Journal of Applied Mathematics and Computational Mechanics 2014, 13(3), 85-99

Transcription

1 Jorna of Aied ahemais and Comaiona ehanis 5-99 COBINED ALGORIT FOR FINDING CONSERVATION LAWS AND IPLECTIC OPERATORS FOR TE BOUSSINESQ-BURGERS NONLINEAR DYNAICAL SYSTE AND ITS FINITE DIENSIONAL REDUCTIONS Arkadii indybaik ykoa Prya Ian Franko Naiona Uniersiy of Li Li Ukraine Absra In he arie he ombined agorihm for finding onseraion aws and imei oeraors has been roosed Using he Noiko-Bogoyaensky mehod he finie dimensiona reions hae been fond The srre of inarian sbmanifods has been eamined aing anayzed hase orrais of amionian sysems aria eriodia soions hae been fond eywords: noninear dynamia sysem onseraion aws imei oeraors mehod of he ndeermined oeffiiens differenia agebrai agorihm ombined agorihm finie dimensiona reion amionian sysem ea soions Bossines-Brgers eaion Genera sheme of he Combined Agorihm Conseraion aws as a oneion ay a signifian roe in he nmeria anaysis of dynamia sysems Sine onseraion aws are fnionas ha remain onsan wih rese o eoion of a dynamia sysem hey are sef for erifiaion of nmeria shemes onsred for a sh sysem een if he ea soion is nknown oreoer aording o he soiary heory he eisene of infinie hierarhy of onseraion aws is onneed wih omee inegrabiiy of a noninear dynamia sysem There are seera ways for finding onseraion aws for a noninear dynamia sysem where : T denoes he Frehé smooh angen eor fied on he reresening a noninear dynamia sysem There are wo main mehods: he asymoia mehod sing La eaion and asymoia eansion and he dire mehod he main idea is soing robem for ndeermined oeffiiens The firs mehod in some ases an no be n smooh -eriodia manifod C R R

2 6 A indybaik Prya aied o a genera dynamia sysem he seond mehod works iky if here are ery few ndeermined oeffiiens When here are many nknowns he mehod roessing beomes sower and sower If fnionas d i Z i i remain onsan aong eor fied : d hey are onseraion aws for sysem Le D be he sae of Frehé smooh fnionas on manifod We define he oeraor grad : D T by F gradf for F D where / is Eer ariaiona deriaie k k d d k k In his aer we roose a ombined agorihm for finding onseraion aws in a few ses: Se Find few onseraion aws by means of ndeermined oeffiiens dire mehod sffiien for finding an imei oeraor Se Consr firs oeraor by means of a differenia-agebrai mehod Se If oeraor saisfies Nöher's eaion find orresonding amionian sing amioniy roery of he sysem grad we obain he firs imei oeraor Se By means of he differenia-agebrai mehod onsr seond oeraor Se 5 Tes oeraor : if oeraor saisfies Nöher s eaion hen sysem is bi-amionian wih he seond imei oeraor and he rersion oeraor an be fond as Λ This rersion oeraor generaes an infinie hierarhy of onseraion aws and heir gradiens are eressed as foows: grad Λgrad i Z i i

3 Combined agorihm for finding onseraion aws and imei oeraors for he Bossines-Brgers 7 Ne onseraion aws are obained eiiy wiho asimoi eansions or soing sysems of inear agebrai eaions This imies fas roessing for finding onseraion aws In he ne seions we wi eamine he aiaion of his agorihm for he Bossines-Brgers noninear dynamia sysem and we wi se obained ress for finding finie dimensiona reions Probem formaion Le C R R be -eriodia smooh manifod We onsider he Bossines-Brgers noninear dynamia sysem gien on manifod : 5 where are fnions : T denoes he Frehé smooh angen eor fied on he manifod reresening he eoion of a noninear dynamia sysem 5 Seing arameer α in Broer-a-ershmi B sysem α яя α α α we obain sysem 5 Imei oeraors and rersion oeraor for Broer-a- ershmi hae been fond in In his arie we sha find onseraion aws for 5 by means of a ombined agorihm oreoer we onsr a orresonding finie dimensiona inarian sbmanifod for a Bossines-Brgers noninear dynamia sysem 5 Differenia- -geomeri roeries of his sbmanifod shod be eamined Aso we sha onsr amionian sysem and on is hase orrai we sha find he se of aroriae iniia daa o obain eriodia soions Frhermore we wi resen a hase orrai for he amionian sysem and show a aria ea soion Aiaion of he ombined agorihm: onseraion aws imei oeraors rersion oeraor In his seion we wi se he ombined agorihm roosed in he firs seion Using he mehod of ndeermined oeffiiens we hae fond he hree firs onseraion aws

4 A indybaik Prya Theorem The foowing fnionas are onseraion aws for sysem 5: d d d 6 Proof Aording o inariane eaiy for fnionas we hae o erify ha d d d Sine manifod is -eriodia we obain / / d d d / d d d / d d d aing fond he hree firs onseraion aws we an onsr imei oeraor by means of a differenia-agebrai agorihm Le denoe a aria deriaie wih rese o ariabe : Proosiion Sysem 5 ossesses imei oeraor : 7 and he orresonding amionian fnion is foowing d Proof This imei oeraor is gien in We show ha we obain his oeraor by means of a differenia agebrai agorihm Le s denoe d and biinear form b a abd b a Using a differenia agebrai agorihm we obain he foowing eressions:

5 Combined agorihm for finding onseraion aws and imei oeraors for he Bossines-Brgers 9 where Le s onsr oeraors * θ aording o : / / * θ Inerse oeraor θ eiss and i is θ Sine oeraor saisfies Nöher's eaion * oeraor is imei where oeraors * are foowing / / / / * Wih rese o amioniy roery of he sysem 5 we obain robem grad ha sh fniona find Ths we hae he foowing eressions: / / / / h Eressions / / / / yied amionian fnion moreoer is he onseraion aw for sysem 5 Ths we hae erformed ye he firs hree ses of he ombined agorihm We erform he ne ses beow Proosiion Sysem 5 is bi-amionian and seond imei oeraor is / / 9 and orresonding amionian is d

6 A indybaik Prya 9 Proof Le s onsider amionian : Aording o he roof of Theorem and we obain oeraors * θ : * * θ Oeraor has been obained from he reaionshi θ : θ Sine oeraor saisfies Nöher s eaion * oeraor is imei and sysem 5 is bi-amionian Theorem Sysem 5 ossesses an infinie hierarhy of onseraion aws { } Z i i whih are generaed by a rersion oeraor Λ wih a gradien reaionshi Λ Z i grad grad i i

7 Combined agorihm for finding onseraion aws and imei oeraors for he Bossines-Brgers 9 Proof Sine oeraors and are imei and eression i i grad grad hods for a Z i aing miied by we obain Z i grad grad i i hene Λ is a rersion oeraor oweer we obain rersion oeraor: Λ Using hese obained ress we hae fond he ne onseraion aws in he hierarhy d d Fniona we wi se for for-dimensiona reion of he sysem 5 Finie dimensiona reion on wo-dimensiona inarian sbmanifod Finding he aroriae se of iniia ondiions for an infinie dimensiona onseraie noninear dynamia sysem for seia soions ike a soiary wae on eriodia fniona manifods is a brning isse for nmeria inesigaion of sh sysems 5 and onneed wih some diffiies We wi se he finie dimensiona reion mehod by Noiko-Bogoyaen- sky 6 for onsring ea soions for a noninear dynamia sysem by means of generaizaions resened in 7-

8 9 A indybaik Prya Le s onsider he inarian sbmanifod as he se of riia oins of he foowing Lagrangian fniona L D : L : d where are arbirary onsans Aording o La roosiion a finie dimensiona fniona sbmanifod for dynamia sysem 5 is deermined by : { : gradl } aing eaaed grad L sing we obain L : : L Consrains yied Using he Gefand-Dikey reaionshi he differenia of is: dl d d d d d whih yieds -form 7 α : d We deermine symei srre ω as an oer differeniaion of -form α : ω : dα d on he sbmanifod oweer sbmanifod is simei wih he srre 5 we inrode anonia ariabes on : Eemens of he hase sae are d 5 : : 6 in anonia ariabes 5 of sbmanifod 7

9 Combined agorihm for finding onseraion aws and imei oeraors for he Bossines-Brgers 9 amionian h whih orresonds o eor fied d/ d has been fond from dh L L he deermining reaionshi d ene he fonded amionian h in oa oordinaes is: h and in anonia oordinaes 6 of sbmanifod has foowing form h Aording o 7 he amionian sysem has been obained d d d 6 d 9 whih has hree fied oins in a hase sae: wo hyerboi oins wih oordinaes and eii oin wih oordinaes Fig Phase orrai of he sysem 9 wih

10 9 A indybaik Prya The hase orrai of he sysem 9 roides imoran informaion for idenifying he se of iniia ondiions for eriodia soions and is shown in Figre If iniia daa for he sysem 9 are aken from he inerna region of imi ye of he sysem 9 we wi obain a eriodia soion for robem 5 Veor fied d/ is amionian oo The amionian h whih orresonds o eor fied d/ has been fond from he deermining reaionshi dh d L L and h has he foowing form in oa oordinaes: h whih in anonia ariabes 6 of inarian sbmanifod is h h Finay we obain amionian sysem: d d 6 whih ossesses hree fied oins in he hase sae: he same as for sysem 9 The hase orrai of sysem is he same as he orrai of sysem 9 If we hoose iniia daa for sysem from he region whih roides eriodia soion we wi obain eriodia soion Theorem Bossines-Brgers noninear dynamia sysem 5 reded on he inarian wo dimensiona sbmanifod is eay eiaen o he se of wo omming anonia amionian fows 9 and ha are omeey inegrabe by adrares sysems The orresonding amionian fnions are gien by eressions аnd 5 Paria soions on he wo-dimensiona inarian sbmanifod We obain a aria ea soion of he dynamia sysem 5 ia ayering eor fieds d/ d 9 and d/

11 Combined agorihm for finding onseraion aws and imei oeraors for he Bossines-Brgers 95 Ea soions for sysems 9 and an be wrien in he foowing form: d d β β where denoe arbirary rea aes and β is arbirary rea arameer Le oeffiiens of he sysem 5 and oeffiiens in he Lagrange fniona are foowing We inegrae sysem 9 whih orresonds eor fied d/ d on he domain wih he se by ariabe wih iniia daa 5 whih are in he imi ye of sysem 9 oweer he foowing reaionshis hod on inarian sbmanifod hen in he iniia momen of ime soions of noninear dynamia sysem 5 are shown in Figre Fig Soion and of he sysem 5 wih in he iniia momen of ime We inegrae he sysem assoiaed wih eor fied d/ on he domain wih se sing he foowing iniia daa 5 whih are in he imi ye of sysem ene we obain ime eoion of he soion in he oin : A eery momen of ime we inegrae sysem 9 wih iniia daa k k k o obain soions Using reaionshis we obain ea soions of noninear dynamia sysem 5 for and whih are gien in Figre

12 A indybaik Prya 96 Fig Soions and of he sysem 5 wih 6 Finie dimensiona reion on he for-dimensiona inarian sbmanifod Le s onsider he foowing Lagrange fniona : d L Simiar o he firs ase we find inarian sbmanifod : { } : grad L as fied oins of Lagrange fniona : : 5 Ths on manifod 5 eiss symei srre d d d d ω in he foowing form d d d d ω 6 Canoni amionian ariabes } { are on sbmanifod

13 Combined agorihm for finding onseraion aws and imei oeraors for he Bossines-Brgers 97 Eemens } { an be rewrien on sbmanifod as foows: 6 6 amionian fnions h and h an be obained by deermining reaionshis: d dh L L d dh L L whih hae he foowing form in he anonia oordinaes of sbmanifod : h 7 / 6 5 h As a res we hae reded or Bossines-Brgers noninear dynamia sysem 5 on he onsred for dimensiona inarian sbmanifod The sysem for he d d/ is obained from 7 and roides informaion for iniia daa for he Cahy robem d d d d d d d d 9

14 A indybaik Prya 9 The sysem for he d/ is foowing: d d d d Theorem Bossines-Brgers noninear dynamia sysem 5 reded on he inarian for-dimensiona sbmanifod is eay eiaen o he se of wo omming anonia amionian fows 9 and ha are omeey inegrabe by adrares sysems The orresonding amionian fnions are gien by eressions 7 аnd Consions Ths in his aer we resen a ombined agorihm for finding onseraion aws and an imei oeraor This mehod has seera adanages: haing fond few onseraion aws we an obain imei oeraors and frhermore onsr a rersion oeraor for eii eaaion of he onseraion aws oreoer we hae shown eisene of he infinie hierarhy of onseraions aws for Bossines- -Brgers noninear dynamia sysem Using onseraion aws he finie dimensiona reion on inarian wo-dimensiona and for-dimensiona sbmanifods has been erformed Iniia ondiions for eriodia soions for dynamia sysem hae been fond

15 Combined agorihm for finding onseraion aws and imei oeraors for he Bossines-Brgers 99 Referenes Prykarasky A Finie-dimensiona reions of onseraie dynamia sysems and nmeria anaysis A Prykarasky S Brzyhzy Vr Samoyenko Ukr ah Jorn 5 - Tao Chen The generaized Broer-a-ershmi sysem and is amionian eension Tao Chen Li-Li Zh Lei Zhang Aied ahemaia Sienes Prykarasky A Agebrai Inegrabiiy of Noninear Dynamia Sysems on anifods: Cassia and Qanm Ases A Prykarasky IV ykyik wer Neherands enosh ОE Differenia-geomeri and Lie-agebrai Fondaions of Inegrabe Noninear Dynamia Sysems on Fniona anifods ОE enosh Pryka А Prykarasky Pbish Cener LNU Franko Li 6 in Ukrainian 5 Arnod VI ahemaia ehods of Cassi ehanis Naka : 979 in Rssian 6 Bogoyaensky OI On he onneion of hamionian formaisms for saionary and nonsaionary robems OI Bogoyaensky SP Noiko Fniona Anaysis and Is Aiaions in Rssian 7 Bakmore D Noninear Dynamia Sysems of he ahemaia Physis: Sera and Differenia-geomeria Inegrabiiy Anaysis D Bakmore A Prykarasky V r Samoyenko Word Sienifi Pb NJ USA 56 yroosky JА Inegrabe dynamia sysems: sera and differenia-agebrai ases JА yroosky NN Bogoibo А Prykarasky VGr Samoyenko Nak Dmka : in Rssian 9 Prykarasky А On he one onsrion of finie dimensiona reions on fniona manifods А Prykarasky OG Bihn ah ehods and Phis-meh Fieds 7- in Ukrainian Prykarasky А Agebrai Ases of Inegrabiiy of Dynamia Sysems on anifods А Prykarasky IV ykyik АМ Samoyenko Nak Dmka 99 in Rssian Samoyenko АМ Agebrai-anayia Ases of Comee Inegrabe Sysems and Their Perrbaions АМ Samoyenko YaА Prykarasky Insie of ahemais of NAN Ukraine К: in Ukrainian

16 A indybaik Prya