【正文】 14120babababahhhbbbaaaa???????????????? ).0(,)( )( 03)2(2)(100 2)(1)2(0)(335 ??? ??? ???? ?? baebebb bebebeaF ????? 36 .,16,02123240421041201?????????????????bbaahhhbbbbaaaaa ).0(,)( )( 22)(212)()2(2)3(1236 ???? ????? ?? baebbb eebebaF ???? 37 .16,4,042240202211431201bahbhhbbbbaaaaaa?????????????????? .)2s in h (24.161643737442224)4(244)2(237???hFFhbaahebbeaF???????為成時(shí)，當(dāng) 38 .,4,011241102240214321?????????????????ababababhhhbbaaaa ).0(,)( )( 111)(0)3(21)()(10)2(2138 ??? ??? ?????? baaeaeab eeaaeaaF ????? 39 .16,16,4,022042022420240202043121bahbababbaahbhhbbaaaaa?????????????????? ).1616/())(16()2(304202420240)2(22)4(2220)2(22039???????????eahaahaahaebebaaeabaF 40 .,4,021412402242024021431aabbababababhhhbbaaa??????????????? ).0(,)( )( 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 ???? 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 in h (2.44434848443234)2(24)(4348???haFFhbaahebebaF??????? 成為時(shí)，當(dāng) 49 .)(4,)(,1,023234144332341102021142201abababhababaabhhbbbaaaaa????????????????? ???? .))c os h ()( s in h (.43494931422341)2(32)(2349????????????baFFaabbbabaeab eaF成為時(shí)，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(duì)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 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(duì) ? 求四階、一階、二階導(dǎo)數(shù) .由方程 (210)中的非線性項(xiàng) uu?? 和最高階導(dǎo)數(shù)項(xiàng) (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(xiàng)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： ,020200: 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 ) 1 2 ( )( ) , , ( 0 , 0 ) 。