THE GENERALIZED KORTEWED DE VRIES (KdV) EQUATION

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1 THE GENERALZED KORTEWED DE VRES KdV EQATON NTRODCTON Te Korteweg-de Vries KdV equatio is a oliear artial differetial equatio of tird order as u t uu u were ad are ositive arameters ad te subscrits deote artial differetiatio Te KdV equatio first aeared i a article writte by Dutc matematicias Korteweg ad de Vries [6] Tey formulated te KdV equatio to describe log wave roagatio o sallow water Te KdV equatio is oe of te simlest oliear model equatios for solitary waves te KdV equatio u u measures te elevatio at time t ad ositio ie te eigt of te water above te equilibrium level Te secod ad te tird terms i te equatio are te oliearity ad te disersio Te term uu describes te sareig of te wave ad u te disersio ie waves wit differet wave legts roagate differet velocities Remarkably te balaces betwee tese two effects allow solutios i terms of roagatig waves wit ucaged form deed te KdV equatio is a secial case of te geeralized Korteweg-de Vries gkdv equatio of te form u t u u u were is a ositive iteger We te gkdv equatio takes te form u t u u u ad it is kow as te Modified Kortweg-de Vries equatio mkdv te study of te KdV equatio Zabusky ad Kruskal [99] idicated tat te wave solutios ersisted after iteractios ad te wave solutios were called solitos Te solito cocet is very imortat i te study of oliear wave eomea Pysically 4

2 we two solitos of differet amlitudes ad ece of differet seeds are laced far aart o te real lie te taller faster wave is o te left of te sorter slower wave te taller oe evetually catces u to te sorter oe ad te overtakes it We tis aes tey udergo a oliear iteractio accordig to te KdV equatio ad emerge from te iteractio comletely reserved i form ad seed wit oly a ase sift Tus tese two remarkable features: i steady rogressive ulse like solitos ad ii te reservatio of teir saes ad seeds cofirmed te article like roerty of te waves May eact solutios of te KdV equatio wit aroriate iitial coditio by usig scatterig teory were give by Garder et al [9] Tey discover te first itegrable oliear artial differetial equatio Garder et al [4] ad Hirota [4849] costructed aalytic solutios of te KdV equatio tat rovide te descritio of te iteractio amog N solitos for ay ositive iteger N Certai oter metods for fidig te eact solutios of te oliear PDEs iclude te Paileve aalysis [6] te Lie grou teoretic metods [7] te direct algebraic metod [5] ad taget yerbolic metod [68] Kruskal ad Zabusky ad te Miura [65] cosidered te equatio of te eistece of coservatio laws of te KdV equatio amely equatios of te form N t J 4 were N N u u u is te desity ad J J u u u is te associated flu f J as or if it is eriodic i sace te te total umber Nd is coserved because d dt N u u u d 5

3 t was coecture tat te KdV equatio ad a ifiite umber of coservatio laws ad tis was latter roved by Kruskal ad Miura ad simultaeously by Garder[4] For it was foud tat te gkdv equatio ad oly tree coservatio laws [7] ad it is also foud o-itegrability [] Miura coecture tat Eq as a ifiite umber of coservatio laws ad tis migt imly tat te KdV equatio ad te mkdv Eq are related He roceed to obtai te trasformatio betwee tem amely if v satisfies Qv v t v v v ad u v v 5 te u satisfies Pu u t uu u Hece u satisfies te KdV equatio Te trasformatio formula 5 is kow as Miura Trasformatio REVEW OF THE NMERCAL METHODS FOR SOLVNG gkdv EQATON Zabusky ad Kruskal [99] were te ioeers i studyig te KdV equatio umerically by usig te leafrog metod as a elicit fiite differece sceme Taert [9] as roosed a slit-ste Fourier metod to solve te KdV equatio umerically Oe of te most imortat umerical metods used to solve te KdV equatio was reseted ad eosed by Gereig ad Morris [45]; tis was te oscotc metod Tis oscotc is a imlicit fiite differece metod Forberg ad Witam [7] gave a etesive study for te gkdv equatio by usig a seudo-sectral metod or wat is called te fiite Fourier trasform 6

4 Saz-Sera ad Criste [84] as alied a sectral metod amely te Galerki metod Taa ad Ablowitz [9] i a series of aers ave studied ad alied te famous ad well kow fiite elemet metod to solve te KdV equatio Tey also gave a etesive comariso betwee te differet umerical metods esecially te fiite differece ad fiite elemet metods Altoug tis is a wide comariso o oe is able to say wic is te best umerical metod for te umerical solutio of te KdV equatio Ca ad Kerkove [4] ave discussed alterative time discretizatio of te KdV equatio Tey sowed tat wit te leafrog metod for te advectio term ad Crak-Nicolso metod for te liear term te stability limit is ideedet of te sace ste for ay fiite time iterval Helal [5] as alied te tau metod for obtaiig a semi-aalytical solutio for te KdV equatio He used te Cebysev ortogoal olyomials as a basis for tis metod A umerical comariso sowed tat tis metod is more efficiet ta te oscotc metod Abdur Rasid [78] as alied te Fourier Pseudosectral metod for solvig te KdV equatio accurately He costructed te discrete reresetatio of te solutio troug iterolate trigoometric olyomial of te solutio at collocatio oits Recetly Ratis Kumar ad Mai Mera [79] ave solved te KdV equatio by usig Wavelet Galerki metod Tey verified te asymtotic stability of te roosed sceme Garder et al [4] as obtaied te solutios of te Modified Korteweg-de Vries mkdv equatio by Galerki metod wit quadratic B-slie fiite elemets Te motio iteractio ad geeratio of solitary waves are studied usig te 7

5 metod T Geyikli ad D Kaya [44] ad used fiite elemet tecique for te umerical solutio of te mkdv equatio Doga Kaya ad El-Sayed [59] ad used Adomai Decomositio Metod ADM for te geeralized KdV equatio Tey also roved te covergece of ADM alied to te geeralized KdV equatio Te remarkable accuracy ad sow by comarig te umerical solutio wit te kow aalytic solutio EXACT SOLTON TO THE gkdv EQATON Te simlest matematical wave is a fuctio of te form u f c wic for eamle is a solutio to te simlest artial differetial equatio u cu were c deotes te seed of te wave For te well kow wave t equatio u c u te famous d Alembert solutio leads to two wave frots tt rereseted by terms f c ad f c Hece we start ere wit a trial solutio u f c f Substitutig te trial solutio ito te gkdv equatio we are led to te ordiary differetial equatio df c f d df d f d d wic ca writte as c df d d d d f f tegratig wit resect to it follows tat cf d f f d d A were A is te costat of itegratio order to obtai a first order equatio of f a multilicatio of df is doe ie : d 8

6 df cf d f df df d f d d d df A d d c d d f f d tegratig bot sides wit resect to leads to d df d d df A d cf f were B is aoter costat of itegratio df d Af B Now it required tat i case df d f we sould ave f d d From tese requiremets it follows tat A B Wit A B equatio ca be writte as df d f c kf By searatio of variables we may write sig te trasformatio df d f c kf were k kf csec so tat kf df c sec d we get c d d c were is aoter costat of itegratio c c f sec ct c c u f c sec ct 4 9

7 Tis is te eact solutio of te gkdv equatio Takig ad i equatio 4 it will give te eact solutios of te KdV equatio ad te mkdv equatio resectively 4 THE PETROV-GALERKN METHOD For coveiece te gkdv equatio is rewritte i te form u t u u 4 Periodic boudary coditios o te regio a b are assumed i te form u a u b Te sace iterval a b is discretized by uiform N grid oits a were N ad te grid sacig is give by b a / N Let deote te aroimatio to te eact solutio u sig Petrov-Galerki metod we assume te aroimate solutio of Eq 4 as N u 4 Te roduct aroimatio tecique [6] is used for te oliear term i te followig maer u N 4 were N are te usual iecewise liear at fuctio give by Te ukow fuctios / [ ] / [ ] oterwise N are determied from te variatioal formulatio were u u u t 44 N are test fuctios wic are take to be cubic B-slies give by

8 oterwise ad deote te usual ier roduct b a d g f g f tegratig by arts ad usig te fact tat b a b a Eq 44 leads to te formulatio t u u u 45 Performig te itegratios o 45 will give te followig system of ordiary differetial equatios ODEs 6 C B A 46 were A B ad C N Now to solve te ODEs we assume to be a fully discrete aroimatio to te eact solutio t u were t t ad t is te time ste size sig te Crak-Nicolso aroac ad te forward differece sceme for te time derivative t Eq 46 is reduced to te system of oliear equatios t t C B A C B A 47 Te oliear system 47 ca be solved by Newto s metod ad te required solutio to te gkdv equatio ca be foud

9 5 STABLTY ANALYSS To aly te Vo Neuma stability for te system 47 we must first liearize tis system Assumig u i te oliear term as locally costat u ~ te liearized sceme is a b c d e e d c u u of te gkdv equatio b a ~ tu t were a d 6 tu~ Substitutig te Fourier mode 6 tu~ t b t tu~ t e 5 66 c e i k ad i 5 were k is te mode umber ad te sace ste size ito te liearized system 5 we obtaied te amlificatio factor g as X iy g 5 X iy were X c a ecos b dcos ad Y a esi b dsi Tus for all values of t ad we ave X Y g gg X Y Hece te roosed metod is ucoditioally stable i te liear sese 6 NMERCAL TESTS t as bee sow tat te gkdv equatio as a aalytic solutio of te form c c u sec ct

10 were c ad are costats t as metio tat te gkdv equatio as tree coservatio laws Tese are give below: b b b ud u d ad u u d a a a tis sectio we reset some umerical eerimets to fid te solutio of sigle solitary wave i additio to te two solito iteractios at differet time levels Also we reset te birt of solitos from a iitial ulse 6 SNGLE SOLTARY WAVE First we take so tat te gkdv equatio is te KdV equatio ad te aalytic solutio takes te form: c c u sec ct were c ad are costat Te L ad 6 L error orms are used to comare te umerical solutios wit te eact solutio ad te quatities ad are sow to measure te coservatio for te sceme We coose c / t ad over te domai [ 8] Tus te solitary wave as amlitude uity ad te simulatios are doe u to t 8 Values of te tree ivariats as well as L ad L error orms from our metod as bee comuted ad reorted i Table Te caces of te ivariats ad from teir iitial values are less ta ad 5 resectively Error deviatios are caged i te rage error 6644 Te motio of a sigle solitary wave usig te roosed sceme corresodig to te above set of arameters as bee comuted ad lotted i Fig

11 Table : variats ad error orms for sigle solitary wave c / t 8 t L L Fig : Sigle solitary wave wit c / 8 at level timet 4 8 Fig : Sigle solitary wave wit c / 8 at level timet 4 8 4

12 Fig : Sigle solitary wave wit c / 5 8 at level timet 4 8 Table : variats ad error orms for sigle solitary wave c / t 8 t L L Secodly we take so tat te gkdv equatio becomes te modified KdV equatio ad te equatio as te aalytic solutio as 6c c u sec ct 6 tis case we cosider te followig arameters for te simulatio of te roblem: c / t ad over te domai [ 8 ] Tus te solitary wave as amlitude uity ad te simulatios are doe u to t 8 5

13 Values of te tree ivariats as well as L ad L error orms from our metod as bee comuted ad reorted i Table Te caces of te ivariats ad from teir iitial values are less ta 5 ad resectively Error deviatios are caged i te rage of 794 error 67 Te motio of a sigle solitary wave usig te roosed sceme corresodig to te above set of arameters as bee comuted ad lotted i Fig Fially we take ad cosider te followig set of arameters for te simulatio of te roblem We take c / 5 t ad for te simulatio of te roblem over te domai [ 8 ] Tus te solitary wave as amlitude uity ad te simulatios are doe u to t 8 Values of te tree ivariats as well as L ad L error orms from our metod as bee comuted ad reorted i Table Te caces of te ivariats ad from teir iitial values are less ta 4 ad 9 resectively Error deviatios are caged i te rage of 6448 error 696 Te motio of a sigle solitary wave usig te roosed sceme corresodig to te above set of arameters as bee comuted ad lotted i Fig Table : variats ad error orms for sigle solitary wave c / t 8 t L L

14 Table : variats ad error orms for sigle solitary wave c / 5 t 8 t L L NTERACTON OF TWO gkdv SOLTARY WAVES Here we study te iteractio of two well searated solitary waves avig differet amlitudes ad travellig i te same directio Te iitial coditio is give by u i ci sec ci i 6 were c ad i i are arbitrary costats tis case also we take i ad For all values of te followig arameters: t ave bee cose wit rage [ 6] 7

15 Fig 4 a Fig 4 b Fig 4c Fig 4d Fig 4e Fig 4 a-d: teractio of two gkdv solitary waves at differet time levels ad Fig 4 e tree dimesioal lot for 8

16 First we take so tat te gkdv equatio becomes te KdV equatio ad te iitial coditio 6 takes te form c i u sec i c i i 64 Here we cose te followig arameters for our simulatio: c / c / 6 ad 6 Te te amlitudes of te two solitary waves are i te ratio : Te simulatio is doe u to te time t Te tree ivariat of motio are tabulated i Table 4 ad Fig 4a-4d sow te iteractio of te two solitos at differet time ad Fig 4e sows its tree dimesioal lot Table 4: variats for two solitary wave iteractio for t Secodly we cosider te case for tis case te iitial coditio takes te form: u i 6c i sec c i i 65 Here we cose te followig arameters for our simulatio: c / c /8 ad 6 Te amlitudes of te two solitary waves are i te ratio : Te simulatio is doe u to te time t Te tree ivariat of motio 9

17 are tabulated i Table 5 ad Fig 5a-5d sow te iteractio of te two solitos at differet time ad Fig 5e sows its tree dimesioal lot Fig 5a Fig 5b Fig 5c Fig 5d Fig 5e Fig 5 a-d: teractio of two gkdv solitary waves at differet time levels ad Fig 5 e: tree dimesioal lots for

18 Fig 6a Fig 6b Fig 6c Fig 6 d Fig 6 e Fig 6 a-d: teractio of two gkdv solitary waves at differet time levels ad Fig 6 e tree dimesioal lots for

19 Fially we take tis case te iitial coditio 6 takes te form: u i c i sec ci i 66 Here we cose te followig arameters for our simulatio: c /5 c / ad 6 Te amlitudes of te two solitary waves are i te ratio : Te simulatio is doe u to time t Te tree ivariat of motio are tabulated i Table 6 ad Fig 6a-6d sow te iteractio of te two solitos at differet time ad Fig 6e sows its tree dimesioal lot Table 5: variats for two solitary wave iteractio for t Table 6: variats for two solitary wave iteractio for t

6 SPLTTNG OF SOLTONS FROM A SNGLE NTAL PLSE Here we study te slittig of solitos from a sigle iitial ulse tis case also we cosider tree cases viz ad First we take Te iitial coditio i tis case takes te

20 6 SPLTTNG OF SOLTONS FROM A SNGLE NTAL PLSE Here we study te slittig of solitos from a sigle iitial ulse tis case also we cosider tree cases viz ad First we take Te iitial coditio i tis case takes te form: f sec 67 8 Te arameter cose are: 4 t ad te movemet of te iitial ulse is cosider over te iterval [ ] Tis simulatio is erformed u to te time stet Te tree ivariats of motio ad are tabulated i Table 7 Fig 7a ad 7b sow te slittig of te solito at time t ad at time t resectively Fig 7c sows te tree dimesioal lot of tis case Fig 7a Fig 7b Fig 7c Fig 7 a ad b: Slittig of gkdv solitary waves from a sigle iitial ulse ad Fig 7 c its tree dimesioal lot for

Secodly we take ad cosider te same iitial coditio as give i equatio 67 for te case Te arameters cose are: 4 t ad te movemet of te iitial ulse is cosider over te iterval [ ] Tis simulatio is erformed

21 Secodly we take ad cosider te same iitial coditio as give i equatio 67 for te case Te arameters cose are: 4 t ad te movemet of te iitial ulse is cosider over te iterval [ ] Tis simulatio is erformed u to te time ste t 8 Te tree ivariats of motio ad are tabulated i Table 8 Fig 8a-c sowed te slittig of te solito from te iitial ulse at differet time t 4 ad 8 resectively Also Fig 8d sows te tree dimesioal lot of tis case Fig 8a Fig 8b Fig 8c Fig 8d Fig 8 a-c: Slittig of gkdv solitary waves from a sigle iitial ulse ad Fig 8 d its tree dimesioal lot for 4

22 Table 7: variace for te slittig of gkdv solitary waves for t Table 8: variace for te slittig of gkdv solitary waves for t Table 9: variace for te slittig of gkdv solitary waves for t Lastly we cosider te case tis case also we cosider te same iitial coditio as above Te arameters cose are: 4 t ad te movemet of te iitial ulse is cosider over te iterval [ ] Tis simulatio is erformed u to te time ste t 8 Te tree ivariats of motio ad are tabulated i Table 9 Fig 9a-c sowed te slittig of te solito from te iitial ulse at differet time t 4 ad 8 resectively Also Fig 9d 5

sows te tree dimesioal lot of tis case t is observed from te Tables 7-9 tat te tird ivariace u u d b a is little fluctuated i all te cases Fig 9a Fig 9b Fig 9c Fig 9d Fig 9 a-c: Slittig of gkdv

23 sows te tree dimesioal lot of tis case t is observed from te Tables 7-9 tat te tird ivariace u u d b a is little fluctuated i all te cases Fig 9a Fig 9b Fig 9c Fig 9d Fig 9 a-c: Slittig of gkdv solitary waves from a sigle iitial ulse ad Fig 9 d: its tree dimesioal lot for 7 CONCLSON Te Petrov-Galerki metod usig te liear at fuctio ad cubic B-sliefuctio as trial ad test fuctios as bee successfully imlemeted to study te solitary waves of te gkdv equatio t as bee sow tat te sceme is accurate 6

24 ad efficiet t as also bee sow tat te sceme so develoed is ucoditioally stable We ave tested our sceme usig sigle solitary wave for wic te eact solutio is kow ad te eteded tis sceme to te study of te iteractio of two solitary waves ad also breakig of te solitary waves from a iitial ulse t is also observed tat te solutio of te gkdv equatio does ot give true solito solutio we We ave also sow te aearace of small tail after te collisio of two solitos i te case of Tus we require furter researc work i tis area troug bot aalytical ad umerical solutio of tis equatio tye 7

x x x 2x x N ( ) p NUMERICAL METHODS UNIT-I-SOLUTION OF EQUATIONS AND EIGENVALUE PROBLEMS By Newton-Raphson formula

x x x 2x x N ( ) p NUMERICAL METHODS UNIT-I-SOLUTION OF EQUATIONS AND EIGENVALUE PROBLEMS By Newton-Raphson formula NUMERICAL METHODS UNIT-I-SOLUTION OF EQUATIONS AND EIGENVALUE PROBLEMS. If g( is cotiuous i [a,b], te uder wat coditio te iterative (or iteratio metod = g( as a uique solutio i [a,b]? '( i [a,b].. Wat