus w hav crad wo solios of h p ha w discussd i h prvious scio. Howvr, hr is o liar suprposiio, so h wo-solio soluio is o h sum of h wo idividual solios i h rgio whr h ovrlap, as o ca s form h plici soluios. I is also s ha hs wo soluios irac i h ara of. I Fig. showd a gaiv ims, h dpr solio, which movs fasr, approachs h shallowr o. A h combi o giv quaio (7) (a sigl rough of dph 6) ad, afr h cour, h dpr solio has ovrak h lowr o ad boh rsum hir origial shap ad spd. Howvr, as rsul of h iracio, h lowr solio prics a dla ad h dpr solio is spdd up. This is also asil s i a Fig. (coour plo. O h ohr had, Th asmpoic soluio for N-solio of KdV qauio rprss spara solios, ordrd accordig o hir spds; as + h alls (ad hrfor fass is a h fro followd b progrssivl shorr solios bhid. All N solios irac a o form h sigl sch which was spcifid as h iiial profil a ha isa. Fiall som plos ar illusrad as D plos, 3D plos & Dsi plos i Fig.3-Fig.6 for diffr valus of N ( i.. N 3,4,5,6 ) whr iracio of N- solios is asil s. 6

10 Mahmaical Thor ad Modlig ISSN (Papr) ISSN 5-5 (Oli) Vol.3, No.3, Coclusios I his papr our aim was o udrsad h mahmaical formalism of h ivrs scarig problms hrough o-liar diffrial quaio. W hav mad all ffors o rprs h mahmaical cocp alog wih ampls as 3D -Plos, Dsi Plos, ad D-Plos for discr valus of im of ivrs scarig problms b usig Compur programmig packag MATHMATICA []. Agai w dal wih h cocio bw ivrs scarig ad h Korwg-d Vris quaio. I his scio w hav dscribd vari of ampls wih Korwg-d Vris quaio: h sigl-solio soluio, h wo-solio soluio ad fiall h N-solio soluio. Rfrcs. Marchko, V.A., O h rcosrucio of h poial rg from phass of h scard wavs, Dokl. Akad. Nauk SSSR 4 (955), C. Gardr, J. Gr, M. Kruskal, ad R. Miura, Mhod for solvig h Korwg-d vris quaio, Phs, L. Rv., 9 (967), D. J. Korwg ad G. d Vris, O h chag of form of log wavs advacig i a rcagular caal, ad o a w p of log saioar wavs, Phil. Mag. (Sr.5), 39 (895), Russll, J. S. (845). Rpor o Wavs. Rpor of h 4h mig of h Briish Associaio for h Advacm of Scic, York, Spmbr 844, pp 3-39, Plas XLVII-LVII). Lodo. 5. N. Zabusk ad M. Kruskal, Iracio of Solios i a collisio lss plasma ad h rcurrc of iiial sas, Phs, L. Rv., 5 (965), Frmi, E Pasa, J ad Ulam, S. Sudis i Noliar Problms, I Los Alamos rpor LA 955. Rproducd i Noliar Wav Moio (Ed. A. C. Nwll). Providc, RI Amr. Mah. Soc B. Forbrg ad G. Whiham, A umrical ad horical sud of crai oliar wav phoma, Phil. Tras. Ro. Soc. A, 89 (978), Lbdv, N.N. Spcial Fucios ad hir Applicaios, Pric-Hall, Eglwood CliffsN.J, Richard S. Palais, A Iroducio o Wav Equaios ad Solios, Th Morigsid Cr of Mahmaics, Chis Acadm of Scics, Bijig, Summr, Sc & 3. (-37).. Macij Duajski, INTEGRABLE SYSTEMS, Dparm of Applid Mahmaics ad Thorical Phsics, Uivrsi of Cambridg, UK, Ma,, Pags Wolfram, S. (3).Th Mahmaica Book, Fifh Ediio. Wolfram Mdia, Ic., ISBN Loard I. Schiff, Quaum Mchaics, McGraw-Hill Kogakusha, Ld, P.G. Drazi ad R. S. Johso, Soliios: a Iroducio, 99. 7

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