Диссертация (1136178), страница 34
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Îêàçûâàåòñÿ, ÷òî òåõíè÷åñêè áîëåå óäîáíî ñòðîèòü àñèìïòîòè÷åñêèå ðàçëî-T (ξ, η), ϕ(ξ, η), à äëÿ ôóíêöèé B(ξ, η), I(ξ, η),÷åðåç êîòîðûå âûðàæàþòñÿ T è ϕ. Íèæå áóäåò ïîëó÷åíà çàäà÷à äëÿíàõîæäåíèÿ cos- è sin-àìïëèòóä.Ïóñòü ôóíêöèÿ S óäîâëåòâîðÿåò óðàâíåíèþ (4.17). Ïðåîáðàçóåì â ðàâåíñòâå (4.12) ôóíêöèè cos Φ è sin Φ ñîãëàñíî ôîðìóëàìæåíèÿ íå äëÿ ïàðûcos Φ = cos ϕ cos S − sin ϕ sin S,sin Φ = sin ϕ cos S + cos ϕ sin S,292à çàòåì ïðèðàâíÿåì ê íóëþ ñëàãàåìûå ïåðåäcos Sèsin S .Òàê êàêh ∂ 2T ∂ϕ 2 ih ∂T ∂ϕ∂ 2I∂ 2Bcos S + 2 sin S =−Tcos Φ − 2+∂η 2∂η∂η 2∂η∂η ∂η∂ 2ϕ i+T 2 sin Φ,∂η∂B∂I∂T∂ϕsin S −cos S =sin Φ + Tcos Φ,∂ξ∂ξ∂ξ∂ξòî ïðèõîäèì ê óðàâíåíèÿì∂ 2B − π∂η 2Z∞∂I20200|η − η | B (ξ, η ) + I (ξ, η ) dη + L(ξ) B + 2S 0 +∂ξ−∞0+S 00 I = 0,∂ 2I − π∂η 2Z(4.21)∞∂B20200|η − η | B (ξ, η ) + I (ξ, η ) dη + L(ξ) I − 2S 0−∂ξ−∞0−S 00 B = 0.Êðîìå òîãî, â ñèëó (4.15), (4.18)(4.22)B(ξ, η) è I(ξ, η) äîëæíû óäîâëåòâî-ðÿòü óñëîâèÿìZ∞B (ξ, η) + I (ξ, η) dη = k/S 0 (ξ),22(4.23)−∞Z∞η B (ξ, η) + I (ξ, η) dη = 0.22(4.24)−∞Èòàê, äîêàçàíàÒåîðåìà 4.2.(4.17),Ïóñòü ôóíêöèÿ S ÿâëÿåòñÿ ðåøåíèåì óðàâíåíèÿà B , I ( à òàêæå L ) ðåøåíèå çàäà÷è(4.21) (4.24)äëÿcos- è sin-àìïëèòóä.
Òîãäà ïðè ξ → +∞ ýéðè-ïîëÿðîí (0.49) óäîâëåòâîðÿåò óðàâíåíèþ (0.45) ñ òî÷íîñòüþ r1 , ãäå r1 èìååò âèä(4.13).2931.3.Àñèìïòîòè÷åñêîå ðåøåíèå çàäà÷è äëÿsin-àìïëèòóäcos-èýéðè-ïîëÿðîíàÏîñòðîèì àñèìïòîòè÷åñêîå ðåøåíèå çàäà÷è (4.21) (4.24) ïðè|η| ξ 1/4â âèäåB(ξ, η) = B0 (ξ, η) + B1 (ξ, η) + B2 (ξ, η) + O B3 (ξ, η) ,(4.25)I(ξ, η) = I0 (ξ, η) + I1 (ξ, η) + O I2 (ξ, η) ,L(ξ) = `0 (ξ) + `1 (ξ) + `2 (ξ) + O `3 (ξ) ,(4.26)(4.27)ãäåBj (ξ, η) = O ξ −(1+j)/3 , j = 0, 1, 2, B3 (ξ, η) = O(ξ −7/6 ), ξ → +∞,Ij (ξ, η) = O ξ −(3+2j)/6 ,`j (ξ) = O(ξ −(1+j)/3 ),j = 0, 1, 2,j = 0, 1, 2,ξ → +∞,`3 (ξ) = O(ξ −7/6 ),ξ → +∞.Bj (j = 0, 1, 2, 3), Ij (j = 0, 1, 2), `j (j = 0, 1, 2, 3) â (4.25)âåùåñòâåííûå, ãëàäêèå; ïðè÷åì, â îáëàñòè (4.3) Bj (ξ, η) èÔóíêöèè (4.27)Ij (ξ, η)ýêñïîíåíöèàëüíî óáûâàþò.Îòìåòèì, ÷òî èñïîëüçóåìàÿ ïðè ïîñòðîåíèè ðàçëîæåíèé òåõíèêà õàðàêòåðíà äëÿ ìåòîäà Óèçåìà[24; 86].Ïîëó÷èì ôîðìóëû äëÿ ãëàâíûõ ÷ëåíîâ àñèìïòîòèêèB, I , LÏîäñòàâëÿÿ ðÿäû äëÿ÷ëåíà ðàçëîæåíèÿ∂ 2 B0 (ξ, η) − π∂η 2Zâ (4.21), (4.23), (4.24), äëÿ ãëàâíîãîcos-àìïëèòóäûïîëó÷àåì çàäà÷ó∞|η − η0−∞ZB0 , I0 , `0 .|B02 (ξ, η 0 ) dη 0+ `0 (ξ) B0 (ξ, η) = 0,(4.28)∞−∞ZB02 (ξ, η) dη = k/S 0 (ξ),(4.29)∞−∞ηB02 (ξ, η) dη = 0.(4.30)Ðåøåíèÿ (4.28) (4.30) âûðàæàþòñÿ ÷åðåç ðåøåíèÿ ñëåäóþùåé, íåçàâèñÿùåé îò äîïîëíèòåëüíûõ ïàðàìåòðîâ, çàäà÷è íà ñîáñòâåííûå294çíà÷åíèÿ äëÿ îäíîìåðíîãî ïîëÿðîíà (0.46), (0.47).
Òàêàÿ çàäà÷à âîçíèêàåò, íàïðèìåð, ïðè ïîñòðîåíèè ëîêàëèçîâàííûõ âáëèçè òî÷êèàñèìïòîòè÷åñêèõ ðåøåíèé îäíîìåðíîãî óðàâíåíèÿ Õàðòðè ñ íåãëàäêèì ïîòåíöèàëîì ñàìîäåéñòâèÿ [46].Îáîçíà÷èì ÷åðåçti (τ ), ρi , (i = 1, 2, . . . )ñîáñòâåííûå ôóíêöèèè ñîáñòâåííûå çíà÷åíèÿ (0.46), (0.47). Äëÿ óïðîùåíèÿ îáîçíà÷åíèéi â ôîðìóëàõ íèæå.B0 áóäåì èñêàòü â âèäåîïóñòèì èíäåêñÔóíêöèþB0 (ξ, η) = y(ξ)t(τ ),ãäå(4.31)τ = β(ξ)η + κ(ξ).Ëåììà 4.3.Åñëèy= k 2/3S0, k 1/3β=S0òî çàäàâàåìàÿ ôîðìóëîé(4.31)åì çàäà÷èÏðè ýòîì(4.28) (4.30).,κ = 0,(4.32)ôóíêöèÿ B0 (ξ, η) ÿâëÿåòñÿ ðåøåíè-`0 = ρ k 2/3S0.(4.33)Äîêàçàòåëüñòâî.
Ïîäñòàâëÿÿ (4.31) â (4.28) (4.30), èìååìβ2dy2βZ ∞t(τ ) y 20 2 00−π|τ − τ |t (τ ) dτ + `0 t(τ ) = 0,dτ 2β 2 −∞2Z∞kt2 (τ ) dτ = 0 ,S−∞Z∞(τ − κ(ξ))t2 (τ ) dτ = 0,−∞à, çíà÷èò,y4= 1,β4Ëåììà äîêàçàíà.`0= ρ,β2y2k=,β2S0κ = 0.295Çàìå÷àíèå4.2.  ôîðìóëå äëÿy = ±(k/S 0 )2/3ìû äëÿ îïðåäåëåííî-ñòè âûáðàëè çíàê ïëþñ. Êðîìå òîãî, íèæå áóäåì ñ÷èòàòü, ÷òîS0 > 0(ñëó÷àék < 0, S 0 < 0k > 0,ðàññìàòðèâàåòñÿ àíàëîãè÷íî).Íåïîñðåäñòâåííûì äèôôåðåíöèðîâàíèåì ïðîâåðÿåòñÿ, ÷òî äëÿðåøåíèÿ çàäà÷è (0.46), (0.47) ñïðàâåäëèâû ðàâåíñòâà(t00 /t)00 = 2πt2 ,((t00 /t)0 )2 = π 2 + 4π t00 t − (t0 )2 .Óðàâíåíèå (4.34) äîïóñêàåò ïîíèæåíèå ïîðÿäêà íà 1.
Åñëèòî ïîñëå çàìåíûY22d Ydx2x0 = Y (x)= x2 − π 2 ,(4.34)defx =(t00 /t)0 ,èìååìx ∈ (−π, π),Y (−π) = Y (π) = 0.(4.35)Èíòåãðèðîâàíèå ýòîé íåëèíåéíîé çàäà÷è ýêâèâàëåíòíî ðåøåíèþ çàäà÷è (0.46), (0.47) äëÿ îäíîìåðíîãî ïîëÿðîíà. ðàáîòå [45] áûëè ÷èñëåííî íàéäåíû ïåðâûå òðè ñîáñòâåííûåçíà÷åíèÿρ1 ≈ −2.45, ρ2 ≈ −5.14, ρ3 ≈ −7.76è òðè ñîáñòâåííûåôóíêöèè çàäà÷è (0.46), (0.47). Ãðàôèêè ñîáñòâåííûõ ôóíêöèé t1 , t2 ,t3 ,à òàêæå ãðàôèêè ñîîòâåòñòâóþùèõ ðåøåíèéY1 , Y2 , Y3âñïîìîãà-òåëüíîé êðàåâîé çàäà÷è (4.35) èçîáðàæåíû íà ðèñ.
4.1 4.3.Ðèñóíîê 4.1Êðîìå òîãî, â ðàáîòå[44] äëÿ çàäà÷è (0.46), (0.47) áûëà äîêà-çàíà ôîðìóëàsρi ≈ −3225 43,π i−242i → ∞,(4.36)296Ðèñóíîê 4.2Ðèñóíîê 4.3äëÿ àñèìïòîòè÷åñêèõ ñîáñòâåííûõ çíà÷åíèé, à òàêæå áûëè ÷èñëåííî íàéäåíû ïåðâûå ñåìü ñîáñòâåííûõ çíà÷åíèé è ñîáñòâåííûõ ôóíêöèé. Ýòî ïîçâîëÿåò ñðàâíèòü àñèìïòîòè÷åñêèå è òî÷íûå ñîáñòâåííûåçíà÷åíèÿ. Îêàçûâàåòñÿ, ÷òî àñèìïòîòè÷åñêàÿ ôîðìóëà (4.36) äàåòõîðîøóþ àïïðîêñèìàöèþ ñïåêòðà óæå ïðèi ≥ 6. Îòìåòèì, ÷òî àíà-ëîãè÷íîå ñðàâíåíèå äëÿ ðàäèàëüíî-ñèììåòðè÷íûõ ðåøåíèé çàäà÷è,îïèñûâàþùåé ïîëÿðîí âR3 ,áûëî ïðîèçâåäåíî â [43].Ïîêàæåì òåïåðü, ÷òî ôóíêöèÿB0 (ξ, η)(4.31) ýêñïîíåíöèàëüíîτ → ±∞. Äëÿ ýòîãî íàéäåì àñèìïòîòèêóôóíêöèé t(τ ) çàäà÷è (0.46), (0.47) ïðè τ → ±∞.Ïóñòü τ → +∞.
Òîãäàóáûâàåò ïðèZ∞−∞Zh|τ − τ 0 |t2 (τ 0 ) dτ 0 = τ 1 − 2τ∞20t (τ ) dτ0iñîáñòâåííûõ∞Zτ 0 t2 (τ 0 ) dτ 0 ,+2τ297è óðàâíåíèå (0.46) ïðèìåò âèäd2 t(τ ) h+ −πτ −ρ+O τdτ 2Z∞020t (τ ) dτ +Oτ∞Z0 20τ t (τ ) dτ0it(τ ) = 0.τ(4.37)Åñëè ïðåíåáðå÷ü â (4.37) ýêñïîíåíöèàëüíî ìàëûìè ïîïðàâêàìè, òîïîëó÷èòñÿ óðàâíåíèå, êîòîðîå ïîñëå çàìåíûz =√3π(τ + ρ/π)ñîâ-ïàäàåò ñ óðàâíåíèåì Ýéðè. Àñèìïòîòèêè ôóíêöèé Ýéðè õîðîøî èçâåñòíû [85].
 ðåçóëüòàòå, ïðèτ → +∞èìååì 1 2√ ρ 3/2 exp −1 + O 3/2π τ+=t(τ ) = p43πττ + ρ/πc+ 2√ 1 ρτ 1/2 c+3/2πτ − √exp −1+O √,=√43τπτãäåc+ êîíñòàíòà. Àíàëîãè÷íî, ïðèτ → −∞ 1 2√ρ|τ |1/2 c−3/2p√π|τ | −1+O pt(τ ) = 4exp −,3π|τ ||τ |ãäåc− êîíñòàíòà.
Îòìåòèì, ÷òî ýêñïîíåíöèàëüíîå óáûâàíèå ñîá-ñòâåííûõ ôóíêöèé çàäà÷è (0.46), (0.47) âûòåêàåò èç òåîðåìû ñðàâíåíèÿ äëÿ îäíîìåðíîãî óðàâíåíèÿ Øðåäèíãåðà[10].cos-àìïëèòóäû ýéðè-ïîëÿðîíà îïðåäåëÿåòñÿ ôîðìóëîé (4.31), â êîòîðîé t = ti (τ ) ñîáñòâåííàÿôóíêöèÿ îäíîìåðíîãî ïîëÿðîíà, à y = y(ξ) çàäàíà â (4.32).Ïîäñòàâèì òåïåðü ðÿäû äëÿ B , I , L â (4.22).
Òîãäà äëÿ íàõîæäåíèÿ ãëàâíîãî ÷ëåíà ðàçëîæåíèÿ sin-àìïëèòóäû I0 (ξ, η) ïîëó÷àåìÈòàê, ãëàâíûé ÷ëåí àñèìïòîòèêèóðàâíåíèåb 0 = 2S 0 ∂B0 + S 00 B0 .RI∂ξÇäåñü ëèíåéíûé îïåðàòîðbR(4.38)çàäàí ôîðìóëîédef ∂ 2 I0bRI0 =− π∂η 2Z∞|η − η−∞0|B02 dη 0+ `0 I0 .(4.39)298Çàìå÷àíèå4.3.
Äëÿ óïðîùåíèÿ îáîçíà÷åíèé â (4.39), à òàêæå â ïî-ñëåäóþùèõ ôîðìóëàõ ï. 1.3 èç 1 ãëàâû 4, íå áóäåì âûïèñûâàòüàðãóìåíòû ó ôóíêöèéBj (j = 0, 1, 2), I0 , I1 , ϕ0 , Ψ, T1 , T2 , T1,0 , Zèèõ ïðîèçâîäíûõ.  óêàçàííûõ ôîðìóëàõ ïåðâûé àðãóìåíò äàííûõôóíêöèé ðàâåíξ , à âòîðîé ëèáî ðàâåí η , åñëè ôóíêöèÿ íå ñòîèò ïîäçíàêîì èíòåãðàëà, ëèáî, â ïðîòèâíîì ñëó÷àå, ñîâïàäàåò ñ ïåðåìåííîé, ïî êîòîðîé ïðîèçâîäèòñÿ èíòåãðèðîâàíèå.ÎïðåäåëèìËåììà 4.4.ϕ0 (ξ, η) = S 00 (ξ)η 2 /6.(4.40)I0 = −B0 ϕ0(4.41)Ôóíêöèÿóäîâëåòâîðÿåò óðàâíåíèþ(4.38).Äîêàçàòåëüñòâî.
Òàê êàêb 0 = 0,RBòî ïîñëå ïîäñòàíîâêè (4.41)â (4.38) ïîëó÷àåì ñîîòíîøåíèå∂B0 ∂ϕ0∂B0∂ 2 ϕ0= 2S 0+ S 00 B0 ,−B0 2 − 2∂η∂η ∂η∂ξêîòîðîå â ñèëó (4.40), (4.31) çàïèñûâàåòñÿ â âèäå0S 00S 00 dtdt0 00β−y t(τ ) − 2y τ (τ ) = 2S y t(τ ) + 2S yτ (τ ) + S 00 yt(τ ).33 dτβ dτ(4.42)Ïîñêîëüêó èç (4.32) âûòåêàåò, ÷òîy02 S 00= − 0,y3Sβ01 S 00= − 0,β3S(4.43)òî (4.42) èìååò ìåñòî. Ëåììà äîêàçàíà.Çàìå÷àíèå4.4.
Ðåøåíèå óðàâíåíèÿ (4.38) â êëàññå ãëàäêèõ, ýêñïî-íåíöèàëüíî óáûâàþùèõ ïðèíîñòüþ äî ñëàãàåìîãî âèäàc(ξ) = 0, ÷òî ïðèâîäèò ê íàèáîëåå ïðîñòûì ðàçëîæåB è I . Îäíàêî ïðè ýòîì `1 (ξ) 6= 0. Óêàçàííûé ïðîèçâîëìû ïîëîæèëèíèÿì äëÿ|τ | → ∞ ôóíêöèé ðàâíî I0 ëèøü ñ òî÷c(ξ)t(τ ). Äëÿ îïðåäåëåííîñòè, â (4.41)299â ðàçëîæåíèÿõ ñâÿçàí ñ íåîäíîçíà÷íîñòüþ ïðè ðàçáèåíèèS(ξ)èϕ(ξ, η)è íå âëèÿåò íà èòîãîâûå ôîðìóëû äëÿΦ(ξ, η) íàG.Îòìåòèì, ÷òî â ôîðìóëàõ (4.31) (4.33), (4.40), (4.41) ôàçàïîêà îñòàåòñÿ íåèçâåñòíîé. Âû÷èñëåíèåSñì. â ï.
1.4 èç 1 ãëàâû 4.Ïåðåéäåì ê âûâîäó ôîðìóë äëÿ ïîïðàâîêâ ñèëó (4.17)S 0 (ξ) ∼íàõîæäåíèÿ ïåðâûõb B1 = πN√[40]. Òàê êàêïðè|η − η−∞Z0|I02 dη 0−∞∂I0− S 00 I0 ,∂ξ(4.44)(I02 + 2B0 B1 ) dη = 0,(4.45)η(I02 + 2B0 B1 ) dη = 0.(4.46)Çäåñü ëèíåéíûé îïåðàòîðbNçàäàí ôîðìóëîédef ∂ 2 B1b−πN B1 =∂η 2ZB0 + `1 B0 − 2S 0∞−∞∞Z−2πB1 , `1ξ → +∞, òî èç (4.21), (4.23), (4.24) äëÿïîïðàâîê B1 (ξ, η), `1 (ξ) ïîëó÷àåì çàäà÷óξ∞ZSZ∞|η − η0−∞∞00|B02 dη 0B1 −|η − η |B0 B1 dη B0 − `0 B1 ,(4.47)−∞à àðãóìåíòû ó ôóíêöèé îïóùåíû â ñîîòâåòñòâèè ñ çàìå÷àíèåì 4.3.ÂìåñòîB1ìû áóäåì èñêàòü ôóíêöèþT1 = B1 + I02 /(2B0 ) = B1 + ϕ20 B0 /2,(4.48)çàäà÷à äëÿ êîòîðîé èìååò áîëåå ïðîñòîé âèä.Ëåììà 4.5.Ôóíêöèè T1 (ξ, η), `1 (ξ) óäîâëåòâîðÿþò çàäà÷åh ∂ϕ 2i00 ∂ϕ0bN T1 = `1 ++ 2SB0 ,∂η∂ξZ(4.49)∞B0 T1 dη = 0,−∞(4.50)300Z∞ηB0 T1 dη = 0.(4.51)−∞Äîêàçàòåëüñòâî.
Óñëîâèÿ (4.50), (4.51) âûòåêàþò íåïîñðåäñòâåííî èç (4.48) è (4.45), (4.46). Äëÿ äîêàçàòåëüñòâà (4.49) ïîäñòàâèìB1 = T1 − ϕ20 B0 /2 â óðàâíåíèå (4.44). Ó÷èòûâàÿ (4.40), (4.41), èìååì∂ 2 ϕ20 bN T1 = 2B0 − π∂η 2Z+`1 B0 + 2S 0∞|η − η−∞0|B02 dη 0+ `0 ϕ202B0 +∂(ϕ0 B0 ) + S 00 ϕ0 B0 .∂ξÒàê êàê çàäàâàåìàÿ ôîðìóëîé (4.31) ôóíêöèÿB0(4.52)óäîâëåòâîðÿ-åò (4.28), òî óðàâíåíèå (4.52) ïðèíèìàåò âèäin (S 00 )2 η 2(S 00 )2 η 2N̂ T1 = `1 + 2SB0 +B0 +B0 +∂ξ66h0 ∂ϕ0io(S 00 )2 3 dt(τ ) S 0 S 00 η 2 2 h 00 dt+η yβ+η y t(τ ) + yβ η (τ ) .9dτ3dτÍàêîíåö, âîñïîëüçîâàâøèñü (4.32), (4.43), ïðèõîäèì ê (4.49). Ëåììàäîêàçàíà.Ïåðåéäåì ê èçó÷åíèþ çàäà÷è (4.49) (4.51).
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