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Дата изменения: Sat Apr 2 01:16:07 2016
Дата индексирования: Sun Apr 10 05:39:04 2016
Кодировка:

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27

28

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H[gZjm`_gh mkbe_gb_ hldebdZ <= \ hdj_klghklb j_ahgZlhjghc fh^u bgZdjZyoaZij_s_gghcnhlhgghcahgu>@JZa\blhn_ghf_gheh]bq_kdh_ hibkZgb_ ]_g_jZpbb <= \ fgh]hkehcghc kljmdlmj_ k jZkij_^_e_ggufb g_ ebg_cgufb bklhqgbdZfb mqblu\Zxs__ fgh]hemq_\mx bgl_jn_j_gpbx baemq_gbc gZdZqdb b <= IhdZaZgh qlh mkbe_gb_ hldebdZ <= gZ qZklhl_ fh^u fbdjhj_ahgZlhjZ b gZ djZyo nhlhgghc ahgu bf__l dZq_kl\_ggh jZa gmx ijbjh^m j_ahgZgk <= \ fh^_ h[mkeh\e_g ehdZebaZpb_cmkbe_gb_f klhyq_c \hegu gZdZqdb \ hdj_klghklb fbdjhj_ahgZlhjgh]h kehy j_ahgZgk <= gZ djZx nhlhgghc ahgu k\yaZg k h^ghjh^guf mkbe_gb_f ihey gZdZqdb \jZkij_^_e_gguoND-a_jdZeZofbdjhj_ahgZlhjZ>@ Bkke_^h\Zgu wnn_dlu mkbe_gby ijb ]_g_jZpbb <= \ k\yaZgguo fbd jhj_ahgZlhjZo gZ hkgh\_ ihjbklh]h dj_fgby>@ < m]eh\uo ki_dljZo bg l_gkb\ghklb <= h[gZjm`_gh j_ahgZgkgh_ mkbe_gb_ \ m]eh\uo iheh`_gbyo khhl\_lkl\mxsbf jZks_ie_gguf fh^Zf k\yaZgguo fbdjhj_ahgZlhjh\ Bamq_gZ aZ\bkbfhklv iheh`_gbc b Zfieblm^ j_ahgZgkh\ <= hl iZjZf_ljh\ nhlhggh]hdjbklZeeZjZa^_eyxs_]hfbdjhj_ahgZlhjgu_kehb Ijb bkke_^h\Zgbb i_jbh^bq_kdbo d\Zglh\uo yf dj_fgbc - ^bhdkb^ dj_fgby ihdZaZgZ qm\kl\bl_evghklv f_lh^Z ki_dljhkdhibb \lhjhc ]Zjfh gbdb d lhesbggh-aZ\bkbfhc we_dljhgghc kljmdlmj_ dj_fgb_\uo yfwg_j ]_lbq_kdhfm iheh`_gbx ih^ahg nhjf_ ebgbb iehlghklb khklhygbc >@ H[gZjm`_g fhghlhgguc k^\b] \ ^ebggh\hegh\mx h[eZklv j_ahgZgkguo wg_j]bcnhlhgZ\lhjhc]ZjfhgbdbgZfw@ Ijh\_^_gu Z[khexlgu_ nZah\u_ baf_j_gby ^ey wlZehgZ b dZeb[ jh\hqgu_ baf_j_gby ^ey ghjfbjh\db gZ ki_dljZevgu_ baf_g_gby k\hckl\ hilbq_kdh]h ijhimkdZgby we_f_glh\ khklZ\eyxsbo hilbq_kdmx ko_fm Hij_^_e_gZ lhqghklv f_lh^bdb Wdki_jbf_glZevgh bkke_^h\Zgh ki_d ljZevgh_ih\_^_gb_d\Z^jZlbqgh]hhldebdZ]jZgbpujZa^_eZ6L-SiO2b Ge(111)-GeO2 >@ Baf_j_gu ki_dljZevgu_ aZ\bkbfhklb Zfieblm^u bg l_gkb\ghklb b nZau <= \ ^bZiZahg_ wg_j]bc nhlhgZ \lhjhc ]Zjfhgbdb hl ^h w< H[gZjm`_gu j_ahgZgkgu_ hkh[_gghklb \ ki_dljZo h[mkeh\ e_ggu_ ijyfufbwe_dljhggufbi_j_oh^Zfb gZih\_joghklb\ hdj_klghklb djblbq_kdhc lhqdb(2 ahgghc kljmdlmju h[tzfZ dj_fgby b ]_jfZgby < jZfdZo nhjfZebafZ djblbq_kdbo lhq_d dhf[bgbjh\Zgghc iehlghklb kh klhygbc jZa\blZ n_ghf_gheh]bq_kdZy fh^_ev ki_dljZevgh]h ih\_^_gby

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d\Z^jZlbqgh]h hldebdZ ihemijh\h^gbdh\ GZ _z hkgh\_ Ziijhdkbfbjh\Zgu wdki_jbf_glZevgu_ ki_dlju bgl_gkb\ghklb b nZau <= hij_^_e_gu iZjZ f_lju j_ahgZgkh\ ?2 ?1
IhdZaZgZ qm\kl\bl_evghklv f_lh^Z d lbiZf djblbq_kdbo lh q_d ahgghc kljmdlmju>@ JZajZ[hlZgZ f_lh^bdZ ^bZ]ghklbdb keZ[h kdh r_gguo ih\_joghkl_c p_gljhkbff_ljbqguo ihemijh\h^gbdh\ k ihfhsvx bgl_jn_jhf_ljbb <= >ey h[jZapZ dj_fgby kh keZ[h kdhr_gghc ih \_joghklvx baf_j_gu ZabfmlZevgu_ Zgbahljhigu_ aZ\bkbfhklb bgl_g kb\ghklb b nZau <= b gZ hkgh\_ j_r_gby h[jZlghc aZ^Zqb hij_^_e_gu kh hlghr_gby f_`^m dhfihg_glZfb d\Z^jZlbqghc \hkijbbfqb\hklb b m]he kdhkZihjy^dZ0 [10]. GZ ijbf_j_ ]jZgbpu jZa^_eZ 6L-SiO2 ieZgZjghc kljmdlmju dj_fgbc - ^bhdkb^ dj_fgby - ojhf ihdZaZgu ^bZ]ghklbq_kdb_ ij_bfms_ kl\Z bgl_jn_jhf_ljbb we_dljhbg^mpbjh\Zgghc \lhjhc ]Zjfhgbdb ^ey bk ke_^h\Zgby aZjy^h\uo oZjZdl_jbklbd kdjuluo ]jZgbp jZa^_eZ ihemijh \h^gbdh\ lZdbo dZd iehlghklvaZj y^Z \ keh_ ^bhdkb^Z Qox ^ey bkke_^h \Zgghc FHI kljmdlmju ihemq_gh Qox = (6 ± 1) 1012 lh]^Z dZd ZiijhdkbfZ pby bgl_gkb\ghklb ^Z_l gbadmx lhqghklvhij_^_e_gby iehlghklb aZjy^Z \ hdkb^_- Qox f_gy_lkyihqlbgZihjy^hdhl.5 1012^h12kf-2 [11]. L< >he]h\Z :B FZc^udh\kdbc F= FZjl_fvygh\ :: N_^ygbg H::dpbi_ljh\Ё=b]ZglkdZylj_lvy]ZjfhgbdZ\nhlhgguodjbklZeeZo bfbdjhj_ahgZlhjZogZhkgh\_ihjbklh]hdj_fgbyґIbkvfZ\@WLN75, 17 (2002). L< >he]h\Z :B FZc^udh\kdbc F= FZjl_fvygh\ = FZjh\kdbc >` FZll_c > RmfZo_j <: Ydh\e_\ :: N_^ygbg H: :dpbi_l jh\ Ё=b]ZglkdZy \lhjZy ]ZjfhgbdZ \ fbdjhj_ahgZlhjZo gZ hkgh\_ nh lhgguodjbklZeeh\ihjbklh]hdj_fgbyґIbkvfZ\@WLN73, 8 (2001). 3. T. V. Dolgova, A. I. Maidykovski, M. G. Martemyanov, A. A. Fedyanin, O. A. Aktsipetrov, G. Marowsky, V. A. Yakovlev, G. Mattei, "Giant microcavity enhancement of second-harmonic generation in all-silicon photonic crystals", Appl. Phys. Lett. 81, 2725 (2002). 4. T.V. Dolgova, A.I. Maidykovski, M.G. Martemyanov, A.A. Fedyanin, O.A. Aktsipetrov, D. Schuhmacher, G. Marowsky, V.A. Yakovlev, G. Mattei, N. Ohta, S. Nakabayashi, "Giant optical second-harmonic generation in single and coupled microcavities on the base of one-dimensional photonic crystals", J. Opt. Sci. Am. B 19, 2129 (2002). 5. D.G. Gusev, I.V. Soboleva, M.G. Martemyanov, T.V. Dolgova, A.A. Fedyanin, and O.A. Aktsipetrov, "Enhanced second-harmonic generation in coupled microcavities based on all-silicon photonic crystals", Phys. Rev. B 68, 233303 (2003).

6. T.V. Dolgova, V.G. Avramenko, A.A. Nikulin, D. Schuhmacher, G. Marowsky, A.F. Pudonin, A.A. Fedyanin and O.A. Aktsipetrov, "Secondharmonic spectroscopy of electronic structure of Si/SiO2 multiple quantum wells", Appl. Phys. B 74, 671 (2002). 7. T.V. Dolgova, A.A. Fedyanin, O.A. Aktsipetrov, D. Schuhmacher, G. Marowsky, "Optical second-harmonic interferometric spectroscopy of Si(111)SiO2 interface in the vicinity of E2 critical points", Phys. Rev. B 66, 033305 (2002). 8. O.A. Aktsipetrov, T.V. Dolgova, A.A. Fedyanin, D. Schuhmacher, and G.Marowsky, "Optical second-harmonic phase spectroscopy of the Si(111)SiO2 interface", Thin Solid Films 364, 91 (2000). 9. T.V. Dolgova, D. Schuhmacher, G. Marowsky, A.A. Fedyanin and O.A. Aktsipetrov, "Second-harmonic interferometric spectroscopy of buried interfaces of Column IV semiconductors", Appl. Phys. B 74, 653 (2002). 10. D. Schuhmacher, G. Marowsky, A.A. Fedyanin, T.V. Dolgova, O.A. Aktsipetrov, "Probe of the vicinal Si(111) surface by second harmonic phase spectroscopy", Materials Science in Semiconductor Processing 4, 51 (2001). 11. T.V. Dolgova, A.A. Fedyanin, and O.A. Aktsipetrov, "dc-electric-fieldinduced second-harmonic interferometry of the Si(111)-SiO2 interface in CrSiO2-Si MOS capacitor", Phys. Rev. B 68, 73307 (2003). Y Ijb \aZbfh^_ckl\bb we_dljhfZ]gblghc \hegu k ijhk\_leyxsbfb lhgdhkehc gufb kljmdlmjZfb ihy\ey_lky hljZ`zgguc kb]gZe k\yaZgguc k baf_g_gb_f \h \j_ f_gb iZjZf_ljh\Zfieblm^u bbeb nZau iZ^Zxs_c \hegu Ihemq_gh ZgZeblbq_kdh_ \ujZ`_gb_ hibku\Zxs__ hljZ`zgguc kb]gZe >Zggh_ y\e_gb_ fh`_l [ulv bkihev ah\Zgh ^ey ZgZebaZ \j_f_gghc aZ\bkbfhklb iZjZf_ljh\ iZ^Zxsbo gZ kljmdlmjm bf imevkh\

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31

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JZkkfhljbf [he__ ^_lZevgh hibkZggh_ y\e_gb_ g_klZpbhgZjgh]h hl jZ`_gby gZ ijbf_j_ ijhkl_cr_c lhgdhkehcghc kljmdlmju khklhys_c ba h^ghc q_l\_jlv\hegh\hc iezgdb k ihdZaZl_e_f ij_ehfe_gby n gZg_kzgghc gZih^eh`dmkihdZaZl_e_fij_ehfe_gbyns>eybaemq_gbyk^ebghc\hegu , iZ^Zxs_]h ba kj_^u k ihdZaZl_e_f ij_ehfe_gby n dhwnnbpb_gl hljZ `_gbydZdba\_klghaZibku\Z_lky\\b^_ r= r0 + r exp(-i20 ) 1 , 1 + rr exp(-i20 ) 01 (1)

JZkkfhljbf \aZbfh^_ckl\b_ k iezgdhc ]Zmkkh\Z bfimevkZ \ \b^_ A(t)=exp[2 (t/)2/2] k d\Z^jZlbqghc fh^meypb_c nZau (t)=-t ]^_ - kdhjhklv qZk lhlghc fh^meypbb GZ jbk ijb\_^_gu j_amevlZlu jZkq_lZ ki_dljZevguf f_lh^hf nhjfu hljZ`zggh]h bfimevkZ hl iezgdb gZg_kzgghc gZ ih^eh` dm k n s ijb ^ebl_evghklb bfimevkZ = 5T b ^ebg_ \hegu g_km s_c fdf \_ebqbgZ 2
]^_ r0 = (n0 n)/(n0 + n b r1 = (n ns)/(n + ns) dhwnnbpb_glu hljZ`_gby hl ]jZgbp jZa^_eZ \g_rgyy kj_^Z-iezgdZ b iezgdZ- ih^eh`dZ khhl\_lkl \_ggh 0 = 2nd/ - nZah\Zy lhesbgZ iezgdb d _z ]_hf_ljbq_kdZy lhe sbgZ n ihdZaZl_ev ij_ehfe_gby iezgdb ;m^_f kqblZlv Zfieblm^m iZ ^Zxs_c \hegu ghjfbjh\Zgghc gZ fZdkbfZevgmx Zfieblm^m Mqblu\Zy qlh \ujZ`_gb_ aZibkZgh \ \b^_ kmffu ^\mo \heg hljZ`zgguo hl iezg db b ih^eh`db k jZaghklvx nZa f_`^m gbfb0=0Ёt, ]^_ 0 kj_^gyy qZklhlZ \hegug_kmsZy b Ёt=2nd/c -- \j_fy aZ^_j`db \lhjhc \hegu hl ghkbl_evgh i_j\hc h[h[sbf _]h ^ey \hegu k baf_gyxsbfbky Zfieblm ^hcbnZahcke_^mxsbfh[jZahf
E 0 (t ) = r0 A(t ) exp [- i(t ) ] + r1 A(t - t ) exp [- i 0 t - i(t - t ) ] . 1 + r0 r1 exp [- i 0 t + i(t ) ] ( 2)

2

n =1

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[2] V. K. Kornev, A. V. Arzumanov, IEEE Trans. Appl. Supercond., vol. 9, pp. 4562-4565, June 1999. [3] V. K. Kornev, A. V. Arzumanov, and N. A. Shcherbakov, Inst. Physics Conf. Ser. No 167, IOP Publishing Ltd, pp. 753-756, 2000. [4] Ryazanov V. V., Oboznov V A, Veretennikov A V, Rusanov A Yu 2001 Phys. Rev. B 65 020501-1-4. [5] Kornev V K, Klenov N V, Ryazanov V V, Oboznov V A, Timofeev A V, and Bol'ginov V V Extended Abstracts of Int. Supercond. Electronics Conf. (ISEC'2003), Owe4, Sydney, Australia, 7-11 July 2003.

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t - , PL

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Kibkhdebl_jZlmju 1. A. Bogdan, M. Kulmala, N. Avramenko, Phys. Rev. Lett. 17, 1042 (1998). 2. C. L. Jackson, G. B. McKenna, J. Chem. Phys. 93, 9002 (1990). 3. A. Iverson, J. Phys. Chem. 68, 515 (1964).

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< ^hdeZ^_ jZkkfZljb\Z_lky kbevgZy koh^bfhklv rj_^bg]_jh\kdhc w\hexpbb d d\Zglh\hc klhoZklbd_ ^ey deZkkZ ]ZfbevlhgbZgh\ \dexqZx s_]h fh^_ev d\Zglh\h]h ^_l_dlhjZ ]jZ\blZpbhgguo \heg >hdZaZgh qlh kbevguf j_ahev\_glguf ij_^_ehf ihke_^h\Zl_evghklb kZfhkhijy`_gguo ]ZfbevlhgbZgh\ y\ey_lky kbff_ljbqgZy djZ_\Zy aZ^ZqZ \ nhdh\kdhf ijh kljZgkl\_>-@ dhlhjZy mgblZjgh wd\b\Ze_glgZ d\Zglh\hfm klhoZklbq_ kdhfm^bnn_j_gpbZevghfmmjZ\g_gbx>@bqlh^eyij_^_evghcw\hexpbb qZklbqgh]h ke_^Z ih kf_rZgghfm khklhygbx g_ kms_kl\m_l _^bgh]h mjZ\ g_gby lbiZ mjZ\g_gby Ebg^[eZ^Z GZijhlb\ dZ`^Zy dhfihg_glZ kf_rZg gh]hkhklhygby\ij_^_e_ihjh`^Z_lkh[kl\_ggucaZdhgw\hexpbb GZqbgZy k wdki_jbf_glh\ J_a_jnhj^Z ih jZkk_ygbx .-qZklbp gZ lhg dhc ahehlhc nhev]_ \ d\Zglh\uo baf_j_gbyo bkihevamxlky fZdjhkdhibq_ kdb_ imqdb d\Zglh\uo qZklbp \aZbfh^_ckl\mxsb_ k fbdjhkdhibq_kdbfb d\Zglh\ufb kbkl_fZfb lZdbf h[jZahf qlh iZjZf_lju imqdZ ^hklmigu_ ^ey baf_j_gby k ihfhsvx deZkkbq_kdbo ijb[hjh\ ^h b ihke_ jZkk_ygby kh^_j`ZlbgnhjfZpbxhkhklhygbbd\Zglh\hckbkl_fuebgZ hilbq_kdh]h imlb nhlhgh\ gZ fgh]h [hevr_ h[eZklb \aZbfh^_ckl\by k d\Zglh\hc kbkl_fhc ihwlhfm lZ dh_ \aZbfh^_ckl\b_ fh`_l [ulv hibkZgh dZd baf_g_gb_ nZau b Zfieblm^u bkoh^gh]h imqdZ ijhbkoh^ys__ \ ij_^_e_ \ h^ghc lhqd_ e_`Zs_c a_jdZe_ bgl_jn_jhf_ljZ mklZgh\e_gghf gZ _]h hilbq_kdhf imlb LZdh_ ij_^_ev gh_ \aZbfh^_ckl\b_ y\ey_lky kbg]meyjguf b \ ij_^klZ\e_gbb \lhjbqgh]h d\Zglh\Zgby hibku\Z_lky nmgdpbhgZeZfb hi_jZlhjh\ jh`^_gby b mgbqlh `_gbyhl^_evlZ-nmgdpbc>bjZdZ Imklva+( xba ( x ) -iehlghklbhi_jZlhjh\jh`^_gbybmgbqlh`_ gby ^_ckl\mxsbo \ kbff_ljbqghf nhdh\kdhf ijhkljZgkl\_ =S(L2(R)) b m^h\e_l\hjyxsb_ dZghgbq_kdbfdhffmlZpbhggufkhhlghr_gbyfb+bb hi_jZlhju jh`^_gby b mgbqlh`_gby ^_ckl\mxsb_ \ l2 , - ]_g_jZlhj ]jmiiuk^\b]h\\=S(L2( R )) : exp{-ct } ( v( x ) )= ( v( x + ct ) ), ( v ) wdkihg_gpbZevguc \_dlhj v L2( R ), f( t ) \g_rgyy kbeZ b , g iZjZf_ljubaf_jbl_evgh]hmkljhckl\Zc, «-nbabq_kdb_dhgklZglu L_hj_fZ (i) Kbevguc j_ahev\_glguc ij_^_e ihke_^h\Zl_evghklb kZfhkhijy`_gguohi_jZlhjh\ H. = «{ b+ b - ic + g ( b + b+ ) / 2 (f( t ) + R dx . (x) a+( x ) a ( x ))} ijb. (x) : (x), . :jZ\_gaZfudZgbxhi_jZlhjZ

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Ih^k_dpbyL_hj_lbq_kdZybfZl_fZlbq_kdZynbabdZ C t = «{ b b+ - ic + g ( b + b+ ) / 2 f( t )},

97

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kbff_ljbqgh]h gZ h[eZklb hij_^_e_gby khklhys_c ba we_f_glh\ijbgZ^ e_`Zsbo DW =S(W21( R \ {0} ) ( I + b+ b)-1 l2 m^h\e_l\hjyxsbo djZ_ \hfmmkeh\bx A = W A+ , W = exp { ig ( b + b+ ) / 2 }, (1)

]^_ A ± ( v ) = v( ±0 ) ( v ) ^eyex[h]h v W21( R \ {0} ). ( ii ) Kms_kl\m_l k_f_ckl\h mgblZjguo hi_jZlhjh\ Ut = exp {i [0,t] Cs ds } (ijZ\uc dhpbde), ihjh`^Z_fh_ k_f_ckl\hf kZfhkhijy`_gguo hi_jZlhjh\ Ct, dhlhjZy mgblZjgh wd\b\Ze_glgZ jZaj_rZxs_fm hi_jZ lhjmd\Zglh\h]hklhoZklbq_kdh]h^bnn_j_gpbZevgh]hmjZ\g_gby d ut = ut {( W ,GW b b+ - ic + g ( b + b+ ) / 2 f( t ) ) dt }, (2) k ijhp_kkhf qbkeZ qZklbp (T) = T dt a+( ct ) a( ct ). MgblZjgZy wd\b\Z e_glghklvmklZgZ\eb\Z_lkyhi_jZlhjhfk^\b]Z ut = exp{-ct }Ut. DjZ_\h_ mkeh\b_ hibku\Z_l kdZqhd nZau dh]_j_glgh]h baemq_gby gZ a_jdZe_ bgl_jn_jhf_ljZ Hlf_lbf qlh mjZ\g_gb_ Rj_^bg]_jZ k ]Zfbev lhgbZghf H . y\ey_lky lhqgh j_rZ_fuf qlh iha\hey_l i_j_clb d ij_^_em y\guf h[jZahf b ihkljhblv k_f_ckl\Z hi_jZlhjh\ ut b Ut Z lZd`_ \u\_klb d\Zglh\h_dbg_lbq_kdh_mjZ\g_gb_khhl\_lkl\mxs__mjZ\g_gbx L_hj_fZ 2. Ij_^iheh`bf qlh gZqZevgh_ khklhygb_ we_dljhfZ] gblgh]h ihey bf__l \b^ kf_rZggh]h khklhygby = n n, n = rn | ( vn ) ( vn ) |, Tr = 1, b imklv Ut \\_^_ggh_ \ur_ k_f_ckl\h mgblZjguo hi_jZlhjh\Lh]^Zkj_^g__ihkhklhygbxihey Pt(B) = Tr ={( I) Ut* ( I B) Ut} = = n rn Tr ={(n I) Ut* ( I B) Ut}} = n rn Pt(n)(B) y\ey_lkykmi_jihabpb_cj_r_gbcd\Zglh\uomjZ\g_gbcEbm\beey t Pt(n)(B) = P
(n) t

Ebl_jZlmjZ 1. Q_[hlZj_\ :F Qlh lZdh_ d\Zglh\h_ klhoZklbq_kdh_ mjZ\g_gb_ k lhqdb aj_gby nmgdpbhgZevgh]h ZgZebaZ FZl_f AZf_ldb l71, \ui k--469. 2. Q_[hlZj_\ :F., Ju`Zdh\ =< H kbevghc j_ahev\_glghc koh^b fhklb rj_^bg]_jh\kdhc w\hexpbb d d\Zglh\hc klhoZklbd_ FZl_f AZ f_ldbl74,\uik--781. 3. Parthasarathy K. R. An Introduction to Quantum

BGL?=J:EVGH?J:KIJ?>?E?GB?I:J?LH >EYB?J:JOBQ?KDHCFH>?EBWDHGHFBDB BF??LIHD:A:L?EV Ijhn_kkhjQ_[hlZj_\:Fklm^_gl=jbgxd>X Ohly jZkij_^_e_gby IZj_lh b]jZxl nmg^Zf_glZevgmx jhev \ wdhgh fbd_ b wdhghnbabd_>-@ h[s_ijbgylu_ fh^_eb hibku\Zxsb_ nhjfbjh \Zgby wlh]h jZkij_^_e_gby hlkmlkl\mxl < ^hdeZ^_ ihdZaZgh qlh bgl_ ]jZevgh_ jZkij_^_e_gb_ IZj_lh k ihdZaZl_e_f \ h[eZklb \ukhdbo ^hoh ^h\ y\ey_lky mgb\_jkZevguf jZkij_^_e_gb_f ^ey deZkkZ b_jZjobq_kdbo fh^_e_c wdhghfbdb kЁijhp_glgufґ \aZbfh^_ckl\b_f Ih^ k\hckl\hf b_ jZjobqghklbihgbfZ_lkyhj]ZgbaZpbylh\Zjhh[f_gZij b dhlh jhclh\Zjubmkem]b^\b]Zxlkyk\_jom\gba\gZijZ\e_gbbm[u\Zgby^hoh ^h\ Z ^_gv]b i_j_f_sZxlky \ ijhlb\hiheh`ghf gZijZ\e_gbb ЁIjhp_gl gucґ oZjZdl_j \aZbfh^_ckl\by wdhghfbq_kdbo km[t_dlh\ khklhbl \ lhf qlh baf_g_gb_ iZjZf_ljh\ oZjZdl_jbamxsbo nbgZgkh\h_ khklhygb_ km[t _dlZ ijhihjpbhgZevgh agZq_gbyf wlbo iZjZf_ljh\ Mgb\_jkZevghklv ha gZqZ_l g_h`b^Zggmx g_aZ\bkbfhklv ihdZaZl_ey hl dhgdj_lgh]h \b^Z Ёijhp_glgh]hґ\aZbfh^_ckl\by. Imklv n(S) iehlghklv wdhghfbq_kdbo km[t_dlh\ jZkiheZ]Zxsbo ^hoh^hf S \ _^bgbpm \j_f_gb b imklv R(S'/S) ^hey jZkoh^h\ wdhghfbq_ kdh]h km[t_dlZ k ^hoh^hf S ihklmiZxsZy km[t_dlm k ^hoh^hf S' > S gZ au\Z_fZy nmgdpb_c ij_^ihql_gby Ijb nbdkbjh\Zgghf kh\hdmighf ^hoh ^_ ^hoh^ S ijboh^ysbcky gZ h^gh]h mqZklgbdZ jugdZ h[jZlgh ijhihj pbhgZevgZ iehlghklb n(S mqZklgbdh\ jugdZ k ^Zgguf mjh\g_f ^hoh^h\ Ihwlhfm ke_^kl\b_f knhjfmebjh\Zgguo \ur_ ij_^iheh`_gbc y\ey_lky bgl_]jZevgh_mjZ\g_gb_hlghkbl_evghiehlghklbn(S): S = n(S)-1
(0,S]

( Lt

(n)

( B ) ),

]_g_jZlhjudhlhjuo\nhjf_Ebg^[eZ^Zbf_xl\b^ Lt
(n)

(B)= Gt
(n)

(n) t

( B ) - (Gt(n))*B B G

(n) t ,

]^_

(n) t

( B ) =|vn( t )|2 W* B W,

= i ( b b+ + g ( b + b+ ) / 2 f( t ) ) - |vn( t )|2 / 2,

n(S') R(S/S') S' dS' S

(S,

R(S'/S) dS',

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]^_i_j\h_keZ]Z_fh_\ijZ\hcqZklbhl\_qZ_lkmffZjgufihklmie_gbyfhl km[t_dlh\ k f_gvrbf ^hoh^hf Z \lhjh_ keZ]Z_fh_ khhl\_lkl\m_l jZkoh^Zf gZ kh^_j`Zgb_ЁklZjrboґ iZjlg_jh\ I_j_oh^y d nmgdpbb F(S) = S n(S), hibku\Zxs_ciehlghklvkh\hdmigh]h^hoh^Z\lhqd_]^_bg^b\b^mZevguc ^hoh^ jZ\_g S b ^_eZy aZf_gu i_j_f_gguo bgl_]jbjh\Zgby S' : x = S/S', S' : y = S'/S \ i_j\hf b \lhjhf bgl_]jZeZo khhl\_lkl\_ggh ihemqZ_f eb g_cgh_h^ghjh^gh_bgl_]jZevgh_mjZ\g_gb_ F(S) = (L F)(S) = S
[1,

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min [Smin,S]

F(S/x) R(x) x-2 dx - S F(S)

[1,

R(y) dy.

(1)

;m^_fbkdZlvj_r_gb_mjZ\g_gby\\b^_jy^Z F(S) = S q
n

Fn S

n

k g_ba\_klgufb dhwnnbpb_glZfb Fn b iZjZf_ljhf q Ih^klZ\eyy wlh jZa eh`_gb_\mjZ\g_gb_ihemqZ_flh`^_kl\h S
q n

Fn S

n

(1 + S m0) = S
q

n

Fn S

n+1

n(S') R(S/S') S' dS' S

[S,

R(S'/S) dS',

(4)

mq+n-2 .
q q-1

(2) = S0(Smin) S
min [Smin,

IjbjZ\gb\Zy dhwnnbpb_glu ijb kl_gi_gyo S , S , S b l^ ihemqZ_f p_ihqdm mjZ\g_gbc i_j\h_ ba dhlhjuo hij_^_ey_l agZq_gb_ ih dZaZl_ey q Z hklZevgu_ mjZ\g_gby h[jZamxl j_dmjj_glgmx kbkl_fm hlgh kbl_evghFn :
q+1

R(S'/Smin) dS' = s0 (Smin)2 m0,

Z gh\h_ mjZ\g_gb_ hlghkbl_evgh n(S [m^_l bf_lv g_kdhevdh [he__ keh` guc\b^ihkjZ\g_gbxkmjZ\g_gb_f F(S) (1 S0(S) / S + S [1, R(y) dy ) = S [1,S / Smin) F(S/x) R(x) x-2 dx , (5) ]^_ s0 = S [Smin,
min

m0 = mq-2,

F0 + m0 F1 = m1 F1, . . . . , Fn + m0 F

n+1

= mn+1 F

n+1

, ....

+ (S

min

)2 m0. I_j_ibr_f mjZ\g_gb_ gZ ihemijyfhc S F(S/x) R(x) x-2 dx = 0. (6)

Ihkdhevdm ba hij_^_e_gby fhf_glh\ mn ke_^m_l qlh mn > mk ^ey ex[uo\_s_kl\_gguon > klhbamkeh\bym0 = mq-2ke_^m_lqlhq Zhk lZevgu_mjZ\g_gbyiha\heyxlhij_^_eblvagZq_gbyFn: Fn = F 0
1 k n

F(S) = L F(S) + ( S0(S) / S 1 ) F(S) - S

[S / Smin,

(mk - m0)-1.

Ke_^h\Zl_evgh kms_kl\m_l j_r_gb_ mjZ\g_gby ^ey iehlghklb jZkij_^_e_gbyqbke_gghklbgZk_e_gby\aZ\bkbfhklbhl\_ebqbgu^hoh^Z n(S) = F0 S ( 1 +
3 n

Ij_^iheh`bf qlh nmgdpby R m[u\Z_l ^hklZlhqgh [ukljhgZijb f_j wdkihg_gpbZevgh lZd qlh[S, R(x) xn dx = O(S- ijb [hevrbo agZ q_gbyo S ^ey ex[h]h n b [m^_f bkdZlv j_r_gb_ mjZ\g_gby ijb [hev rboS\\b^_ n(S) = sa (1 s )b r(s), s = S
min

S

n

1 NQ(mk

- m0)

-1

/ S,

).

(3)

Ihwlhfm ]eZ\guc qe_g jZkij_^_e_gby N(S) = [S, n(s) ds m[u\Z_l dZd O(S 2) g_aZ\bkbfhhl\b^Znmgdpbbij_^ihql_gby R.

C(S) = S S0(S) + S2 = S2 ( 1 s) c(s), S -q
n

Fn S

n

(1 + S m0) = S -q

n

Fn S

n+1

(mq+n-2 + O(S-)).

Ih^k_dpbyL_hj_lbq_kdZybfZl_fZlbq_kdZynbabdZ

101

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IjbjZ\gb\Zydhwnnbpb_gluijbh^bgZdh\uokl_i_gyoS q+1, S q, S q b l^ ^ey Zkbfilhlbq_kdh]h j_r_gby mjZ\g_gby fu ihemqZ_f lm `_ p_ihqdm mjZ\g_gbc b lhl `_ \b^ j_r_gby \ h[eZklb [hevrbo ^hoh^h\ qlh b \ kemqZ_ s0 A =eZ\guc qe_g Zkbfilhlbdb \ h[eZklb \ukhdbo ^hoh ^h\ g_ baf_gblky _keb[S, R(x) xn dx = O(S-2 ^ey \k_o n JZkkfhljbf ih\_^_gb_ jZkij_^_e_gby n(S \ hdj_klghklb lhqdb S = Smin Ij_^iheh`bf, qlh
1

S0(S) - s0 = kCk (S - S

min

)k , C1 5[\ 51 +

k+j>0

Rkj x ky j(7)

b[m^_fbkdZlvZkbfilhlbdm]eZ\gh]hqe_gZmjZ\g_gby\\b^_ n(S) = (S - Smin) d k (S - Smin) k Nk. (8) L_hj_fZ Imklv [S, R(x) xn dx = O(S-2) b \uiheg_gu mkeh\by L_h j_fu b ij_^iheh`_gby (7). Lh]^Z ^ey ex[uo iheh`bl_evguo agZq_gbc iZjZf_ljh\ R1, Smin, m0 b -C1 \ deZkk_ iehlghkl_c \_jhylghklb gZ (Smin, kms_kl\m_l j_r_gb_ ki_dljZevghc aZ^Zqb (6), ZkbfilhlbdZ dhlhjh]h bf__l \b^ \ h[eZklb \ukhdbo ^hoh^h\ b bgl_]jbjm_fmx kbg]meyjghklv H(S - Smin) db n(S) = N0 (S - Smin) d + O(S - Smin)
d+1

< ^Zgghc jZ[hl_ jZkkfhlj_gZ aZ^ZqZ ihkljh_gby dhffmlZlb\guo ]Z fbevlhgh\uob_jZjobc>@w\hexpbhgguomjZ\g_gbcZkkhpbbjh\Zgguok ]bi_j[hebq_kdbfb kbkl_fZfb mjZ\g_gbc Wce_jZ-EZ]jZg`Z \ qZklghklb ^\mf_jgufbkbkl_fZfbLh^u>@ 1. @ JZkkfhljbfr-f_jgmx ]Zfbevlhgh\m ^bgZfbq_kdmx kbkl_fm k aZ\bkbfufb i_j_f_ggufb ui bfimevkZfb mj ijhkljZgkl\_ggufb dhhj^b gZlZfbxb\j_f_g_ftaZ^Zggmxkdh[dZfbImZkkhgZ {ui(x,t),mj(x',t)}A=Aij (x-x'), (1) {ui,uj}A={mi,mj}A=0, ]^_A wlhrвr)-fZljbqguchi_jZlhj\iheguoijhba\h^guoihxZkdh[dZ {·,·}A y\ey_lky ^bnn_j_gpbjh\Zgb_f ih dZ`^hfm ba Zj]mf_glh\ Imklv H=[H dx] ]ZfbevlhgbZg k iehlghklvx H(x)=H(u(x),u(x);m(x),Dm(x ]Z fbevlhgh\Z ^bgZfbdZ ut={u(x), H(u(x'), m(x'))}A, mt={m(x),H(u(x'),m(x'))}A i_j_f_gguo u b m ihemqZ_lky ba fgh]hdjZlguf bgl_]jbjh\Zgb_f bg l_]jbjh\Zgb_fihqZklyfut=A(H/m(x)), mt=-(H/u(x)). 2. JZkkfhljbf eZ]jZg`bZg L=[L(u,ux,uy;x,y) dx dy@ i_j\h]h ihjy^dZ k iehlghklvx L=Ѕi,ji j uix ujx+H(u;x,y ]^_ g_\ujh`^_ggZy ihklhyggZy kbff_ljbqgZyrвr)-fZljbpZ Ke_^my>@\u[_j_f \dZq_kl\_\j_f_gbg_aZ \bkbfmx i_j_f_ggmx y b q_j_a mj=L/ujy h[hagZqbf bfimevk khhl\_lkl \mxsbc j-c aZ\bkbfhc i_j_f_gghc uj ijb dZ`^hf j[1,r@ >bnn_j_gpbZev guc oZjZdl_j k\yab mi = Ѕ i,ji j ujx y\ey_lky hkgh\guf bgkljmf_glhf \ ihkljh_gbb ]Zfbevlhgh\uo kljmdlmj ^ey ]bi_j[hebq_kdh]h mjZ\g_gby Wc e_jZ-EZ]jZg`Z ={L/u ` Ihevamykv khhlghr_gb_f u=-2-1Dx-1(m), ij_^klZ\bf aZ^Zggmx ij_h[jZah\Zgb_f E_`Zg^jZ H dx dy = m, L/uy L iehlghklv H(u;x,y \ \b^_ kmffu ^\mo jZ\guo keZ]Z_fuo H=ЅH[u]+ЅH[m@ \ wlhf kemqZ_ mjZ\g_gb_ ={L/u ` wd\b\Ze_glgh ]Z fbevlhgh\uf mjZ\g_gbyf uy = H/m, my = - H/u hlghkbl_evgh dZghgbq_ kdhc]Zfbevlhgh\hckljmdlmjuA=1>bgZfbq_kdb_mjZ\g_gbyjZkiZ^Zxlky \kbemkhhlghr_gbyЅ /m = -1 Dx-1°/u: uy=A1°([H[u] dx])/u, my=-Ѕ1°([H[m] dx])/m, (2)

, d = -1 + R1 Smin / (1 + 2 Smin m0 C1) > -1 .

\h[eZklbfbgbfZevgh]h^hoh^ZS

min

MjZ\g_gb_ F(S) = F(S bf__l j_r_gb_ ZkbfilhlbdZ dhlhjh]h \[ebab lhqdb Smin bf__l \b^ H(S - Smin) d - , d > - Ihwlhfm lhqdZ y\ ey_lky fZdkbfZevguf agZq_gb_f iZjZf_ljZ ^himkdZxs_]h j_r_gby n(S)=H(S - Smin) d - - 1bf_xsb_kfukeiehlghkl_c\_jhylghklbijbex[uo iheh`bl_evguoagZq_gbyoiZjZf_ljh\R1, Smin, m0 b-C1. Ebl_jZlmjZ 1. Reed J.W. The Pareto law of incomes - an explanation and an extension// Physica A, 319 (2003), p.469 -486. 2. Sornette D., Zajdenweber D, Economic returns of research: the Pareto law and its implications// Eur. Phys. J., B 8, (1999), p.653-664. 3. Ehsbgbg F; AZdhg IZj_lh ihlj_[ghklv i_j_hldjulby MdjZbgkdbc`mjgZeЁWdhghfbklґNk-68.

HG?DHLHJUO=:FBEVLHGHU

]^_\aZbfghh[jZlgu_]Zfbevlhgh\uhi_jZlhjukmlvA1=-1 Dx-1b1= Dx . Ijbf_j ?keb r=1, ____ b ]ZfbevlhgbZg H ljb\bZe_g lh \aZ bfgh h[jZlgu_ ]Zfbevlhgh\u hi_jZlhju khhl\_lkl\mxsb_ \hegh\hfm mjZ\g_gbxsxy=0, kmlvA1=-1 Dx-1b1=Dx .

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Z T ebg_ZjbaZpby Hi_jZlhj R ihjh`^Z_l dhffmlZlb\gmx ]Zfbevlhgh\m b_jZjobx ={k=Rk(ux)|k` gzl_jh\uo kbff_ljbc mjZ\g_gbc Lh^u>@ ZgZeh]h\ ihl_gpbZevgh]h fh^bnbpbjh\Zggh]h mjZ\g_gby Dhjl_\_]Z-^_ NjbaZ bf__f A1=-1 Dx-1 b A2= °Dx-1° B_jZjobb b k\yaZgu hlh[jZ `_gb_f : kk+1 b ^himkdZxl h^gm b lm `_ ihke_^h\Zl_evghklv ]Zfbev lhgbZgh\b_jZjobbDhjl_\_]Z-^_NjbaZkemqZck [uejZkkfhlj_g\>@ :\lhju [eZ]h^Zjgu BKDjZkbevsbdm b :F<_j[h\_pdhfm aZ ih e_agu_ h[km`^_gby bkke_^h\Zgby :D \uihegyebkv ijb ih^^_j`d_ klb i_g^bb IjZ\bl_evkl\Z Jhkkbckdhc N_^_jZpbb b ]jZglZ INTAS YS 2001/233. Ebl_jZlmjZ. 1. Magri F. A simple model of the integrable equation // J. Math. Phys. 1978. 19, n.5. C. 1156-1162. 2. H\qbggbdh\ : < Kbkl_fu Lh^u Zkkhpbbjh\Zggu_ k Ze]_[jZfb Eb b W-Ze]_[ju \ g_dhlhjuo aZ^ZqZo fZl_fZlbq_kdhc nbabdb >bkk dnfg 1996. FF=Mk 3. Dbk_ez\ : < H gzl_jh\uo kbff_ljbyo mjZ\g_gbc Lh^u <_klgbd Fhkdh\kdh]h mgb\_jkbl_lZ K_j NbabdZ :kljhghfby
NHJFBJH<:GB?DHGLJ:KLGUOKLJMDLMJ KYSBFBG:=J:GBPMH;E:KLB :kkbkl_glG_^_evdhB< JZkkfhljbfaZ^Zqm
X - X W = I X [ [ W ' в +
X_' = J [ W + X [ = X [ [ '

(1) (2) (3)

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: Kms_kl\mxl nmgdpbb X [ b X [ ba C 2 ( D) lZdb_ qlh X [ < X [ , [ ' b \ h[eZklb = ^X [ X [ X X [ [ '` nmgdpby f (u, x,0) h[jZsZ_lky \ ghev lhevdh gZ ih\_joghklyo u = i ( x), i = 0,1,2, ijbq_f

ijb

xD:

X [ < [ < [ < [ < X [ ;

f u ( i ( x), x,0) > 0, i = 1,2,

f u ( 0 ( x), x,0) < 0; f (u, x, ) ^hklZlhqgh ]eZ^dZy nmgdpby \ h[eZklb 1 в [0, 0 ],

]^_ 1 g_dhlhjZy hldjulZy h[eZklv kh^_j`ZsZy Z 0 > 0 g_dhlhjh_ qbkeh <\_^_fnmgdpbx J ( x) =
2 ( x) 1 ( x )

: nhjfbjh\Zgb_ \ aZ^Zq_- g_klZpbhgZjghc DKLK Zkbfilhlbq_ kdb g_hlebqbfhcl_ h[eZ^Zxs_c l_fb `_ ij_^_evgufb k\hckl\Zfb ijb ijb^hklZlhqgh[hevrbo W hlDKLK u s ( x, ) . Hlf_lbf qlh jZg__ \hijhk nhjfbjh\Zgby fgh]hf_jguo dhgljZkl guo kljmdlmj [ue jZkkfhlj_g \>@ Fh`gh kdZaZlv qlh \ gZklhys_c jZ[h l_ ih^oh^ ba>@ jZkijhkljZg_g gZ kemqZc dh]^Z djb\Zy fh`_l bf_lv h[sb_lhqdbk]jZgbp_c ' h[eZklb ' <\_^_flj_[h\ZgbydgZqZevgufnmgdpbyf
X [ X [ & % ' ^Y [ & ' Y [ = J [ [ '` b u ( x) < u 0 ( x) < u ( x), x D.

f (u , x,0)du.

:.

: Imklv kms_kl\m_l djb\Zy ' \hafh`gh bf_xsZy h[sb_ lhqdb k ]jZgbp_c ' h[eZklb ' b lZdZy qlh - [ = [ Imklv ^Ze__ djb\Zy jZa^_ey_l h[eZklv D gZ ^\_ ih^h[eZklb ' + b ' - b imklv kms_ kl\mxl^hklZlhqgh]eZ^db_aZfdgmlu_djb\u_ & ' + b & ' - h]jZgb qb\Zxsb_ h^ghk\yagu_ h[eZklb ' ' + b ' ' - khhl\_lkl\_ggh lZ db_qlh
- [ > [ '
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J ( x) 0.

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-

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[ J [ [ [ ' .

Ba\_klgh>@ qlh ijb \uiheg_gbb mkeh\bc:-: ^ey ^hklZlhqgh fZeuo kms_kl\m_l klZpbhgZjgh_ j_r_gb_ u s ( x, ) aZ^Zqb m^h\e_ l\hjyxs__ij_^_evghfmkhhlghr_gbx
[ [ ' + (4) OLP X V [ = - [ [ ' Ba ke_^m_l qlh \ hdj_klghklb djb\hc ba mkeh\by: j_r_gb_ u s ( x, ) bf__l \gmlj_ggbc i_j_oh^guc kehc ihwlhfm ke_^my l_jfbgheh

I_j_^ l_f dZd knhjfmebjh\Zlv hkgh\ghc j_amevlZl ^Zgghc jZ[hlu \\_^_f _s_ h^gh lj_[h\Zgb_ K wlhc p_evx \u^_ebf \ h[eZklb ' - \k_ hl^_e_ggu_ ^jm] hl ^jm]Z fgh`_kl\Z b hl^_evgu_ lhqdb _keb lZdh\u_ bf_xlky ]^_ J ( x) 0 Z \ h[eZklb ' + \k_hl^_e_ggu_ ^jm] hl ^jm]Z fgh`_kl\Z b hl^_evgu_ lhqdb _keb lZdh\u_ bf_xlky ]^_ J ( x) 0. DeZkk \hafh`ghimklhcmdZaZgguowe_f_glh\h[hagZqbfq_j_a N . : Imklv eb[h deZkk N imkl eb[h hg kh^_j`bl dhg_qgh_ qbkeh we_f_glh\ ijbq_f ex[hc we_f_gl deZkkZ N g_ y\eyxsbcky lhqdhcij_^ klZ\ey_l ba k_[y eb[h ^hklZlhqgh ]eZ^dmx aZfdgmlmx djb\mx [_a kZfhi_ j_k_q_gbceb[hh[eZklvk^hklZlhqgh]eZ^dhc]jZgbp_c L_i_jv hkgh\ghc j_amevlZl ^Zgghc jZ[hlu fh`gh knhjfmebjh\Zlv \ \b^_ke_^mxs_cl_hj_fu L_hj_fZImklv\uiheg_gumkeh\by:-: Lh]^Z ijb ^hklZlhqgh fZeuo kms_kl\m_l _^bgkl\_ggh_ j_r_gb_ X [ W aZ^Zqb-b W >+ kijZ\_^eb\hij_^_evgh_khhlghr_gb_
[ [ ' OLP X [ W + _ OQ _ = [ [ '
+ -

(5)

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[5.] Egorov Yu. V., Kondratiev V. A. On blow-up solutions for parabolic equations of second order, in the book: Differential Equations, Asymptotic Analysis and Mathematical Physics, Editors: M. Demuth, B.-W. Shulxe, Akademie Verlag, Berlin, Serie: Mathmatical Research, 100, P. 77-84, 1997. [6.] Fblb^b_jb W Ihoh`Z_\ K B :ijbhjgu_ hp_gdb b hlkmlkl\b_ j_r_gbc ^bnn_j_gpbZevguo g_jZ\_gkl\ \ qZklguo ijhba\h^guo Ljm^u FB:G >@ Dhjimkh\ FH K\_rgbdh\ := H jZajmr_gbb aZ dhg_qgh_ \j_ fy j_r_gby gZqZevgh-djZ_\hc aZ^Zqb ^ey ihemebg_cgh]h mjZ\g_gby kh klZ\gh]hlbiZ@\uqbkefZl_fbfZl_fnbaL. 40. N 11. C. 16471654. >@ Dhjimkh\ FH D \hijhkm h jZajmr_gbb aZ dhg_qgh_ \j_fy j_ r_gby aZ^Zqb Dhrb ^ey ik_\^hiZjZ[hebq_kdh]h mjZ\g_gby $XBW )X >bnn_jmjZ\g_gbyL112. C. 1-6. >@Dhjimkh\ FH K\_rgbdh\ := H jZaj_rbfhklb kbevgh g_eb g_cgh]h mjZ\g_gby ik_\^hiZjZ[hebq_kdh]h lbiZ k ^\hcghc g_ebg_cghklvx @\uqbkefZl_fbfZl_fnbaL. 43. N 7. K. 944-962. >@Dhjimkh\ FH JZajmr_gb_ j_r_gby ik_\^hiZjZ[hebq_kdh]h mjZ\g_gby k ijhba\h^ghc ih \j_f_gb hl g_ebg_cgh]h weebilbq_kdh]h hi_ jZlhjZ@\uqbkefZl_fbfZl_fnbaL. 42. N 12. K. 1717- 1724.

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