【正文】 ( )( 422)(1)2(0)4(24 2)2(2)(10)2(240 ???? ???? ?? ???? baaeaeaeab aeaeaaeaF ??? ??? 41 .16,0 ,0240212 10431221 ?????? ????????? ??? hhhbbb bbaaaaaa ).0(,404)4(041 ??? babeaF? 42 .25,0 ,0240212 10432210 ?????? ????????? ?? hhhbbb bbaaaaaa ).0(,414)5(142 ?? ??? babeaF? 43 .,9,042112402110243210bbaahhhbbbbaaaaaa?????????????????? ).0(,)( )( 412)2(44)2()(2)(4143 ?????? babebb eebebaF ???? 紅河學(xué)院本科畢業(yè)論文（設(shè)計） 9 44 .36,9,042142202110431220bahbhhbbbbaaaaaa?????????????????? .3636)2(214)4(24)(4144?????? eahebebaF 45 .,16,02420240211043211bbaahhhbbbbaaaaa????????????????? ).0(,)()(222)2(42)2()2(24245 ???????? babebbeebbaF??? 46 .36,0 24022 110431210 ?????? ????????? ?? hhhbb bbbaaaaaa ).0(,424)6(246 ?? ??? babeaF? 47 .,9,0211421122402104320?????????????????ababababhhhbbbaaaa ).0(,)( )( 122)(1)3(11)()2(2)(1)(1247 ??? ??? ????????? baaeaeab eeaeaeaaF ?????? 48 .4,1,042342202110421210bahbhhbbbbaaaaaa?????????????????? .)s i n h (2.44434848443234)2(24)(4348???haFFhbaahebebaF??????? 成為時，當(dāng) 49 .)(4,)(,1,023234144332341102021142201abababhababaabhhbbbaaaaa????????????????? ???? .))c o s h ()( s i n h (.43494931422341)2(32)(2349????????????baFFaabbbabaeab eaF成為時，當(dāng) 50 .1,02324131403142304140242??????????????haabbaabba abba abbhhbaa /)( )(2)(10)(1)3(3350 ???? ???? ????? eaeaaeaeaaF).0() ) ,(( 43)(21)(0)2(13)4(4 ????? ??? baeaaeaeaaeb ???? 51 .36,9,042102244212032101abhahhbbbbbaaaaa?????????????????? ).0(,36 )36( 1414)()2(210)4(2451 ????? baba eebheaF??? 52 .25,0 24042 120322101 ?????? ???????? ??? hhhbb bbbaaaaaa ).0(, 141 )5(452 ?? ?? babeaF ? 53 .36,0 24042 110322101 ?????? ???????? ??? hhhbb bbbaaaaaa ).0(,242)6(453 ?? ?? babeaF? 第二 章 Benjamin Ono 方程的精確解 10 54 .,16,4,02002204222040244121311bbaabbaabaahhhbbbbaaa?????????????????? /)()2(0020)2(04)4(2454 ??? ????? ebabaebaebaF).( 0)2(22 bebb ?? 55 .9,9,18,0242444242242402210132101abhbbaabahhbbbbaaaaa??????????????????? ).0(,)()(44)2(2)4(44)4(4)2(2455 ???????? baebebbebebaF???? 56 .425,425,225,0242444141242402221032102abhbbaabahhbbbbaaaaa??????????????????? ).0(,)( )( 44)(1)4(44 )(1)4(4456 ???? ?? ?? baebebb ebebaF ?? ?? 57 .,16,4,01234121021130241240212bbaabbaabaahhhbbbbaaa??????????????????? .)( )(1)(21)()(1121)3(13)4(2357 bebb eebabaebaebaF ? ????????????? 58 .,9,00134240421221021bbaahhhbbbbaaaaa????????????? ???? ).0(,030)3(358 ?? babeaF? 59 .,16,0,0024201432404211021bbaabbaahhhbbbaaaa???????????????? ).0(,)( )( 04)2(2)(100)2(2)3(1)4(0459 ??? ??? ???? ?? baebebb ebebebaF ????? 60 .,25,0,02134240421021021?????????????????bbaahhhbbbbaaaaa ).0(,)()(23)(212)()3(2)4(1360 ??????????? baebbeebebaF???? 61 .,4 2 134240421 ? ??? ??????? b baahhhbaa )( )( )2(2)(10)(1)2(22)2(2)(1)2(0)3(1)4(2461 ???????????????????? ???? ebebbebebb ebebebebebaF).0(, 24 ?ba Benjamin Ono 方程的求解及對解的變換和分析 Benjamin Ono 方程的 一般解 為 求 (11)方程的 一般解 ，作 行波 變換 紅河學(xué)院本科畢業(yè)論文（設(shè)計） 11 ,),(),( btaxutxu ??? ?? (29) 其中 ,ab為待定常數(shù)，將 (29)式帶入 (11)式， 可將 (11)式的左端化為關(guān)于 ()u?的 ODE ,0))((2 222)4(4 ????????? uuuaubua ?? (210) 其中 (4),u u u? ? 分別表示 u 對 ? 求四階、一階、 二階導(dǎo)數(shù) .由方程 (210)中 的非線性項 uu?? 和最高階導(dǎo)數(shù)項 (4)u 齊次平衡，得 2n? .故可設(shè) 20 1 2( ) ( ) ( )u A A F A F? ? ?? ? ?, (211) 其中 0 1 2,A A A 為待定常數(shù)，且 2 0A? ， ()F? 滿足 廣義 Riccati方程 (25). 利用 (211) 式 與 (25)式可將 (210)的左邊化為關(guān)于 ()F? 的多項式 .令 ()F? 的各次冪的系數(shù)為 零，得到關(guān)于 0 1 2,A A A , ba, , 024,h h h 的代數(shù)方程組： ,020xx0: 422224246 ?? hAahAaF ?? ,02424: 421224145 ?? hAAahAaF ?? ,01 2 0612616: 42244224202421222224 ????? hhAahAbhAAahAahAaF ???? ,0418220: 4102221241242143 ???? hAAahAAahAbhhAaF ??? ,0724481612: 042422122222202222402222 ?????? hhAahAahAbhAAahAahAaF ?????,012212: 02122102041422142121 ????? hhAahAAahhAahAahAbF ???? ,02482: 0220202022402120 ???? hAbhAAahhAahAaF ??? 解上述代數(shù)方程組得到： .6,0,12 )4( 2 42144 22420 A haAha bhaAA ???? ????? (212)把 (212)式 代 入 (211)式可得：4 2 4 22444444( 4 ) 12 ( )( ) , , ( 0 , 0) 。xt? ? ? ? ? ?圖 (b)光滑孤立波 解 ,3,4,2,5:)( 2425 ????? hhbaAu ? 。b b x t?? ? ? ? ? ? ? ?圖 (c)緊孤 波解 ,5,1,4:)( 23 ???? baAu ? 2h ? ,6,7 4?h 。0 0 0 0 ,1010,10,4,3,2,1 4211 ???????????? txhhbbaa 圖 (e)周期波 解 ,3,4,1,8,10,1:)( 214223 ???????? kkhhAu ?? 。 圖 (g)周期 爆破 波解 ,1,5,8:)( 14226 ??????? ?? khhAu 2k ,2,3 0 ?? a 。? 圖 (i)周期 爆破 波