MANBALI UMUMLASHGAN NOCHIZIQLI SHREDINGER TENGLAMASINI BIRINCHI INTEGRAL USULIDA YECHISH

TENGLAMASINI BIRINCHI INTEGRAL USULIDA YECHISH

Jumaniyozova Xayriniso

Urganch Davlat Universiteti

Annotatsiya: Ushbu maqolada nochiziqli Shredinger tenglamasining o‘zgarmas amplitudali yechimlari haqida ma’lumot berilgan.

Kalit so‘zlar: nochiziqli Shredinger tenglamasi, soliton yechim, birinchi integral usuli

Ma’lumki, chiziqli bo‘lmagan murakkab fizik hodisalar fizikadan biologiya, kimyo, mexanika va boshqalarga qadar ko‘plab sohalarda ishtirok etadigan chiziqli bo‘lmagan xususiy hosilali differensial tenglamalar bilan bog‘liq. Hodisalarning matematik modellari sifatida bu tenglamalarning yechimlarini tekshirish bu hodisalarni yaxshiroq tushunishga yordam beradi.

Xususiy hosilali differensial tenglamalarning aniq yechimlarini olishning ko‘plab samarali usullari yaratilgan va ishlab chiqilgan, masalan, Li simmetriyalari usuli [1], exp-funktsiya usuli [2, 3], sin-cos usuli [4, 5], kengaytirilgan tanh-coth usuli [6, 7], proyektiv Rikkati tenglama usuli [8, 9] va boshqalar.

Birinchi integral usul Feng tomonidan [10] da kommutativ algebraning halqa nazariyasiga asoslangan bo‘lib, dastlab Burgers-KdV tenglamasini yechishda taklif qilingan. So‘nggi paytlarda bu usul ko‘pchilik tenglamalar uchun keng qo‘llanilgan.

Mazkur ishda biz ushbu    ,

22 tXi

xxxxxxt keuuidicuubuauiu

 



(1)

manbali umumlashgan nochiziqli Shredinger tenglamasini yechimini topishning birinchi integral usulini ko‘rib chiqamiz.

U quyidagi qadamlarda amalga oshiriladi:

1-qadam. txuu , funksiya (1) tenglamaning yechimi bо‘lsin. Quyidagi   

ftxu ,

,,ctx 

 (2)

almashtirishni kiritamiz. Bu almashtirish bizga quyidagi o‘zgarishlardan foydalanish imkonini beradi:               

2 2 2 2

. . , . . , . . , . . ,c c

x t x t x

       

     

        

(3)

Bu almashtirishlardan foydalanib (1) nochiziqli hususiy hosilali differensial tenlamani ushbu.,...,, 





 



uu

uH

(4)

nochiziqli oddiy diferensial tenglamaga keltiramiz. Bu yerda u u

 noma’lum funksiya,H esau o‘zgruvchili va uning hosilalaridagi ko‘phad.

2-qadam. Deylik (4) ning oddiy differensial tenglama yechimini quyidagicha yozish mumkin bo‘lsin:   ,u x t f



. (5)

Budan tashqari biz yangi mustaqil o‘zgaruvchini ham kiritamiz       , f

X f Y

 

 

  

. (6)

3-qadam. (5) va (6) ga ko‘ra (4) tenglamani nochiziqli birinchi tartibli differensial tenglamaga quyidagicha almashtirish mumkin.   

      

YXF

,

,

1

(7)

Agar (7) tenglamaning integrallarini topa olsak, u holda (7) tenglamaning umumiy yechimlarini bevosita topish mumkin. Bunda tx

  haqiqiy qiymatli funksiya va

 ,,,,,,, kdcba larning barchasi haqiqiy.

Ushbu      ,, ti

etxu



 

(8)

ko`rinishdagi tekis to`lqin yechimlarini ko`rib chiqaylik, bu yerda 

 haqiqiy funksiya. Qulaylik uchun0x



 , bu yerda va0x haqiqiy o`zgarmaslar va     tx

deb olamiz. Shundan keyin (8) funksiya hosilalari ustida tegishli shakl almashtirishlarni bajarib, natijaning haqiqiy va mavhum qismlarini ajratib, biz quyidagi ikkita oddiy differensial tenglamani hosil qilamiz:  ,0332 22323 





 dcac

(9)

387      .03 3332232  kdbcaca







(10)

(10) ni ga nisbatan bir marta integrallab, quyidagi      ,032 222  Mcac



(11)

tenglikni hosil qilamiz. Bu yerdaM ihtiyoriy integral o`zgarmas. Ushbu  funksiya (10) va (11) tengliklarni qanoatlantirgani uchun biz quyidagi cheklash shartini kiritamiz:.

3 22

M

k

d

db

ca

ca

ca 











(12)

(6), (7) tengliklardan foydalanib quyidagi yechimlarni hosil qilamiz:   ,

 YX 

(13)      .3



 c

M

a

d

Y 



 



 

(14)

Birinchi integral metodiga ko`ra, 

X va Y mos ravishda (13) va (14) ning notrivial yechimlari hamda,    im

i i YXaYXP  

 0

, YXC , kompleks sohadagi qisqarmas ko`phad. Shunday deb faraz qilamizki:          ,0, 0

 

i

m

i

i YXaYXP

(15)

bu yerda   miXai ,...,2,1,0, X ning ko`phadlari va  0Xam .

Ushbu (15) tenglik (13) va (14) tengliklarning birinchi integrali deb ataladi. Bo`linish teoremasi tufayli YXC , kompleks sohada   YXgXh  ko`phad mavjud bo`lib,       .

i

m

i

i YXaYXgXh

d

dP

P

d

dX

P

d

dP 





(16)

Yuqorida biz1m va2m deb faraz qilib, ikkita holatni ko`rib chiqdik.

1-hol. Faraz qilaylik1m  bo`lsin. U holda (16) ning ikkala tomonidagi)0,1,2( iY i

koeffitsiyentlarini tenglashtirib ushbu     1 1a X g X a X 

, (17)         0 1 0a X h X a X g X a X  

(18)

388     3 2

(19)

tengliklarni hosil qilamiz.   0,1,2ia X i 

ko‘phad bo‘lgani uchun (17) ga asosan 1a X o‘zgarmas va  0g X 

degan xulosaga kelamiz. Soddalik uchun 1 1a X  deb olamiz. h X , va 0a X xulosaga kelamiz. Faraz qilaylik, h X AX B  va0A  boʻlsin, u holda biz Xa koʻphadni  2

a X X BX D  

(20)

koʻrinishda topamiz. BundaD ihtiyoriy integral o`zgarmasi.

(19) ga 0a X , 1a X va h X ni qo`yib,X ning barcha darajalari oldidagi koeffitsiyentlarini nolga tenglashtirib, nochiziqli algebraik tenglamalar sistemasini hosil qilamiz va uni yechish orqali ushbu yechimlarni olamiz:2 2

2 2 3 ,

0, , 0,

i c d D a c

i d

M A B

   

(21)2 2

2 2 3 ,

0, , 0.

i c d D a c

i d

M A B

  

(22)

(15) da (21) va (22) shartlardan foydalanib, biz quyidagiga ega bolamiz   22i d

Y X D

   

. (23)

Biz (23) ni (13) bilan birlashtirib, (13) va (14) ning aniq yechimlarini oldik. harakatlanuvchi to`lqin yechimlarini quyidagicha yozishimiz mumkin:

389   

     

1 4 1 4

1 4 3 4 1 4

11 4

, 2

tanh 1 2 2

exp ,

u x t i c D

       

     2 2

2 2 3i c d D a c

(24)   

     

1 4 1 4

1 4 3 4 1 4

11 4

, 2

exp ,

u x t i c D

       

     2 2

2 2 3i c d D a c

(25)

2-hol. Faraz qilaylik2m  bo`lsin. U holda (16) ning ikkala tomonidagi)0,1,2( iY i

koeffitsiyentlarini tenglashtirib ushbu     2 2a X g X a X 

, (26)         1 2 1a X h X a X g X a X  

, (27)   

       

3 2

0 2 2 2 2

1 0

a X a X X X

h X a X g X a X

                   

 

(28)     3 2

2 3 .

(29)

tengliklarni hosil qilamiz.   0,1, 2ia X i 

ko‘phad bo‘lgani uchun (26) ga asosan 2a X o‘zgarmas va  0g X 

degan xulosaga kelamiz. Soddalik uchun 2 1a X  deb olamiz. h X , va 0a X xulosaga kelamiz. Faraz qilaylik, h X AX B  va0A  boʻlsin, u holda biz 1a X va 0a X koʻphadlarni ushbu

390  2

1 2

a X X BX D

   

(30)   

4 3

8 2 2

A d

a X X AB X

AD B a M

c X BD X F

c c

    

   

         

  

(31)

koʻrinishda topamiz. Bunda, ,A B D vaF ihtiyoriy integral o`zgarmasi.

(29) tenglikga 0a X , 1a X 2a X va h X ni qo`yib,X ning barcha darajalari oldidagi koeffitsiyentlarini nolga tenglashtirib, nochiziqli algebraik tenglamalar sistemasini hosil qilamiz va uni yechish orqali ushbu yechimlarni olamiz:2 2

0, 2 4 6 ,

, , 0,

M i c d D a c

D i d

F A B

      

(32)2 2

0, 2 4 6 ,

, , 0,

M i c d D a c

D i d

F A B

     

(33)

(15) da (32) va (33) shartlardan foydalanib, biz quyidagiga ega bolamiz   2

i d X cD

Y c

 

. (34)

Biz (34) ni (13) bilan birlashtirib, (13) va (14) ning aniq yechimlarini oldik. harakatlanuvchi to`lqin yechimlarini quyidagicha yozishimiz mumkin:   

  

1 4 1 4 1 4

tanh 2 2

  

       

  

     

3912 21 2 2 3

2 i c d D a c

, (35)   

    

11 4 1 4 3 4 1 4

d D x t c c

   

       

     2 21 2 2 3

2 i c d D a c

. (36)

(24)-(25) va (35)-(36) tenglamalar (1) manbali umumlashgan nochiziqli Shredinger tenglamasining aniq harakatlanuvchi to`lqin yechimlaridir

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