歐陽(yáng)正勇,鄭 珊
(1.佛山科學(xué)技術(shù)學(xué)院理學(xué)院,廣東佛山528000; 2.廣州航海學(xué)院基礎(chǔ)部,廣東廣州520725)
淺水波方程的暗孤子解
歐陽(yáng)正勇1,鄭 珊2
(1.佛山科學(xué)技術(shù)學(xué)院理學(xué)院,廣東佛山528000; 2.廣州航海學(xué)院基礎(chǔ)部,廣東廣州520725)
研究了一個(gè)非線性淺水波動(dòng)方程的孤立子解,運(yùn)用微分方程定性理論,證明了向左遷移的暗孤子解的存在性,并分析了暗孤子解的一些定性特征:該解具有對(duì)稱性,其振幅隨著波速的增大而增加,不同波速的暗孤子解必相交于對(duì)稱的兩點(diǎn),在無(wú)窮遠(yuǎn)處呈指數(shù)衰減到零.
淺水波方程;微分定性理論;暗孤子
考慮淺水波動(dòng)方程[1-2]
圖1 一維水波圖形Fig.1 One鄄dimensional water wave profile
文獻(xiàn)[2]研究了當(dāng)波速c>1時(shí),方程(1)存在孤立子解,本文對(duì)方程(1)(見(jiàn)圖1)進(jìn)行了進(jìn)一步的研究,證明了當(dāng)波速c<-1時(shí),方程(1)存在暗孤立子解,并給出了暗孤子解的一些定性特征.
作變換[2]
方程(1)可轉(zhuǎn)換成如下常微分方程
注波速c大于0時(shí)表示右行波,波速c小于0時(shí)表示左行波,本文研究的孤立子解滿足條件φ, φ(n)→0(ξ→∞),其中n∈N.
對(duì)方程(3)兩邊關(guān)于ξ積分,得
其中C為積分常數(shù).因?yàn)楸疚闹锌紤]的暗孤子解滿足φ,φ′→0(ξ→∞),故方程(4)中的積分常數(shù)C為0.
將方程(4)轉(zhuǎn)變成對(duì)應(yīng)的平面系統(tǒng)如下:
系統(tǒng)(5)的首次積分為
其中h為積分常數(shù).
定理1當(dāng)波速c<-1時(shí),方程(1)存在整體暗孤子解.
證明當(dāng)波速c<-1時(shí),系統(tǒng)(5)有兩個(gè)平衡點(diǎn)P1(0,0)和P2(φc,0),其中φc是多項(xiàng)式
唯一的負(fù)實(shí)根.事實(shí)上P(φ)關(guān)于φ是單調(diào)遞減的,且P(φ)→+∞(φ→-∞),P(φ)→-∞(φ→+∞),故P(φ)有唯一的零點(diǎn).又因?yàn)楫?dāng)c<-1時(shí),
所以φc<0.
下面計(jì)算系統(tǒng)(5)的線性化系統(tǒng)在平衡點(diǎn)P1(0,0)和P2(φc,0)處的特征值并判斷奇點(diǎn)的類型,為了方便計(jì)算,對(duì)系統(tǒng)(5)作變換,設(shè)dξ=(1+c+7φ)dτ,系統(tǒng)(5)可轉(zhuǎn)換為
除了奇直線φ=-(1+c)/7,系統(tǒng)(5)和(9)的分支相圖是一樣的,因而可用系統(tǒng)(9)代替(5)進(jìn)行計(jì)算.設(shè)奇點(diǎn)的特征值為λ,則λ滿足
當(dāng)c<-1時(shí),平衡點(diǎn)P1(0,0)有一正一負(fù)兩個(gè)不同的實(shí)特征值,因而是鞍點(diǎn).在平衡點(diǎn)P2(φc,0)處的特征值為在平衡點(diǎn)P1(0,0)處的特征值為
其中f′(φ)=12(c-1)-36φc+18φc2-18φc3,因?yàn)棣誧<0,所以f′(φc)>0,又c<-1,從而λ3,4為一對(duì)純虛根,故P2(φc,0)為中心.
圖2 暗孤子解對(duì)應(yīng)的同宿軌Fig.2 Honoclinic orbit of dark soliton
系統(tǒng)(5)的同宿軌對(duì)應(yīng)著方程(1)的孤立子解,為了證明方程(1)的暗孤子解的存在性,需要在φ-y平面上找出一條從鞍點(diǎn)P1(0,0)向 φ軸負(fù)方向出發(fā),環(huán)繞中心 P2(φc,0)之后再回到P1(0,0)的同宿軌.在φ-y平面的下半平面,因?yàn)棣铡洌統(tǒng)<0,φ隨著ξ的增大而減小,所以存在一條軌線從P1(0,0)向左出發(fā),且必定穿過(guò)直線φ=φc.事實(shí)上,
其中MP是φP(φ)的最大值,由式(13)可知y′有下界,故軌線必穿過(guò)直線φ=φc.當(dāng)軌線穿過(guò)φ=φc之后 (即φ<φc),有φ′=y(tǒng)<0,y′>0,軌線接著向左上方走,且穿過(guò)φ軸,不然,假設(shè)y單增趨于某常數(shù)(y′→0),那么φ→-∞,由式(5)可推出y′→∞,矛盾.再利用系統(tǒng)(5)關(guān)于φ軸的對(duì)稱性,即系統(tǒng)(5)在變換ξ→-ξ下是不變的,在上半平面,軌線按相同的方式回到P1(0,0).這樣就證明了存在一條同宿軌,而該軌線就對(duì)應(yīng)著方程(1)的一個(gè)暗孤子解.
雖然不能直接求出暗孤子解的表達(dá)式,但仍可以應(yīng)用微分方程定性理論分析暗孤子解的一些特征,然后畫出暗孤子解的平面圖.本文討論的暗孤子解具有如下定性特征:
定理2方程(1)的暗孤子解具有對(duì)稱性,其波峰隨著波速的增大而增大,且在無(wú)窮遠(yuǎn)處呈指數(shù)衰減到0.
證明由于本文所求的暗孤子解滿足φ′,φ→0(ξ→∞),因而式(6)中的積分常數(shù)h=0,上節(jié)中的同宿軌滿足如下等式
即
由式(17)可近似解得
這表明暗孤子解在無(wú)窮遠(yuǎn)處按指數(shù)衰減到0,也能反映出對(duì)應(yīng)的同宿軌從鞍點(diǎn)P1(0,0)發(fā)出時(shí)與縱軸的夾角.
定理3設(shè)φ(ξ,c1)和φ(ξ,c2)表示在不同波速c1,c2下的兩個(gè)暗孤子解,那么這兩個(gè)解必相交于兩點(diǎn).
證明本小節(jié)考慮不同波速下波形的比較,設(shè)φ(ξ)是方程(1)的暗孤子解,且在ξ=0處達(dá)到波峰.由定理2知,波峰的高度是波速的函數(shù),定義如下函數(shù)
圖3 不同波速的波形比較Fig.3 Comparison of different wave velocity waveforms
本文從方程本身的結(jié)構(gòu)出發(fā),結(jié)合微分方程定性理論,分析并證明了在波速c<-1時(shí)暗孤子解的存在性,給出了暗孤子解的定性特征,拓展了文獻(xiàn)中的相關(guān)結(jié)論.當(dāng)波速-1≤c≤1的時(shí)候,孤立子解的存在性有待進(jìn)一步研究.
[1]CONSTANTIN A,LANNES D.The hydrodynamical relevance of the Camassa-Holm and Degasperis-Processi equations[J].Arch Ration Mech Anal,2009,192:165-186.
[2]GEYER A.Solitary traveling water waves of moderate amplitude[J].J Nonlinear Math Phys,2012,19:104-115.
[3]CAMASSA R,HOLM D.An integrable shallow water equation with peaked solitons[J].Phys Rev Lett,1993,71:1 661-1 664.
[4]DEGASPERIS A,PROCESI M.Asymptotic integrability[C]//Symmetry and Perturbation Theory.London:World Scientific Publishing Company,1999:23-37.
[5]CONSTANTIN A.On the scattering problem for the Camassa-Holm equation[J].Math Phys Eng Sci,2001,457:953-970.
[6]CONSTANTIN A,GERDJIKOV V S,IVANOV R I.Inverse scattering transform for the Camassa-Holm equation[J]. Inverse Problems,2006,22:2 197-2 207.
[7]BOUTET DE MONVEL A,KOSTENKO A,SHEPELSKY D,et al.Long-time asymptotics for the Camassa-Holm equation[J].SIAM J Math Anal,2009,41:1 559-1 588.
[8]EL DIKA K,MOLINET L.Exponential decay of H1-localized solutions and stability of the train of N solitary waves for the Camassa-Holm equation[J].Math Phys Eng Sci,2007,365:2 313-2 331.
[9]LIU Z R,QIAN T F.Peakons of the Camassa-Holm equation[J].Appl Math Model,2002,26(3):473-480.
[10]ZHANG W L.General expressions of peaked traveling wave solutions of CH-gamma and CH equations[J].Sci China Ser A Math,2004,47(6):862-873.
[11]BRESSAN A,CONSTANTIN A.Global conservative solutions of the Camassa Holm equation[J].Arch Ration Mech Anal,2007,183:215-239.
[12]CONSTANTIN A,ESCHER J.Analyticity of periodic traveling free surface water waves with vorticity[J].Ann of Math,2011,173:559-568.
[13]CONSTANTIN A,ESCHER J.Wave breaking for nonlinear nonlocal shallow water equations[J].Acta Math,1998,181:229-243.
[14]CONSTANTIN A,MOLINET L.Global weak solutions for a shallow water equation[J].Commun Math Phys,2000,211:45-61.
[15]CONSTANTIN A,STRAUSS W.Stability of peakons[J].Comm Pure Appl Math,2000,53:603-610.
[16]CONSTANTIN A,STRAUSS W.Stability of the Camassa-Holm solitons[J].J Nonlinear Sci,2002,12:415-422.
[17]OUYANG Z Y,ZHEN S,LIU Z R.Orbital stability of peakons with nonvanishing boundary for CH and CH-gamma equations[J].Phys Lett A,2008,372:7 046-7 050.
[18]BENJAMIN T B.The stability of solitary waves[J].Proc R Soc Lond,1972,328:153-183.
[19]EL DIKA K,MOLINET L.Exponential decay of H1-localized solutions and stability of the train of N solitary waves for the Camassa-Holm equation[J].Math Phys Eng Sci,2007,365:2 313-2 331.
[20]EL DIKA K,MOLINET L.Stability of multipeakons[J].Ann I H Poincar AN,2009,26:1 517-1 532.
The Existence of Dark Solitons of Shallow Water Wave Equation
OUYANG Zheng-yong1,ZHENG Shan2
(1.Science School,F(xiàn)oshan University,F(xiàn)oshan 528000,China;2.Department of Basic Courese,Guangzhou Maritime College,Guangzhou 510725,China)
This paper studied the soliton solutions of a nonlinear shallow water wave equation.By using the qualitative theorem of differential equations,we prove the existence of dark soliton solutions and discussed some of their qualitative characteristics.The dark soliton solutions are symmetric on both sides of the crest,and the amplitude increases with the increase of wave speed.Dark soliton solutions of different speeds intersect each other in two symmetrical spots and decay exponentially to zero in infinity.
shallow water wave equation;differential qualitative theory;dark soliton
O193
A
1673-4432(2015)05-0089-05
(責(zé)任編輯 曉 軍)
2015-01-06
2015-06-04
國(guó)家自然科學(xué)基金項(xiàng)目 (11401096);廣東省教改項(xiàng)目 (GDJG20141204)
歐陽(yáng)正勇 (1978-),男,講師,博士,研究方向?yàn)槲⒎址匠?E-mail:zyouyang_math@163.com