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 ÿâíîì âèäå ïîëó÷åíî îáùåå ðåøåíèå äëÿ îäíîé èç òàêèõ ìîäåëåé, à èìåííî, ìîäåëè èíäóöèðîâàííîéãðàâèòàöèè ñî ñòåïåííûì ïîòåíöèàëîì.12. Ðàçâèò ìåòîä ïîñòðîåíèÿ ñ ïîìîùüþ òåñòà Ïåíëåâå ðåøåíèé íåèíòåãðèðóåìûõ ñèñòåì â âèäå ðÿäîâ Ëîðàíà è ïðèìåíåíèÿ äàííûõ ðåøåíèé äëÿïîèñêà òî÷íûõ ýëëèïòè÷åñêèõ ðåøåíèé. Ñ ïîìîùüþ ïðåäëîæåííîãî ìåòîäà äîêàçàíî îòñóòñòâèå ýëëèïòè÷åñêèõ ðåøåíèé â âèäå ñòîÿ÷èõ âîëíó êóáè÷åñêîãî êîìïëåêñíîãî óðàâíåíèÿ ÃèíçáóðãàËàíäàó è ïîëó÷åíîýëëèïòè÷åñêîå ðåøåíèå óðàâíåíèÿ ÃèíçáóðãàËàíäàó ïÿòîé ñòåïåíè.
Â18äâóõ íåèíòåãðèðóåìûõ ñëó÷àÿõ îáîáù¼ííîé ñèñòåìû ÕåíîíàÕåéëåñàïîëó÷åíû òð¼õïàðàìåòðè÷åñêèå ðåøåíèÿ â âèäå ñõîäÿùèõñÿ ðÿäîâ Ëîðàíà è íîâûå òî÷íûå äâóõïàðàìåòðè÷åñêèå ðåøåíèÿ, ÿâëÿþùèåñÿ ýëëèïòè÷åñêèìè ôóíêöèÿìè ÷åòâ¼ðòîãî ïîðÿäêà. Ìåòîä ïîèñêà ýëëèïòè÷åñêèõðåøåíèé íåèíòåãðèðóåìûõ ñèñòåì ñ ïîìîùüþ ôîðìàëüíûõ ðåøåíèé ââèäå ðÿäîâ Ëîðàíà àâòîìàòèçèðîâàí è îáîáù¼í íà ìíîãîçíà÷íûå ðåøåíèÿ, ðàçëîæèìûå â ðÿä Ïþèç¼.èçëîæåííûõ â ðàáîòå ðåçóëüòàòîâ îáåñïå÷èâàåòñÿêîððåêòíîñòüþ ïîñòðîåíèÿ ìàòåìàòè÷åñêèõ ìîäåëåé, âíóòðåííåé ñîãëàñîâàííîñòüþ è ñîãëàñèåì ïîëó÷åííûõ â äèññåðòàöèè ðåçóëüòàòîâ ñ èçâåñòíûìèðåçóëüòàòàìè èññëåäîâàíèé äðóãèõ àâòîðîâ.ÄîñòîâåðíîñòüÀïðîáàöèÿ ðåçóëüòàòîâÎñíîâíûå ðåçóëüòàòû äèññåðòàöèè äîêëàäûâàëèñü íà ñëåäóþùèõ ìåæäóíàðîäíûõ êîíôåðåíöèÿõ è ñåìèíàðàõ:Mathematical physics meetings: Summer School and Conference onModern Mathematical Physics (Belgrade, Serbia, 2014; 2010; 2008); InternationalSeminars on High Energy Physics ("Quarks-2014", Ñóçäàëü, 2014; "Quarks2012", ßðîñëàâëü, 2012; "Quarks-2006" ÑàíêòÏåòåðáóðã, 2006; "Quarks2004", Ïóøêèíîãîðüå, 2004; "Quarks-2002", Âåëèêèé Íîâãîðîä, 2002);International conference "Zeldovich-100" on Cosmology and RelativisticAstrophysics (Ìîñêâà, 2014); Ìåæäóíàðîäíàÿ çèìíÿÿ øêîëà-ñåìèíàð ïîãðàâèòàöèè, àñòðîôèçèêå è êîñìîëîãèè "Ïåòðîâñêèå ÷òåíèÿ - 2014" (Êàçàíü, 2014) The 20-th International Conference on General Relativity andGravitation "GR20" and the 10-th Amaldi Conference on Gravitational Waves"Amaldi10" (Warsaw, Poland, 2013); The International School-Seminars "TheActual Problems of Microworld Physics" (Gomel, Belarus, 2013; 2011; 2005);International Workshop High Energy Physics and Quantum Field Theory("QFTHEP'2013", ÑàíêòÏåòåðáóðã, 2013; "QFTHEP'2011", Ñî÷è, 2011;"QFTHEP'2010", Ãîëèöûíî, Ìîñêîâñêàÿ îáëàñòü, 2010; "QFTHEP'2004",ÑàíêòÏåòåðáóðã, 2004; "QFTHEP'2003", ÑàìàðàÂîëãîãðàä, 2003); TheInternational Workshops "Supersymmetries and Quantum Symmetries"(SQS'2013, Äóáíà, 2013, SQS'2011, Äóáíà, 2011, SQS'2009, Äóáíà, 2009);II Russian-Spanish Congress, Particle and Nuclear Physics at all Scales andCosmology (ÑàíêòÏåòåðáóðã, Ðîññèÿ, 2013); IV Jornadas CPAN (Granada,Spain, 2012); Ginzburg Conference on Physics (Ìîñêâà, Ðîññèÿ, 2012);Dual year Russia-Spain: Particle Physics, Nuclear Physics and AstroparticlePhysics (Barcelona, Spain, 2011); Workshops on Geometric Methods in Physics19(Bialowieza, Poland, 2010; 2006); 4-th International Sakharov Conference onPhysics (Ìîñêâà, 2009); Second International Conference on String Field Theoryand Related Aspects (Ìîñêâà, 2009); The 15-th International Symposium onParticles, Strings, and Cosmology (PASCOS'2009, Hamburg, Germany, 2009);International Conferences "Symmetry in Nonlinear Mathematical Physics"(Kiev, Ukraine, 2009; 2007; 2005; 2003); Ìåæäóíàðîäíàÿ êîíôåðåíöèÿ "Ñîâðåìåííûå ïðîáëåìû ìàòåìàòèêè, ìåõàíèêè è èõ ïðèëîæåíèé" (Ìîñêâà,2009); International Workshops "Classical and Quantum Integrable Systems"(Ïðîòâèíî, 2008; 2006); The XVII-th International Colloquium on IntegrableSystems (Prague, Czech Republic, 2008); International Symposium on QuantumTheory and Symmetries ("QTS-5", Valladolid, Spain, 2007, "QTS-4", Varna,Bulgaria, 2005); The International Conferences on Geometry, Integrability andQuantization (Varna, Bulgaria, 2006, 2005); The International Conferences"Computer Algebra in Scientic Computing" (CASC'2005, Kalamata, Greece,2005; CASC'2004, ÑàíêòÏåòåðáóðã, 2004); The XXVI-th InternationalWorkshop on the Fundamental Problems of High Energy Physics and FieldTheory (Ïðîòâèíî, 2003); The XVII International Conference "Order and Chaosin Stellar and Planetary Systems" (AGAVA'03, ÑàíêòÏåòåðáóðã, 2003); TheXth International Conference "Symmetry Methods in Physics" (Åðåâàí, Àðìåíèÿ, 2003); International Workshop on Computer Algebra and its Application toPhysics CAAP-2001 (Äóáíà, 2001); à òàêæå íà ñåìèíàðàõ ïî êîìïüþòåðíîéàëãåáðå â Ìîñêâå (ÂÌÊ ÌÃÓ) è Äóáíå, íà ñåìèíàðàõ ïî ãðàâèòàöèè èêîñìîëîãèè èì.
À.Ë. Çåëüìàíîâà â ÃÀÈØ ÌÃÓ, íà ôèçè÷åñêèõ ñåìèíàðàõ âÍÈÈßÔ ÌÃÓ, Èíñòèòóòå êîñìè÷åñêèõ èññëåäîâàíèé (Áàðñåëîíà, Èñïàíèÿ),Áîëîíñêîì è Ïèçàíñêîì óíèâåðñèòåòàõ (Èòàëèÿ).ÏóáëèêàöèèÎñíîâíûå ðåçóëüòàòû äèññåðòàöèè îïóáëèêîâàíû â 35 íàó÷íûõ ðàáîòàõ,èç êîòîðûõ 23 ñòàòüè èçäàíû â ðåôåðèðóåìûõ æóðíàëàõ èç ñïèñêà Web ofScience è/èëè Scopus è ðåêîìåíäîâàííûõ ÂÀÊ. Ïîëíûé ñïèñîê ïóáëèêàöèéïðèâåä¼í â êîíöå àâòîðåôåðàòà.Îáú¼ì è ñòðóêòóðà ðàáîòûÄèññåðòàöèÿ ñîñòîèò èç ââåäåíèÿ, øåñòè ãëàâ îñíîâíîãî òåêñòà, çàêëþ÷åíèÿ, òð¼õ ïðèëîæåíèé è ñïèñêà ëèòåðàòóðû.
Îáùèé îáú¼ì äèññåðòàöèèñîñòàâëÿåò 321 ñòðàíèöó. Äèññåðòàöèÿ ñîäåðæèò 21 ðèñóíîê. Áèáëèîãðàôèÿâêëþ÷àåò 455 íàèìåíîâàíèé.20Ñîäåðæàíèå ðàáîòûÂî Ââåäåíèè îáîñíîâûâàåòñÿ àêòóàëüíîñòü ïðîâåä¼ííîãî èññëåäîâàíèÿ, ñôîðìóëèðîâàíà öåëü ðàáîòû, ïåðå÷èñëåíû èñïîëüçóåìûå ìåòîäû èññëåäîâàíèÿ, îïèñàíà ñòðóêòóðà äèññåðòàöèè è äàíî êðàòêîå ñîäåðæàíèå å¼ ãëàâ. Ãëàâå 1 èññëåäîâàíû êîñìîëîãè÷åñêèå ìîäåëè ñ íåëîêàëüíûì ñêàëÿðíûì ïîëåì, êîòîðûå îïèñûâàþòñÿ ñëåäóþùèì äåéñòâèåì( 2)∫MP1′4 √R + ϕ F (α ) ϕ − V (ϕ) − Λ,(3)Sf = d x −g22ãäå α′ êâàäðàò õàðàêòåðíîé äëèíû ñòðóíû.
Òàê êàê ïðîèñõîæäåíèå íåëîêàëüíîãî ïîëÿ ñâÿçàíî ñ òåîðèåé îòêðûòûõ ñòðóí, äåéñòâèå ñîäåðæèò è áåçðàçìåðíóþ êîíñòàíòó âçàèìîäåéñòâèÿ îòêðûòûõ ñòðóí go . Ñêàëÿðíîå ïîëå ϕ,êîòîðîå àññîöèèðóþò ñ òàõèîíîì îòêðûòîé ñòðóíû, ÿâëÿåòñÿ áåçðàçìåðíûì.Ôóíêöèÿ F ÿâëÿåòñÿ àíàëèòè÷åñêîé íà âñåé êîìïëåêñíîé ïëîñêîñòè,èíûìè ñëîâàìè, îíà ÿâëÿåòñÿ öåëîé ôóíêöèåé îïåðàòîðà Äàëàìáåðà. Ñëåäîâàòåëüíî, îíà ìîæåò áûòü ïðåäñòàâëåíà â âèäå ñõîäÿùåãîñÿ ðÿäà∞∑F(J) =fn J n ,(4)n=0à òàêæå, ïî òåîðåìå Âåéåðøòðàññà, â âèäå ñëåäóþùåãî ïðîèçâåäåíèÿ:( )pk) J J2∞ (1J∏JJ + 2 +···+ pJF(J) = J m eY (J)1−k=1Jkek2Jkkk,(5)ãäå m ïîðÿäîê êîðíÿ J = 0 (m ìîæåò ðàâíÿòüñÿ íóëþ), Jk íóëè ôóíêöèèF(J), Y (J) ÿâëÿåòñÿ öåëîé ôóíêöèåé. Íàòóðàëüíûå ÷èñëà pn âûáðàíû òàê,∞ ( )pn +1∑J÷òî ðÿäñõîäèòñÿ àáñîëþòíî è ðàâíîìåðíî.Jnn=1 ñëó÷àå êâàäðàòè÷íîãî ïîòåíöèàëà äåéñòâèå (3) ïîëó÷àåòñÿ äîáàâëåíèåì ãðàâèòàöèè â ìîäåëü, êîòîðàÿ â ïðîñòðàíñòâå Ìèíêîâñêîãî ÿâëÿåòñÿëèíåàðèçàöèåé ìîäåëè, ïîëó÷àåìîé ñ ïîìîùüþ ïîëåâîé òåîðèè ñòðóí.
Îòìåòèì, ÷òî â ïðîñòðàíñòâå ñ ïðîèçâîëüíîé ìåòðèêîé ïîëó÷àåìûå óðàâíåíèÿíå ÿâëÿþòñÿ ëèíåéíûìè, îäíàêî îáëàäàþò íåêîòîðûìè ñâîéñòâàìè ëèíåéíûõ óðàâíåíèé, ÷òî ïîçâîëÿåò ñóùåñòâåííî óïðîñòèòü èõ àíàëèç, ñâåäÿ åãî êàíàëèçó ëîêàëüíûõ óðàâíåíèé Ýéíøòåéíà. äèññåðòàöèè ïîêàçàíî, ÷òî óêàçàííàÿ ìîäåëü ñ êâàäðàòè÷íûì ïîòåíöèàëîì ïîðîæäàåò ìíîæåñòâî, êîíå÷íîå èëè áåñêîíå÷íîå, ëîêàëüíûõ ìîäåëåéñî ñêàëÿðíûìè è ôàíòîìíûìè ñêàëÿðíûìè ïîëÿìè.
Ðåøåíèÿ ïîëó÷åííûõ ëîêàëüíûõ ìîäåëåé ÿâëÿþòñÿ ÷àñòíûìè ðåøåíèÿìè èñõîäíîé íåëîêàëüíîé ìîäåëè. Òàêèì îáðàçîì, íàéäåí ìåòîä ïîèñêà òî÷íûõ è ÷èñëåííûõ ÷àñòíûõ ðåøåíèé íåëîêàëüíîé ìîäåëè ñ ïîìîùüþ àíàëèçà ëîêàëüíûõ êîñìîëîãè÷åñêèõ21ìîäåëåé. Äàííûé ðåçóëüòàò îáîáù¼í íà ìîäåëè f (R) ãðàâèòàöèè. Îñíîâíîéèäååé ïîèñêà ðåøåíèé ÿâëÿåòñÿ ïðåäïîëîæåíèå, ÷òî ôóíêöèÿ ϕ ÿâëÿåòñÿ ñîáñòâåííîé ôóíêöèåé îïåðàòîðà Äàëàìáåðà â ïðîñòðàíñòâå, ìåòðèêà êîòîðîãîgµν áóäåò íàéäåíà ïîçæå. Ïðè ïîèñêå òî÷íûõ ðåøåíèé äàííîé ìîäåëè ðàçëîæåíèå àíàëèòè÷åñêîé ôóíêöèè ïî òåîðåìå Âåéåðøòðàññà îêàçûâàåòñÿ î÷åíüïîëåçíûì.Òàêæå â ýòîé ãëàâå ðàññìîòðåíî íåëîêàëüíîå óðàâíåíèå ÊëåéíàÃîðäîíà â ñëó÷àå ïðîèçâîëüíîãî ïîòåíöèàëà:F()ϕ = V ′ (ϕ),(6)ãäå øòðèõ îáîçíà÷àåò ïðîèçâîäíóþ ïî ϕ.
×àñòíûì ðåøåíèåì (6) ÿâëÿåòñÿðåøåíèå ñëåäóþùåé ñèñòåìû ëîêàëüíûõ óðàâíåíèéN−1∑fn n ϕ = V ′ (ϕ) − C,fN N ϕ = C,(7)n=0ãäå N − 1 ÿâëÿåòñÿ íàòóðàëüíûì ÷èñëîì, à C ïðîèçâîëüíàÿ êîíñòàíòà. äèññåðòàöèè ïîêàçàíî, êàê ìîæíî ðåøèòü ñèñòåìó (7) â ñëó÷àå ìåòðèêè ÔËÐÓ. Ïðåäëîæåííûé ìåòîä ïðèìåíèì äëÿ ïðîèçâîëüíîãî ïîòåíöèàëà, çàèñêëþ÷åíèåì ñëó÷àåâ ëèíåéíîãî è êâàäðàòè÷íîãî ïîòåíöèàëîâ, è ïîçâîëÿåòïîëó÷èòü â êâàäðàòóðàõ ðåøåíèÿ, çàâèñÿùèå îò äâóõ ïðîèçâîëüíûõ ïàðàìåòðîâ. Òî÷íûå ðåøåíèÿ íàéäåíû äëÿ ïðîèçâîëüíîãî êóáè÷åñêîãî ïîòåíöèàëà,ðàññìîòðåíèå êîòîðîãî ìîòèâèðîâàíî ïîëåâîé òåîðèåé ñòðóí, à òàêæå äëÿýêñïîíåíöèàëüíîãî, ëîãàðèôìè÷åñêîãî è ñòåïåííîãî ïîòåíöèàëîâ.
Ïîêàçàíî,÷òî äîáàâëåíèåì âñïîìîãàòåëüíîãî ñêàëÿðíîãî ïîëÿ ìîæíî ïîëó÷èòü òî÷íîåðåøåíèå, óäîâëåòâîðÿþùåå âñåì óðàâíåíèÿì Ýéíøòåéíà.Ãëàâû 2 è 3 ïîñâÿùåíû ìîäåëÿì íåëîêàëüíî ìîäèôèöèðîâàííîé ãðàâèòàöèè, ìîòèâèðîâàííûì òåîðèåé ñòðóí.  íà÷àëå ýòèõ ãëàâ îïèñàíà ñâÿçüäàííûõ ìîäåëåé ñ ïîëåâîé òåîðèåé ñòðóí. Ãëàâå 2 ðàññìîòðåíà ìîäåëü íåëîêàëüíîé ãðàâèòàöèè, îïèñûâàåìàÿäåéñòâèåì[ 2()]∫√M1PS = d4 x −gR + RFR − Λ + LM ,(8)222M∗ãäå M∗ ìàññà, ïðè êîòîðîé âûñøèå ïðîèçâîäíûå â äåéñòâèè ñòàíîâÿòñÿ âàæíûìè, Λ êîñìîëîãè÷åñêàÿ êîíñòàíòà, LM ëàãðàíæèàí ìàòåðèè. Äàííàÿìîäåëü ïðåäëîæåíà â 2005 ãîäó Áèñâàñîì (Biswas), Ìàçóìäàðîì (Mazumdar)è Ñèãåëîì (Siegel) è ïðèâëåêàòåëüíà íàëè÷èåì òî÷íîãî ðåøåíèÿ òèïà îòñêîêà(bounce solution) äëÿ ìîäåëè ñ ðàäèàöèåé è êîñìîëîãè÷åñêîé êîíñòàíòîé. Ýòîðåøåíèå áûëî ïîëó÷åíî ñ ïîìîùüþ íàëîæåíèÿ äîïîëíèòåëüíîãî óñëîâèÿR = r1 R + r2 ,(9)22ãäå r1 ̸= 0 è r2 ÿâëÿþòñÿ êîíñòàíòàìè.Åñëè ñêàëÿðíàÿ êðèâèçíà R óäîâëåòâîðÿåò óñëîâèþ (9), òî óðàâíåíèÿíåëîêàëüíîé ãðàâèòàöèè ñòàíîâÿòñÿ ëîêàëüíûìè.
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