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BIRINCHI TARTIBLI CHIZIQLI DIFFERENSIAL TENGLAMALAR SISTEMASI UCHUN SOCHILISH NAZARIYASINING TO'G'RI MASALASI
Zufarbek Shodlik o'g'li Bekdurdiyev
Urganch davlat universiteti magistranti
ANNOTATSIYA
Ushbu maqolada birinchi tartibli oddiy differensial tenglamalar sistemasi uchun sochilish nazariyasining to'g'ri masalasi o'rganilgan.
Tayanch so'zlar: Chiziqli differensial tenglamalar sistemasi, Yost yechimlari, to'g'ri masala, sochilish nazariyasi, o'tish koeffitsiyentlari.
Quyidagi chiziqli differensial tenglamalar sistemasini qaraymiz
U)f = f t > 0 .
ox
(1)
Bu yerda va keyinchalik ushbu
=
r0 1
v
0
=
r0 -i^
i 0
Pauli standart matritsalaridan foydalanamiz. f funksiya
&3 =
'1 0^
0 -1y
noma'lum 2 x 2 kvadrat matritsa
vv
V
U =
0
iu (x, t)
-iu( x, t) 0
t > 0.
u (x, t) funksiya
0 to
J (1 - x) u(x, t) - peia~2iph dx + J (1 + x) u(x, t) - peiß~2 dx < 00 , p> 0 (2)
-X 0
shartni qanoatlantiruvchi funksiya bo'lsin.
Butun o'qda Dirak operatori uchun sochilish nazariyasi teskari masalasini A.B.Zaharov, A.B.Shabat [1], V.E.Grin, M.G.Gasimov, B.M.Levitan, I.S.Frolov, L.P.Niznik, Fam Loy Vu, L.A.Taxtadjyan [2], L.D.Levitan, A.B.Hasanonov [3] va boshqalar tomonidan o'rganilgan.
Sochilish nazariyasining teskari masalasi usuli Gelfand-Levitan-Marchenko integral tenglamalari sistemasi deb nomlanuvchi ikkita integral tenglamalar sistemasini yechishga keltiriladi. Akslanuvchi potensiallar xolida bu sistemaning yechimini topish murakkab bo'ladi. Biror bir taqribiy yechimni topish uchun sonli usullar qo'llaniladi. [4], [5], [6], [7], [8], [9], ishlarda diskret nochiziqli Shredinger tenglamasini turli xil funksiyalar sinfida teskari masalalar usulida integrallash o'rganilgan.
£ e R \(-p, p) da f- va f + orqali (1) tenglamaning quyidagi asimtotikani
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f - ~ E- (x,Ç, t), x ^-w, f+ ~ E+ ( x,Ç, t ), x ^œ, qanoatlantiruvchi yechimini belgilaymiz. Bu yerda
(3)
E - ( x,Ç, t ) =
1
i(Ç - P)cia-2i(2t
v P
r
E+ ( x,Ç, t ) =
i(Ç- P)c-ia+2ip2t
P
j(Ç- P)c-iß+2ip1t
-ipa^x
P
i (Ç- P )ciß-2ip1t P
-ipa^x
№ = 4~Ç
2 P2
(4)
Bunday yechimalar (1) tenglamali Yost matritsali yechimlari deyiladi. Ushbu
d det f± ( x,Ç, t ) = 0, dx
tenglikning o'rinli bo'lishini ko'rsatish mumkin. Bundan hamda (3) shartlardan
2P(Ç- P)
det f ± (x,Ç, t) =
P
(5)
(6)
Bundan shunday S(Ç,t) matritsa mavjud bo'lib, f+ (x,Ç,t) va f (x,Ç,t) yechimlarni uchun
f - (x,Ç, t) = f + (x,Ç, t)S (Ç, t ), (7)
tenglik o'rinli bo'ladi. Bunda
J a(Ç, t ) b (Ç, t ) " ( , ) [b(Ç,t) a(Ç,t)) a(Ç, t) va b(Ç, t) koeffisentlarini o'tish koeffitsiyentlari deyiladi. (6) va (7) tengliklardan quyidagi
|a(Ç, t)|2 - \b(Ç,t)|2 = 1,
va
a(Ç, t )
b(Ç, t ) =
P
2 P (Ç- P )
det(f " ( x,Ç, t ), f2+ ( x,Ç, t ))
(8) (9)
P
-det(fi+ ( x,Ç, t ), fr ( x,Ç, t ))
2P(Ç- P) www.scientificprogress.uz
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tengliklarning o'rinli bo'lishi kelib chiqadi.
a(£, t) funksiya £ boyicha analitik va faqat chekli sondagi nollarga ega. Bu nollarni £,£,...,£ bilan belgilaymiz. (9) tenglikdan agar £ = £ da a(£n,t) = 0 bo'lsa, f- (x,£, t) va f+ (x,£, t) ustunlar chiziqli bog'lanishda bo'ladi, ya'ni
f;(x,£n,t) = ^(t)f+ (x,£,t), n = 1,2,...,N (10)
Shuning uchun £ = £ soni (1) tenglamalar sistemasining xos qiymati bo'lishi kelib chiqadi. (1) tenglamalar sistemasi o'zaro qo'shma, demak uni xos qiymatlari, ya'ni a (£, t) funksiyaning nollari haqiqiydir. Bu nollar (-p, p) oraliqda joylashgan.
a(£,t) funksiya nollari oddiy (karrasiz)ligini ko'rsatamiz. (9) dan £ boyicha hosila olamiz
2
Pn(£n Pn)
+ det(/r(x,£„,0,/2+(x,£„,0). (11)
f~(x,£,t) va /2+(x,£,i) ustunlar (1) tenglamalar sistemasini, f~(x,£,t) va
d
/2+(x,£,/) ustunlar esa z<t3 —v = v + £v tenglamani qanoatlantiradi.
" dx
Bu tenglamalardan quyidagi tengliklar kelib chiqadi.
d
—det( (x, £, t), (x, £, t)) = i det((x, £, t), f* (x, £, t)), ox
d
—det((x,£,t), f*(x,£,t)) = -zdet(aJT(^0,№,£,0) • ox
£ = £„ da f" (x,£,t) va f2+ (x,£,t) ustunlar proporsional va |x| ^to ekspotensial kamayishini hisobga olib,
oo
detC/j" (x, £n, t), /2+ (x, £n, 0) = ~icn {t) J A(s, £n, t )ds, (12)
x
X
det(/r(*,£,0,/2+C*,£,0) = -Kit) J ACv,£„,0ds, (13)
—to
hosil bo'ladi. Bu yerda
A( x,£, t) =e^f (x,£, 11.
£n + Pn
va ||-|| - C2 da oddiy vektor normasini ifodalaydi. (12),(13) formulalarini (11) tenglikga qoyib , ä{£n,t) uchun ushbu
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-iß+2ip1t œ
à(Z, t ) = - c{t)ppé' ' HI f+ Mn, t )
ç + p J 11
~n -fi
2
ds
ifodani hosil qilamiz. Demak, à{Çn,t) nolgateng emas. Quyidagi
{à(z,t), b(z,t),4(t),cM(t),n = 1,2,...,(14)
to'plamga (1) tenglamalar sistemasining sochilish nazariyasining berilganlari deyiladi. (1) tenglamalar sistemasi uchun sochilish nazariyasining to'g'ri masalasi potensial berilganda (14) berilganlarni topishdan iborat.
Misol. (1) tenglamalar sistemasining u ( x, t ) potensiali
p e,pe~'PlX + ce,aemx u = pe p -:-:-
e-pix + ceipix
bo'lsin. Bunda, p,a, ß va c berilgan musbat sonlar. Sochilish nazariyasining berilganlarini topamiz.
Yost yechimlari x ^œ
KZ- P)„-ß+2P e~'Plx + caelß~iae'Plx
- „ -ß2-p .------eipx
1 p e-Pl x + ceiPl x
e-pix + cae-Pix lpx
¥2 = —:-— • eP
T2 e-Pix + ce-Pix
(2)
x ^-œ da
ae pix + cepix wx
<P\ =—:-:— e ipx
^ e-pix + ceipix
= l(Z-ß)ca-2-p2t aelß-iae~iPix + ce-Pix p e~ipix + ceipix
bo'ladi. Quyidagi tengliklarning o'rinli bo'lishini oson ko'rsatish mumkin:
Z±Ä = p
p Pi,
Z + P-Pi = Zi-Pi.P-Z + Pi Z + P-Zi + Pi Z + Pi Z + P-Zi-Pi Haqiqiy Z va p sonlari uchun a(t,Z), b(t,Z) koeffitsiyentlarni hisoblaymiz.
(3)
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a (t ,f) =
ae-ipix + ceipix - ipx i(f - p) „-iß+2P eipix+caeß-iaepix ^
P2 e-mx + Ce^pi x P e-ipix +cemx
2p(f- p ) i(f- p) £ ia - 2iP*taeiß-iae -w + Ceipix e-ipix + caeipi x lvx cpx
P e-ipix + ceipi x e-px + ceipix
P
2p(f- p) (e~ipx + cepx)2
P
i(f p) e~ iß+2iplt ( e~ Px + cae '^-iae Px ) P
i(f p) eia-2 p2t (ae iß-iae~ipix + ce plX )
P
1
2p(f- p) (e-pix + ceipx )2 (f- p)2
[(ae~ipix + ceipi x )(e~ipix + caeipi x ) -
eia-lß{aelß-iae~ipix + ceip x ) x (e~ip x + caelß-iaeip x )] =
P'
iß-ia ipi x -
P
1
[ae 2 ipiX + c + caL + cz ae
2p(f- p) (e-px + ceipx )2
ia-iß / iß -ia -2 ipxx
2 _i_ r*2nr>2ipx
(f p) ia-iß r „ iß -ia -2 ipx . . „2 2iß-2ia . 2 „ iß-ia 2ipx\-i
--e ß (ae ß e pi + c + ca e ß + c aeß e pi )] -
P2
P
2
1
(f- p )2
(f- p )2
2p(f- p) e-pix + ce
px
[a(1 - (f p ) )e"2ipiX + c2a(1 - (f p) ) x
P
P
xe2ipix + (c + œ2 _ (f p)2 ^„ia-iß*- (f p)2 ¿ß-ia2
)eia-ißc-^ P' eip-iaca2]; P P
Quyidagilarni hisoblaymiz:
(1
(f- p )2 P2
eiß ia )ca2 + (1
(f- p )2 P2
eia-iß)c
= ca(1 - p) 2 )a + (1 - (f p)2 )a ) =
(fi + p 1)2
(ff- p1)2
= ca((f1 + p)2 a + p)2 a")
(fi + A) (f- A)
(f- p -f1- pi)(f- p + f1 + pi) ^ia-iß f- p-f1+ A
(fl + pi )2
■e
f-p-fi-px
+
, (f- p-f1+ pi)(f- p + f1 -pi ^-ia+iß . f-p-fl -pi
" f-p-f1 + pi
(fl- pi)2
(f- p-fi + pi)(f- p + fi + pi) fi + pi
(fl + p l)2
f1 -pi
1
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(£-p + £-pi)(£-ppi) Pi _
(Si - Pi )2
S + pi
= ,(S - p)2 + (Si + A)(S - p ) - P2 - (S - p)(Si - A) +
P2
(S - p)2 + (S - p )(S - pi) - p2 - (S - p)(Si + pi)
P2 ) "
ca{(£ - p)2 + (S - p )2 p -p2 + (S - p)2 + (S - p )(-2 pi) -p2}
P2
2ca(i - (S -p) )
Bunga asosan,
a(t ,S) =
Pz
i
2p(S - p) (e~ipx + ceipx )2
P
a ■ (i - (S 2p) )(e"2lpx + 2ca + e2lpx ) =
P
S - p - Si - pi
S + p-Si + pi'
Demak, a(t,S) funksiyaning noli S = Si va px = iyjP2 -
b =
ae"ipix + cej^ lpix
P2 e~ipix + ce
2 p (S - p ) i(S~ p) „a -2P aeiß~
e P e~
P2 e-2ipx i(S- p) e
2p(S - p) e -ipix + ceipx
P
-ipx
e- + ca eipix e~'pix + celpix
-ipx
xe~'pix + ce'pix c_px i(S- p)jp-2é-x + câe'px &_ipx
■px + ceipix
ae-ipix + ceipx
P
e-x + ceipx
ae-x + caepx
U-p)
P
eip-2 2 (e- +caeipx )
-2 ipx
b
2p(S - p) e~ipx + ce P2 i (S- p )
ip x
ae~ipx + ceipx
ae~ipx + caeipx
aelß~iae~ip x + ceip x eiß~iae~ip x + caeip x )
i(s- p)
-iß-ia
P
>iß-ia[aelß~ia e~2ipx + caa + ceiß~ia + c2 ae
2p(S - p) P
-caaeiß-ia- c - c2ae2ipx ] = 0 Endi normallovchi o'zgarmasni topamiz. S = Si da
iß-ia | ^2—^2ipx Qßiß-aç-2px
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c
% =-:-:-,
^ e~>Pix + ceiPiX % = ^ P) C
p e~ipx + ceipix
va
... _ 1 pi) P~iß+2ip2t W\----e
p e~ipix + ceip x
1
¥2 - —
e ipx + ceipx
bo'lgani uchun p — c¥ tenglikdan
Ç_— c -pi) . e-rß+2P . 1 )
1 ^ _ —ipx . ipx ' '
e _ ipix + ceipix p e~ipix + ceipx
. P
A
c — c-^- e~lß+2pt
yoki
c — + pi) eiß_ 2P
p
kelib chiqadi. f = f,P = P a(f,t) funksiyaning noli bo'lgani uchun f = f soni xos qiymat bo'ladi. Demak, (1) tenglamalar sistemasining sochilish nazariyasining berilganlari
a(f) = f + P~ f ~ P , b(f) = 0, f +p-f+p
q = ci(f±P) , f = f p
berilganlardan iborat bo'ladi ekan. REFERENCES
[1] V. Zakharov and A. Shabat, "Exact theory of two-dimensional self- focusing and one-dimensional self-modulation of waves in nonlinear media," JETP, vol. 61, no. 1, p. 118-134, 1971.
[2] Л. Тахтаджян и Л. Фаддеев, Гамильтонов подход в теории солитонов, Москва: Наука, 1986, p. 528.
[3] А. Хасанов, «Об обратной задачи теории рассеяния для системы двух несамосопряженных дифференциальных уравнений первого порядка,» ДАН, т. 277, № 3, pp. 559-562, 1984.
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[4] M. Ablowitz and J. Ladik, " Nonlinear differential-difference equations and Fourier analysis," J. Math. Phys., no. 17, pp. 1011-1018, 1974.
[5] M. Ablowitz and J. Ladik, "Nonlinear differential-difference equations," J. Math. Phys., no. 16, pp. 598-603, 1975.
[6] M. Ablowitz and J. Ladik, "A nonlinear difference scheme and inverse scattering," Stud. Appl. Math., no. 55, p. 213-229, 1976.
[7] F. Demontis and C. van der Mee, "Exact solutions to the integrable discrete nonlinear Schrodinger equation under a quasiscalarity condition," Commun. Appl. Ind. Math., vol. 2, no. 2, p. 21, 2011.
[8] F. Demontis and C. van der Mee, "Closed form solutions to the integrable discrete nonlinear Schrodinger equation," J. Nonlin. Math. Phys, vol. 2, no. 19, p. 22, 2012.
[9] F. Demontis h C. van der Mee, «An alternative approach to integrable discrete nonlinear Schrodinger equations,» Acta Appl. Math, № 131, p. 29-47, 2014.
