A remark on the stability of solitary waves for a 1 D Benney Luke equation
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(2) # ,Ò #E ~ #5Ws~ "sR J sà I > I 5R r F 5R Z?"p uc F uc > Í R Z 1 " p F ] : ~ U R s ] ~ \ > : d W 5 & 1 3 3 I P I * I # & 5 s δ 2 F (uc ) 5W < ~EeeZ ~##Í ~g5 J uOc?"p ZWb"p UOWb\5Ws5&I~]g O 9 ~T 5R PX# "p Xs 5 g35R R ×R~s g O ~X .]5& s ]:U~~Wå d X3u 5c& 5Wse "p: d 9 d(c) = F(uc ) , R I(# 3 I] s +& 2 & +{ F J 5Ws ]3f ~Z ~j P ~ A . %Xs 3 I~# L,\ eZF# ,dO s utt = uxx +. 1 un+1 xx + uxxtt . n+1. - 0. E 5L O 2 , E/5 ~ 5Ws# Zs~I] ~1 "]5& sO ~d "piI R IU~ d 2Ì á FI#5& s RF s]1~ > ~ δ F1E(ω ~c) ωc 5R #"p W R $ 5&R2 +p#NN2E 1L I Y ~z# Ú ωcJ+e£& s 5I 9 ~s I~/R 5R * ~g 5R 6?"p# %X: 7s ~ ; 5&I]S 9 9 I3 !~\]"p# ~ \5 [> 2 &+{ 2 +p# I~/R RO=3Í N 5R * A 9 5 R ~zd O5 I s N#,dO s I]P ~4Q£2Es JR 5&] R~ 4Ò, OR( * OB<5Ws "pWg - #0 3 #5&B4 2 I]ß 2EJR N1 RO R # 5 R S 5 R * VI ~] &"e sN/5 1> 2 9 &+{E #RE]"p ~435 R *å 5R X #p" #VW5 s3 ] I]P ~jI~/R M©A s ~jX % s3 I~#3, #Ò, O s R~r]& 2EI I]`3E£ Q 2E s JR 5& ] Q &p" ?> ~ /5 Rr~ VW5 s p" W9 #3 #5&X \ 2 ]p" OR ~W5 s# Z s~I] Ò, OR( IO 2 &s+ X] # (3~] 5;R{ 2 N1e3. s ~<I~/RO 5 R * 5R 9 X#p" #X A #d, O s >( ~I]ÌÒ 2 5/ &"= ]W5 ·§ ] 5 R R 5R #p" # ~ ~!O- R~0 > I]S ~ # 3Ì J +e> O5 rR~ 5R3YI 9 A #Ò, O s FIHJ* > I( I F KJ ~? W ON5 5R ?p" ; #F < ~!A % ]~&e'*§ ) p+ Õ p RÌs 5 5 R j~ R~ s 5á ] 5 R 4~ -/.0 0 < c < a/b < 1 p 0 < c < a/b < 1 9 R~ 2 3 ~ W , O Z W5 s ] ~/5 c>0 c > max (1, a/b) e I( SI F KJ C \2 p+ R RO ! 2EJR f ~ s~ /5 R W s Ì ( 2 &{+ I`F J 9 A W ~ O Z . E 9 ÕsdR M/W ID ~ 7 ?I]bX # 1D ~ #W5 s~ #5& s ]W\W5 ~ £ Q 2E s JR 3 5& ]Õ < ~ % ]~&e'*9 § ) p+ Õ#Ò, O s > Y~U ~ 3 5R EX #p" # I s R~g N ~? ~~sá 5/R a -/~.N0 5 s J +=?> O5 gR~ 5RL , ~ I #5& s i2 p+ ~ I~/R ; A A 9 RO ¿ 5 R * 1 ~j. s] i#5& s j] # eV ~£Ò 2 5/ s + 9 9. !. "$#&%`('Vý/ÿü)#~ü+*Zý/þ{ý_þ-,. , s]_F353 ~VQÚ2Es JR\35&]1>[_5Wse"p (_U~33T / 7s 2 Ig#,dO s ~ 7 ?I]Z53OR]p j"sR JR # > W Ò?> q = Φx r = Φt 9 -/Z.#0 5W 2 #X ~j &2 # ,ÒO s. -/.0. qt = r x rt = Aqx − rqx − 2rx q + brxxt ,.
(3) - . . . . . , OR 5 JRN _ Í]:U~~Y ~z Z5s > ~ A = I − a∂x2 9 B = I − b∂x2 O?p" 4 O _,ÒO s. -/.0. . r+B. −1. . . 1 2 q 2. = ∂x B −1 (Aq − rq). t. X ~ N /'*)lR{5R]pÍ#,dO si .s ~z 5& si "]5& sO S=. ~j ~)lR{5R]1JR 1"p( 1 L(Φ, r) = 2. Ú?GjI(\5W4. Z. R. Z. t1. t0. L(Φ, Φt )dt,. (rBr − Φx A(Φx ) + r(Φx )2 )dx.. ~5Ws! ]dj2 2 e 2 p=r+B. −1. . "{R JR . 1 2 q . 2. , 2P¿ ~#5T"sR JR #?>~ Q£2Es JR 5&]j .s #,ÒO s. e. H. q R = R rBpdx − L(q, r) p R = 12 R p − B −1 21 q 2 B p − B −1. 1 2 2q. . -/.0. X1"p. + qAq dx.. 5I1e/ .s XRY5/5 J s ~?X Oa ~_U~3 a"{R J sÌ ~_Q£2Es JR 1"p( ". δH Ú. j O. i ~z ~XOR~. . q p. . . =. − p − B −1 . B p. ∂x B −1 B p − B −1. ∂x B −1 − p − B −1. 1 2 2q. . 1 2 2q. − B −1 1 2 2q. . . q + Aq. 1 2 2q. . .. . -[0. = rx = qt .. q + Aq = ∂x B −1 (−qr + Aq) = pt .. W j2E #T Oj ~E%X]~&e'*)§ +pÕ#,ÒO s F#,d "{ (j ~ &2 / . 0 I /5 R~s 5/QÚ2Es JR .s 2 . q p. . = t. . 0 ∂x B −1 ∂x B −1 0. . δH. . q p. . .. h - 0. ~N! ~R5R3J s I("{R JR5W O ~£#,dO s 2 sa># & ~?78W ~W' ^&2 L]# ~ :d 5WF ~T . I]g5Ws "pW ,dORe * N (q, p) =. Z. B(p)qdx. R.
(4) h. W i,ÒOR( 5 5&JjI ~6 ~ 5R X#"p# G WW c > 0 Z#5/RY ~& 5/R Z 5LOR5 5& M/W R ~Y 5 sOR Ie3Ú B ~4 .]5& sO Ú j O ·#,d "{ (R ~T . I]g &2. F = H + cN 9. 2. . δF = 0 −[p − B −1 ( 21 q 2 )]q + Aq + cBp 0 . = 0 1 2 −1 + cBq B p−B 2q. ~4#5Ws~#,ÒO sa> p&. p = −cq + B −1. 1 2 2q. . .. N I]E ~j]&"= s·#,ÒO siI(P ~TU~3 ¿#,ÒO sa> Fp& −cBp = − p − B −1. W ~: 7s. 1 2 2q. . q + Aq = cq 2 + Aq.. −c −cBq + 12 q 2 = cq 2 + Aq.. % X ¿#,ÒO si ~52 . s·#,Ò "{ ( . 3 c2 Bq − Aq − cq 2 = 0 2. H 3 (bc2 − a)qxx − (c2 − 1)q + cq 2 = 0. - 0 2 I5W ß L 2 ! O R~ 2 > ~6 5/Rð]:U~~ 2 c>0 c > max(1, a/b) λ = A > R~ W Ò4 ~ .]5& s ]:U~~Wð( (bc2 − a)β 2 λ2 − 1 = c2 − 1 3cα = 1 9 η > ~ > 9 /UO# ~ #,ÒO s q(x) = αη(z) z = βx 5Li 7s. 1 λ2 η 00 + (1 − λ2 )η + η 2 = 0, 2. λ>1. O ~ I s. 2. 1 2. η(z) = 3(λ2 − 1) sech. "√ # ! λ2 − 1 z . λ. W ~: 7s1>Z#,dO s H O V 7s R~ 2 ~j 7 ?I] c>0 c > max(1, a/b) - 0 I s. A. I5W. c2 − 1 2 qc (x) = sech c. 1 2. "r. # ! c2 − 1 x . bc2 − a. > ~4 5 sOR Z I(· . . p = −cq + B −1 (q 2 /2) F qc qc . = φc = −cqc + B −1 (qc2 /2) pc. -[0.
(5) - . . . H. . " ! 2 ( s~WDID ~ I( \5& sa> J+e> O5àR~ 5R3ß# 5R' ~Wßp~5# V 5LS5LOR5 5& M/; #Ús 5A 5R .sA P5&J 3 _ /' &2P T3 #5&JQ£2Es JR\ 5&]1> IE ~5/ 5Ws ]WE~ C WP OY ~ # Í J+eÌ&Í_ $4I 5/ ,6 $ 5&?>9 I#5& ; ~j 5R W5 2E] 5 sæ 2 Rs]~i ?p" # ~XX O9 . δ F φc R (R, S)δ 2 F(φc ) = S Z cq + B −1 (qc (·))qc + A −qc + cB R (R, S) dx. −qc + cB B S R. W V2 s / ~ W . 5Lf 2E UO#. ~. Ú?4>. . R S. . . =. . ~j5R .s 2 s. I 0 B −1 (qc (·)) − c I. . R W. . ,. δ2 F Z T R R R 2 e dx, = (R, W )A δ F (φc ) D D W W W R. Ae =. . −L 0 0 B. . ,. . L = −(bc2 − a)∂xx + (c2 − 1 − 3cqc ).. /5 Ræ/ i"p OTM/\£R&Ipe"sI~Õ æ&Ip "]5& s Z ~å ~EOR~> 1OR{~GF JX ~XE OE ~L S I]1 å&Ip' (qc )x 9 .]5& s ~d "p &Ip("{I~ " 5 > ~Õ#3 J_ Z#5& 2 L s ~TZ 1L "pÚ/l Kd=XZs]~]W#X# 9 . 2 M/ß( 2 #5/ 2 c −1 c > 1 0:9 W ~:7 s\ g5 WE O 3 5& 6 1 "p:dW5 & . sY* I#5& s! 53 L Rz L e5&J W ~T* ]{W#¿ " WW] 2 ~j 5R ?p" #?]> O2 & O R~ Z WW IW5 Z 1 "på]:~U Z 5 sP f W O 2 ~d "pSI R 9 IA ~U Bd Ì 2 z I#5& s W Ò /5 R ~ fδ ~F 5 s J +eÌ&Í _ #35 R s 5 95 R *ð X 5R ?p" #7 s ; % ]~&e'*§) p+ z#Ò, O s X 9 .ü , #]þ$*Zý #(' Vý B , s]#5Ws{ M/P ~\2 # 53OR 2 X 5R *6\ 5R e R O= I]f ~ 5R r R ~ I~/RPOR T ~ %X]~&('*) +p#,dO s C W5 s ] E ~ 9 #,ÒO s. 5L ¿#,Ò "s eX. Φtt − Φxx + aΦxxxx − bΦxxtt = 0, (1 − b∂xx )Φtt = (1 − a∂xx )Φxx .. -10.
(6) . C EBW Oj 7sj z#,ÒO sb ~ :d 3z ,ÒOs\5 5E 7s 2 ~1 5L6j5Ws "pW If 2 1>] NI~]:U~ W T2 /RT O E - 5/Ræ~ 0 9 E E Z Wf s 5 I 5R R ~I~/RR &"p& Q &"p?> W4 ON ~ 9 Q 2E s JRf 5& ]4¿~s]1i # 5R 3 5R * £ C \W5 s ]z#Ò, O s I g. 52 X : .5W\9 2 ?"=I]iIb ~ Z1 "p - 10 I#5& sf Z WW > O c. t=τ. ~ 5Ws ] ". ,. Ψ(τ, z) = Φ(t, x) 9. . - 0. z = x − ct.. % §( J I]g Fp&X O. Φx = Ψz , Φxx = Ψzz Φt = Ψτ − cΨz , Φtt = Ψτ τ − 2cΨzτ + c2 Ψzz .. N I]E X4WT ~ #,ÒO s. -10. W\5WW . - [0 W U~33· &2E XÕ# 5R i 5R ~T "=Ja I s ~¿~1!2 & ~ N I] s ] <5R/ .s 2 7s]~ O&#,dOΨ =s 0 1"p# 9 -[0 O b 9 /UO# (1 − b∂zz )(Ψτ τ − 2cΨzτ + c2 Ψzz ) = (1 − a∂zz )Ψzz .. Ψ. b τ τ − 2ckiΨ b τ − c2 k 2 Ψ b + bk 2 Ψ b τ τ − 2bck 3 iΨ b τ − bc2 k 4 Ψ b = −k 2 (1 + ak 2 )Ψ b Ψ. s·#,Ò "{ ( . . b τ τ − 2cki(1 + bk 2 )Ψ b τ + 1 + ak 2 − c2 (1 + bk 2 ) k 2 Ψ b = 0. (1 + bk 2 )Ψ. s#,Ò~( . Ú?. b τ τ − 2ckiΨ bτ + Ψ. . 1 + ak 2 2 b = 0. − c k2 Ψ 1 + bk 2. -/.[0. W ~#,ÒO s 1T O b -/.[0 Ψ = u + iv 9 uτ τ + ivτ τ − 2cki(uτ + ivτ ) +. . 1 + ak 2 2 − c k 2 (u + iv) = 0, 1 + bk 2. 5La>I 2P· ~F/_R~ ~42 R1IOR OR ?>~ #5W 2 #V ~4 &2 uτ τ vτ τ. W+p. τ =k. q c2 −. uτ τ = k. 2. . 1 + ak 2 2 −c u = 0 + 2ckvτ + k 1 + bk 2 2 2 1 + ak 2 − 2ckuτ + k − c v = 0. 1 + bk 2. 1+ak 2 1+bk 2. 2. s. R~]:U~~. 1 + ak 2 c − 1 + bk 2 2. . . Uss ,. u(τ ) = U (s) vτ τ = k. 2. . . v(τ ) = V (s) 9 1 + ak 2 c − 1 + bk 2 2. W ~ . Vss ..
(7) - W ÒX 5Ws5&I~]4 O. U. R~. V. . . . . . 5 g ~4 &2. Uss + 2αVs − U = 0, Vss − 2αUs − V = 0,. -/.1.0. ~ > I5W R~ 2 \ OÌ( Ú 2 −1/2 α = c c2 − 1+ak a ≤ b c > 19 > 1 2 1+bk ]:~ U I] ~ >4WT OV3= &2 # 5W 2 # -/.1.0 W = Us Z = Vs . U W V Z W = U − 2αZ Z V + 2αW s. . . . -/. 0. P +~? 4> ~æRO=3 5R à 4 ~iM/Z I s ] &Z~ s ~ &Ipe"sI~#N ~ 5Ws 5Re5W \5& ( &2 3=5&JW ~BI~/R M© s , X5/ 1> ~4@Ò 5Ws JR 2 d 1"p( ". -/. 0:9. . 0 0 1 0. ~153OR5 5& 5Ì#,dO sf. 0 1 0 0 0 1 , 0 0 −2α 1 2α 0. -/. [0. λ4 + 2(2α2 − 1)λ2 + 1 = 0,. W #,dO siO X ~T . I] 3 λ2+ = 1 − 2α2 + 2. p. α2 (α2 − 1), λ2− = 1 − 2α2 − 2. p α2 (α2 − 1).. W ~PU~3 s "{ s`j OY EO?"pP O 2 > I5W sW?"p?> λ− < 0 α > 19 W 7 ? X · ( ~ 5 & I ] \ O λ2+ < 0 9 s 2 1 p 2 2 1 1 2 2 = α4 − α2 + > α2 (α2 − 1), α − > α (α − 1), α − 2 4 2. p 5L 2E #X O 2 λ+ = 1 − 2 α2 + α2 (α2 − 1) < 0. , ~4s ?> ~ &Ip("{I~#ß O?"pPM/å/BOR ?>_R~ / . [ 0 # Z #5&_ T ~¿&5& ]/R ~s 2i>[ ~V5 sP&~s]1\F#35R \ 5R ß ~R "eJ I sa> ] ~ 3e2E 5N 5R * R OBVO?"pN ~:d/' 9 W5 F ~jW5 s p" W~1 . s ~T &2 ]:U~~W ~j,ÒOs\5 5 -/.1.0 E.
(8) . . 7s 2 U V E W = Z . =. h -/. 0. 1 1 W 2 + Z2 − U2 + V 2 2 2 . T U −1 0 0 0 0 −1 0 0 1 V 0 1 0 2 W 0 Z 0 0 0 1. . U V W , Z. C ß s "p OT j,ÒOs\5 5 7s 2 TI~]:U~ > j5/Ræ~ £ ÍWf 9!A # 5 R 3 5R *á ~T "eJO5 I s ~j &2 W X2 /R O / 1 . . : 0 9 ~ß~1 2 & ~æg 7N s 5I 35R * B ~Y "=J I sð < ~ &2 W ~ J z ]p 5L`T W j]s &2 O Q£2Es JR 5& ]-/g.1.0:9 53r ?XÌ 4 ] ?p" g ~ ]# IW 5R *ð# Ì ~P "=J I s ~j &2 C j35 R ( 5Ws ] I]\ ~£~& 5&¿ "{R JR # > > R~ ~ -/.1.0:W5 9 s# s~ B ß4 eZ · a ~R2 2 ( 2 5Ws!*]d U V P1 "{R JR Ú jP 2OX ~jP i &2 O ~j. I]g£ Q 2E s JR 5& ] 9 -/.1.0 H dA = JδH1 (A) , -/. 0 ds ~ ~zR( 3e2E2 & 5Ì 5 s T R~ A = (U, V, P1 , P2 ). R~. H1. ¿ ~j .s 2. J . 0 0 J = 1 0. 0 −1 0 0 0 −1 , 0 0 0 1 0 0. 1 2 1 P12 + P22 + α − 1 U 2 + V 2 + αN1 , 2 2 Z&I] N1 (A) = V P1 − U P2 . H1 (A) =. , $ 5&?> " 2 . ~. H -/. 0. U 0 V 0 P1 = 1 P2 s 0. 0 −1 0 0 0 −1 0 0 0 1 0 0. ∂H2 ∂U ∂H2 ∂V ∂H2 ∂P1 ∂H2 ∂P2. Us = − (P1 + αV ) ,. . . 2 − ∂H ∂P1 ∂H2 − ∂P2 = ∂H2 , ∂U . Vs = − (P2 − αU ) , P1s = α2 − 1 U − αP2 , P2s = α2 − 1 V + αP1 .. ∂H2 ∂V. -/. [0 /- . 10 -/. 0 -/.[0.
(9) - A. "=I]E 7s. P1. I. -/. [ 0. R~§( J I] Ú?G2. . . . jO?"pj O P1 = −Us − αV,. - [ 0. (P1 )s = −Uss − αVs .. - .0. - [ 0. 35L I]#,ÒO s. R~i] I]. . P2. R~. - [ 0. - .0. 45Ws5&I~]4 O. α2 − 1 U − αP2 = −Uss − αVs ,. I( #,ÒO s. -/. 10. jU~~ O. Uss + 2αVs − U = 0.. ,E 2EJRX#. Vss − 2αUs − V = 0.. N I]#,ÒO s ' E"p S O RE5Ws "pW ,dORe # -/. [0 -/.[0 " X s ~j I s &2P 5Ws N 5#1,d~H 5W11>TO#"pj O ]:U~~W -/.1.0:9. e. Kα. 1 2 1 P12 + P22 + α − 1 U2 + V 2 2 2. - 0 ÕðS5Ws "pWD,dORe * sàR( 5 I sÕ R ~ &2 1 ~ -/.1.0:9 s 7s· = K (A(s )) s ∈ R9 W ~Kα2 (A(s)) 1 \2EZs 5αR(g 7/0 ]iP O\ .s > ]:U~~#áð~s 2 7s α > 1 Kα 7 sN 5Lb ~! "eJ I s &2 3e2E 5/ 5R , -/.1.0 9 R4 ~B s ?>e O?p" 7 s]~E O< ~X2 #53OR 2 35R * _ ~I~/RB &"p& FO 5 W s6 ~ :d W5 ¿ Ú Q 2E s JRZ 5& ] R~6 ~ :d 5W ¿ 1 "p W5 s p" W Ò, OR( Z ~s 2 53 ]&Z ~ s ~ Ò, OR( N1 OXVW5 s p" W 7 sX I sK α - &2 0 · / 1 . . : 0 9 C f Re # 5 R 2 f s 5 s 5 R * 4 ~f 5R #p" g. s ~\X X % ]~&('* ) p+ #Ò, O s 7 s Z WWb ?p" / 34. 2 0 _Ig ~XI~/R /5 c >¿e1EI~/R M© s ~·d 2 5/ Ie "= & C X] =W5 W-/.W 9 Ì #Ò, O s Rs]~ ~S 5R X #p" I s W5 S g ]s~1>X -/.0 ]& 2EI~ ~ :d=3 W5 N £ Q 2E s JRf3 5& ]z9 R~RO spsNW5 s p" W Ò, OR( # > R~ W ~j~U O_ RO=3l ]&Z ~s ~U ~I] !2 /R I] H̃ Ñ K̃ 9 7 s Z&I] W5 s p" Wfd, ORe * Ñ " 5I1e/. s X R W5 2E] 5 s9 2 Rs]~ ~i 5R Z #p" f ~?X δ H O Kα = H1 − αN1 =. 2. δ H. . q p. . =. . cq + B −1 (q(·))q + A −q −q B. . .. W ~i ~4I~/R M© s h Rs]~R( I s 1"p( - 0 . Q P. . t. =. . 0 ∂x B −1. ∂x B −1 0. . cqc + B −1 (qc (·))qc + A −qc −qc B. . Q P. . .
(10) -. . , 2P· ~j"{R JR # . Q P. . = τ. (τ, z). ]:U~~Wå(. . 0 ∂z B −1 −1 ∂z B 0 Q +c∂z P. . &2. - 0. - [0. 5+p#V ~N .s 2. cqc Q + B −1 (qQ)qc + AQ − qc P −qc Q + BP. . ~ XOR E5WT OX ~4Q£2Es JR 1"pe H̃. ~. . Q P. . =. Z. 1 Q cqc Q2 + qc QB −1 (qc Q) + QAQ + P BP dz + Ñ , P 2. R. Ñ Ú?G45Ws. . Q P. ]X ~z 5s. . =. Z. R. (cBQ − qc Q) P dz.. ]:U~~W(. K̃ Q Q Q − Ñ = H̃ K̃ P P P Z 1 = cqc Q2 + qc QB −1 (qc Q) + QAQ + P BP dz. R 2. C ßs "p! OF RI ~YI~/Rj5/ 1> NZ1 "p ] _ßZe J 9 K̃ # Z #5&X > 5Ws5&I~]4 O τ. dH̃ dτ. = 0.. dÑ dτ. Z. = =. ZR. R. {(cBQτ − qc Qτ ) P + (cBQ − qc Q) Pτ } dz 1. 2 (qc )z P. +. Z. R. . 2. + 12 c2 (qz )z Q2 + c(qc )z QB −1 (qc Q) dz.. ∂z (qc Q) B −1 qc B −1 (qc Q) + AQ dz.. ,6/5 Oj RÕ]/I]i 6 ~ "=J_ I s 53` qc = 0 5?"p&I]gX#"p4 I s > jO?"p 0. % 2E # O R~. K̃. . Q P. . dÑ = 0. dτ. K̃. 5Ws "pW á# Z#5& ! ~R 2 N"{R JR . = =. Z 1 (QAQ + P BP ) dz 2 R Z 2 2 1 Q2 + a Q0 + P 2 + b P 0 dz, 2 R. τ. >.
(11) . . . . . . . -. . .1.. ]:~ U ~#F ~s 2 I 2 sf 5L ~ "=J5 I sS C Í s "p! O H (R) 5 R 1> I ~jK̃ I~/RV5/ z353WRZ?"p 9 Ú? I 5/ 5P R]/I]f i "eJ ~s 2 ~g 5R 6?"pg U~~ O dÑ dτ. =. Z. R. 1. 2 (qc )z P. +. Z. . R. 2. + 12 c2 (qz )z Q2 + c(qc )z QB −1 (qc Q) dz.. ∂z (qc Q) B −1 qc B −1 (qc Q) + AQ dz.. 5L ß~ Õ~#5W#35R ZM/ > =5#5]WDI ~ s + 2 &+{ W ~ A >a .s9 j Z ]O s R~ RÕ] +~? b .]5& sj Oz]&Z~ P Q τ z / sE,Ò X\5 _£# 2 1 ~X~:d5#5& s! ~? dÑ /dτ 9 Ò 2 5Õ&"e ]5Wß _I 5R < ~! 5R iX#"p#N 7sN5W 5Iæ5R]pß _ ~ #p" 3 WW X 9. % ,dþ #&' ,dþ (% ,ü ý 5 6 5 sOR Ie j ~ .]5& sO 5/ æRdI 2 ~d "p IxR IU~T = δ T (φ ) Ò 2zZ¿ lI#5& s <X\~R ~V$ 5&< O c ~ Z5s ;~d "p −B ]:U~ "pfIi V3O sa> ~4V 2 j s +i]{p# I]\ OX ~Y I{ 9 5/RZFWi # 5 R S 2 4 s · 5 R sXI 5R * )/dc ~jdN I~(φ /RcV &"p& WEF J*>F J , ~R5/ Ú- a ~N. % ]~. & e0:'*§)9 p+ £#Ò, O s >e ~Nd, ORe * &{" I O W I -/.0 N ~45 5R gX#"p4¿1"p( Ú. i O. φc = (qc , pc )T , −F = −(H + cN ) 9. N (φc ). =. Z. B(pc )qc dx R. = −. )&· ~ 5W4 O. 3/2 1/2 8 2 c −1 bc2 − a 3c2 + 2 c−3 15 5/2 −1/2 −1 8 2 c −1 bc2 − a bc . − 15. 1/2 −3/2 −4 d 8 2 c Y (a, b, c), N (φc ) = − c −1 bc2 − a dc 15. ~j ~j "]5& s. Y. 5W 2 #·]:U~~Wåe. 3c2 + 2 +bc4 c2 − 1 (b − a) + bc2 c2 − 1 bc2 − a 2c2 + 3 2 +4bc4 c2 − 1 bc2 − a + 6c2 c2 − 1 bc2 − a 2 = 3 bc2 − a 3c2 + 2 + b2 c4 c2 − 1 4c2 + 1 − 5a/b 2 +bc2 c2 − 1 bc2 − a 2c2 + 3 + 6c2 c2 − 1 bc2 − a. Y (a, b, c) = 3 bc2 − a. 2.
(12) - . ". I#5&Fs "{ sb ~XT Oj . Y (a, b, c) > 0 9. c2 > b/a ≥ 1. s. c2 > 5a/4b − 1/4. > ~. " 5#5WsI]!j ~V s + ,P ~V* Y]&"= sB5/ #< VO#"pX O dN (φc ) < 09 _p R~ C &I &I`F J*> X2 /RX OVdc5R å ~j 5R X#"p#· #,ÒO s B+p&PÌ=. 5#5]?>\B / XB ~RI~/R &"p& ?I]E ~N52 / . 0 ]/ > Ìs "p! ~Í:d=35WÍ 53 ON 7s 9 2 > YO?"p c0 (a, b) 1 < c < c20 > 53f s 2E gI 5R I~/RV*= dN (φc ) 9 >0 dc Ú? 4>e V 5Ws ] 2 ¿: d 2 e 3<~ ·# 5R 3P ~¿Ò 2 5/ &"= ] 5W I 5R *E 7s 1g ~XU~3 B J 5W1 >eU d=I] R~ >e N5/R\U~~ a > b9 a b Ò 2 5/ 5Lf O dN (φc ) C Y I 35 > 09 c0 > 1 0 < c < c0 dc p 5 2 : d] 2E # XIi 53 R~. a>b. ,Òþ(% ,ü. c>. a/b 9. 5 5+p R ~ >aI6 a = 0.7 b = 0.1 1 ~Õ{5 5E E 5+p R~ dN (φc )/dc > 0 0 < c < 2.8 9 a = 0.5 $> Ii X5/ 5 b = 0.3333 dN (φc )/dc > 0 0 < c < 1.247 9 5/ . ýæú 1 ~Í{5. . 40 600 30 400. 20. 10 200 0 0. -10 3. 4 4.5 3.5 Graphic a. . . 5. . dN/dc. . 1.4. 1.6 1.8 Graphic b. 2. c . . W_ 5W 2E &2 (zs]Fs "{ sðsf ~Íd 2 5/&&"= ]5Wß <I 5R [> ~< . I]{5 ~;&"p I s IY 2 ; X5 5R j#"pX %X]~&('*) +p #Ò, O s 7sN ~Y"{I~#£ < ~ß:d=Z 2 (3 R~ RY]#5(W s 1b 9 ~ÕÒ 2 -/5./0 B2E &2 e5 sD E&5E ~Õ1a # 3 ¿ 1 MEI F J< ~j *X R W P ~\ ~j%X]~&e'*)§ +p #,ÒO s 2 d 2 5/ _ -/.0 MATLAB 9.
(13) . . . . . . . -. . .. t=100. t=80. q(x,t). t=30. t=0. −10. −5. 0. 5. . . 10. 15. 20. x−Ct. a=0.5, b=0.3333, C=1.23. -. . . . . . 4 "pj ~j &2 qt = r x rt = Aqx − rqx − 2rx q + brxxt ,. ] #5&X ~jI J5Ws~ s. c2 − 1 2 sech q(x, 0) = (1 + ) c. ~. r(x, 0) = −cq(x, 0),. 1 (1 + ) 2. "r. # ! c2 − 1 x bc2 − a. ~ OR52 & O&]& 2EI~# ~ I(/ 75W $ ~ 5R Y#"p 9 I{]# ']h ~c ~ß{5 XRæ] 5R E 5R f#"pE .sF§(4"{I~# > R~ I ~jIi ~T2 "eI]\ "52 a b c x − c t9 ,Òþ(% ,ü ý ! .ü Wý #(' Vý ,SU~{] P5+p a = 0.5 > b = 0.3333 > R~ 1f T/5 4Ís5 "p ON ~I J 5R iX#"p 9 . s(R OV2 I(5 IN3X .s 2 ] ?>3V3 WW I5/ 5# c = 1.23 ]/+ ] Iæ= 0.02 ?p" 4 R~fN3~?XF ] O#5& s6W5 2EZ s~( W £] O#5& sSÚ\~Y \ ~7 5&R O 9 £ ~ Z /U 5&JX s Ið #/5 R å Z&I] R~D ~ Z WWr R ~ a > b ] O#5& s &I]\32 9 , ~ U {] S5 p+ > > ,Òþ( % ,ü ý . ü Wý #( ' J ý R~ W ~Ú &O?"e s¿ ~N5 I s 32EJaR= Y0.7 O b =~U {0.1 ] c = 2.7 W ~7 5&V O=R0.02 ~9 #p" ! Z WW · I{pR ORSI ~] &"e sN/5 /5 R5 # ~9 ]#W5 2EZ 1 s ~j 5R cX #p" 4 Õ=5#5 ]XIE3~s 2 9 , ~ U {]ShZ S5 p+ > > ,Òþ( % ,ü ý . ü Wý #( ' J ý a = 0.7 b = 0.1 R~ ,\ /5 R ~ &O?"e s a ~N 5R P ?p" j53OR]p# c = 2.7 = −0.02 9 W5 2E & & * 1 #X3 . s 2 R~ 2 I(5 IV ~4 Z WW 9 9.
(14) .. - h. t=15. t=22.5. t=7.5. t=15 q(x,t) q(x,t). t=7.5 t=0. t=0. −15. −10. −5. -. . . ülÿ. 0. 5. 10. 15. 20. 25. . . 30. −15. −10. −5. 0. 5. . . . . . 10. 15. 20. x−Ct. a=0.7, b=0.1, C=2.7. x−Ct. a=0.7, b=0.1, C=2.7. . . . . . ' ,( % ,Òü ý . W ~Í# 3Ú < NO FRYOR Ú < ~Ì#/R35LSs +]s~Y .sR2f2 3 ~# gF J "p3 ~s ]& ; Ú2 "p {5: . <$& @d1 1 Ie >2Y ~# zI#5&s?>NR~69 ] 7#3s4@pOR R 1z 9 1 M[>a 7s2j ~ { ~RW5 1>e ~ ~& R~ ~ 5Ws]5R12 ( ]?"= ]W ;&"p 5Rp 2 2! s +. , ,dþ ,Òü, . 9. F J .. ". ]{ X@ d=35W;R~ 5R * 5R TC #"p I sa Ú s I~/R A 5/ " Z3 "p9 F"p I sæ_,ÒOA s N] A 5/ 5 #¿æ&2. ^ 9 9 9 9 õ IOR §( J _,dO s &s J<%XpR F Y J % I +=~ >· 5 ;N 9 9 % J]& E ]9 I]; 9 2EO Re ^ 9O. 9 F J I]1 sæ >$l ) &"eI s W ~Ws IOR §( JN_,ÒO s S5 5? '*QV9 9 ^ 9 H ; 9. 9 F h J 5M#I >Rp@ ] s s R I( \5& s 2z] \p6W_:d=3 I ^" Wæ 9 ~&2 5# A 9 A 9 9O. 9 FIHKJ J +eß > O5 ZC R~ 5R3!C " 5 R ` ~Ws R 5R ;X 9 A 9 A 9 9 A " #p" #RI ~j] # W5 =2 & á ]5 O h > HK']h A 9 9 -/. 10 . 9 F Ì J ,OR{~ s]4J Ì#35 R ~s]TJ ¿ s~~ X 5R J XO R5!J ß#5 O 5& s ]` % ]~&e'*9 §) A p+ ð]~1' 2 3 sO l W #3]ðæs# J æ & 2 5/ ?> 9 "p3 ~s]& ; H - 0:9 F Ì J Mb@ _5 O 5& s~# 3 "{ O R5rs~~ 5 O 5/ N 1 ©M "{R123 5R9 J 9 > 9 2 ' h H 0 . 9 . 9 9 F KJ Ie R@ s I~/R 5 R * ~ s~:1' 2 sOX % s3 I~#3£ , #Ò, O s 9 d@ s]O_ 9 O2E5# R~ §( J _d, O s> & H> ^ ^ . 9.
(15) . . . . . . . -. . .. H. F J p R~ I( @ Waå2 ( sO_ 5R #"p#j .sF\%X]~&e' ) p+ z#,Ò9 O s ( 5/ 9 > ih ']h 9 ^ . -/. [0 9 F J p >C &Ie &I Ipe"sI~#;R~!I 5R #; 5R Y#"p# . l 9 9 W 5R e5 l ) s~ 9 " > h h ' 9 9A 9 9 -/. 0 9 " z,( \5& s ×R J F J ORF > ·sp3 §( J_,ÒO s .1. ] I]p/' &9 J R> 9 ^ 9 A . 9 F J VÒ]~ % > ;s]][ s )I~/R ]5& sO " O I]p/' ] . & J R 9 9 9 A 9 9 F J 2 &{+ " &2 R + s ~S 5R #p" 5 R 7 sá X % s3 I~#3, . #A Ò, O s 9 s I~/R Z 3 "p_X #p" =3 &2P J R~] > H1HK' 9 ^ A . .0:9 >C s 5& & U 5 N] 3 I]> X"p?]> ; *1> @ A 9 9 . 9. . . / 1 0 . . . . . 2 3". ! #54"%$ '& )(+* "-, .. - - - . . . . . 1 2.
(16)
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