一类辅助常微分方程的亚纯函数通解（英文）
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摘要
考虑一类常微分方程Aw″+Bw'+Cw+Dw~2+E=0,然后运用一个新方法——复方法,得出此类辅助常微分方程的亚纯函数通解,并运用这些结果得到Fisher方程、Oskolkov方程、BBMPB方程和OBBMB方程的所有亚纯行波精确解.结果显示,Fisher方程、Oskolkov方程、BBMPB方程和OBBMB方程的所有单周期函数解为孤立行波解,而且复方法比其他方法更简捷有效.
In this paper,we consider a class of ordinary differential equation Aw″ + Bw' + Cw + Dw2+ E = 0,and then we use a new method named complex method to derive the general meromorphic solutions for this auxiliary ordinary differential equation. At last,all traveling wave exact solutions for the Fisher equation,Oskolkov equation,BBMPB equantion and OBBMB equation can be found by our results. Our results shows that all simple periodic exact solutions in the Fisher equation,Oskolkov equation,BBMPB equantion and OBBMB equation are solitary wave solutions,and the complex method is simpler and more precise than other methods.
In this paper,we consider a class of ordinary differential equation Aw″ + Bw' + Cw + Dw2+ E = 0,and then we use a new method named complex method to derive the general meromorphic solutions for this auxiliary ordinary differential equation. At last,all traveling wave exact solutions for the Fisher equation,Oskolkov equation,BBMPB equantion and OBBMB equation can be found by our results. Our results shows that all simple periodic exact solutions in the Fisher equation,Oskolkov equation,BBMPB equantion and OBBMB equation are solitary wave solutions,and the complex method is simpler and more precise than other methods.
引文
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[8]TANG S Q,ZHENG J X,HUANG W T.Travelling wave solutions for a class of generalized Kd V equation[J].Appl Math Comp,2009,215(7):2768-2774.
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[10]TANG S Q,XIAO Y X,WANG Z J.Travelling wave solutions for a class of nonlinear fourth order variant of a generalized Camassa-Holm equation[J].Appl Math Comp,2009,210(1):39-47.
[11]WANG M L,Solitary wave solutions for variant Boussinesq equations[J].Phys Lett A,1995,199(3/4):169-172.
[12]FAN E.Uniformly constructing a series of explicit exact solutions to nonlinear equations in mathematical physics[J].Chaos Soliton Fract,2003,16(5):819-839.
[13]KUDRYASHOV N A.Meromorphic solutions of nonlinear ordinary differential equations[J].Comm Nonl Sci Num Simul,2010,15(10):2778-2790.
[14]DEMINA M V,KUDRYASHOV N A.From Laurent series to exact meromorphic solutions:the Kawahara equation[J].Phys Lett A,2010,374(39):4023-4029.
[15]DEMINA M V,KUDRYASHOV N A.Explicit expressions for meromorphic solutions of autonomous nonlinear ordinary differential equations[J].Comm Nonl Sci Num Simul,2011,16(3):1127-1134.
[16]KUDRYASHOV N A,SINELSHCHIKOV D I,DEMINA M V.Exact solutions of the generalized Bretherton equation[J].Phys Lett A,2011,375(7):1074-1079.
[17]YUAN W J,LI Y Z,LIN J M.Meromorphic solutions of an auxiliary ordinary differential equation using complex method[J].Math Meth Appl Sci,2013,36(13):1776-1782.
[18]YUAN W J,HUANG Y,SHANG Y D.All travelling wave exact solutions of two nonlinear physical models[J].Appl Math Comp,2013,219(11):6212-6223.
[19]YUAN W J,SHANG Y D,HUANG Y,et al.The representation of meromorphic solutions to certain ordinary differential equations and its applications[J].Sci Sin(Math),2013,43(6):563-575.
[20]LANG S.Elliptic functions[M].2nd ed.New York:Springer Verlag,1987.
[21]CONTE R,MUSETTE M.Elliptic general analytic solutions[J].Stud Appl Math,2009,123(1):63-81.
[22]QI J M,CHEN Q H,XIONG W L,et al.A note about the general meromorphic solutions of the Fisher equation[J].Math Probl Eng,2014,2014:1-4.
[23]ABLOWITZ M J,ZEPPETELLA A.Explicit solution of Fisher's equation for a special wave speed[J].Bull Math Biol,1979,41(6):835-840.
[24]FENG Z S,LI Y.Complex traveling wave solutions to the Fisher equation[J].J Phys A,2006,366:115-123.
[25]GUO B Y,CHEN Z X.Analytic solutions of the Fisher equation[J].J Phys A,1991,24(3):645-650.
[26]The tanh-coth method for some nonlinear pseudoparabolic equations with exact solutions[J].Adv Differ Equ,2013,143:1-13.
[27]JEFFREY A,XU S Q.Exact solutions to the Korteweg-de Vries-Burgers[J].Wave Motion,1989,11(6):559-564.
[28]JEFFREY A,MOHAMAD M N B.Exact solutions to the Kd V-Burgers’equation[J].Wave Motion,1991,14(4):369-375.
[29]KUDRYASHOV N A.On new travelling wave solutions of the Kd V and the Kd V-Burgers equations[J].Commun Nonlinear Sci Numer Sim,2009,14(5):1891-1900.
[30]WANG M L.Exact solution for a compound Kd V-burgers equation[J].Phys Lett A,1996,213(5/6):279-287.
[31]WAZZAN L.A modified tanh-coth method for solving the Kd V and the Kd V-Burgers equations[J].Commun Nonlinear Sci Numer Sim,2009,14(2):443-450.
[32]XIE Y X,TANG J S.New solitary wave solutions to the Kd V-Burgers equation[J].Internat J Theoret Phys,2005,44(3):293-301.
[33]XIA T C,ZHANG H Q,YAN Z Y.New explicit and exact traveling wave solutions for a compound Kd V-Burgers equation[J].Chin Phys,2001,10(8):694-699.
[2]HIROTA R,SATSUMA J.Soliton solutions of a coupled Korteweg-de Vries equation[J].Phys Lett A,1981,85(8/9):407-408.
[3]MATVEEV V B,SALLE M A.Darboux transformations and solitons[M].Springer Series in Nonlinear Dynamics,Berlin Germany:Springer,1991.
[4]OLVER P J.Applications of lie groups to differential equations.Graduate Texts in Mathematics[M].2nd ed.Vol 107,New York,USA:Springer,1993.
[5]LI J B,LIU Z R.Travelling wave solutions for a class of nonlinear dispersive equations[J].Chin Ann Math,2002,23(3):397-418.
[6]TANG S Q,HUANG W T.Bifurcations of travelling wave solutions for the generalized double sinh-Gordon equation[J].Appl Math Comp,2007,189(2):1774-1781.
[7]FENG D H,HE T L,LYU J L.Bifurcations of travelling wave solutions for(2+1)-dimensional Boussinesq-type equation[J].Appl Math Comp,2007,185(1):402-414.
[8]TANG S Q,ZHENG J X,HUANG W T.Travelling wave solutions for a class of generalized Kd V equation[J].Appl Math Comp,2009,215(7):2768-2774.
[9]MALFLIET W,HEREMAN W.The tanh method:I.Exact solutions of nonlinear evolution and wave equations[J].Phys Sci,1996,54(6):563-568.
[10]TANG S Q,XIAO Y X,WANG Z J.Travelling wave solutions for a class of nonlinear fourth order variant of a generalized Camassa-Holm equation[J].Appl Math Comp,2009,210(1):39-47.
[11]WANG M L,Solitary wave solutions for variant Boussinesq equations[J].Phys Lett A,1995,199(3/4):169-172.
[12]FAN E.Uniformly constructing a series of explicit exact solutions to nonlinear equations in mathematical physics[J].Chaos Soliton Fract,2003,16(5):819-839.
[13]KUDRYASHOV N A.Meromorphic solutions of nonlinear ordinary differential equations[J].Comm Nonl Sci Num Simul,2010,15(10):2778-2790.
[14]DEMINA M V,KUDRYASHOV N A.From Laurent series to exact meromorphic solutions:the Kawahara equation[J].Phys Lett A,2010,374(39):4023-4029.
[15]DEMINA M V,KUDRYASHOV N A.Explicit expressions for meromorphic solutions of autonomous nonlinear ordinary differential equations[J].Comm Nonl Sci Num Simul,2011,16(3):1127-1134.
[16]KUDRYASHOV N A,SINELSHCHIKOV D I,DEMINA M V.Exact solutions of the generalized Bretherton equation[J].Phys Lett A,2011,375(7):1074-1079.
[17]YUAN W J,LI Y Z,LIN J M.Meromorphic solutions of an auxiliary ordinary differential equation using complex method[J].Math Meth Appl Sci,2013,36(13):1776-1782.
[18]YUAN W J,HUANG Y,SHANG Y D.All travelling wave exact solutions of two nonlinear physical models[J].Appl Math Comp,2013,219(11):6212-6223.
[19]YUAN W J,SHANG Y D,HUANG Y,et al.The representation of meromorphic solutions to certain ordinary differential equations and its applications[J].Sci Sin(Math),2013,43(6):563-575.
[20]LANG S.Elliptic functions[M].2nd ed.New York:Springer Verlag,1987.
[21]CONTE R,MUSETTE M.Elliptic general analytic solutions[J].Stud Appl Math,2009,123(1):63-81.
[22]QI J M,CHEN Q H,XIONG W L,et al.A note about the general meromorphic solutions of the Fisher equation[J].Math Probl Eng,2014,2014:1-4.
[23]ABLOWITZ M J,ZEPPETELLA A.Explicit solution of Fisher's equation for a special wave speed[J].Bull Math Biol,1979,41(6):835-840.
[24]FENG Z S,LI Y.Complex traveling wave solutions to the Fisher equation[J].J Phys A,2006,366:115-123.
[25]GUO B Y,CHEN Z X.Analytic solutions of the Fisher equation[J].J Phys A,1991,24(3):645-650.
[26]The tanh-coth method for some nonlinear pseudoparabolic equations with exact solutions[J].Adv Differ Equ,2013,143:1-13.
[27]JEFFREY A,XU S Q.Exact solutions to the Korteweg-de Vries-Burgers[J].Wave Motion,1989,11(6):559-564.
[28]JEFFREY A,MOHAMAD M N B.Exact solutions to the Kd V-Burgers’equation[J].Wave Motion,1991,14(4):369-375.
[29]KUDRYASHOV N A.On new travelling wave solutions of the Kd V and the Kd V-Burgers equations[J].Commun Nonlinear Sci Numer Sim,2009,14(5):1891-1900.
[30]WANG M L.Exact solution for a compound Kd V-burgers equation[J].Phys Lett A,1996,213(5/6):279-287.
[31]WAZZAN L.A modified tanh-coth method for solving the Kd V and the Kd V-Burgers equations[J].Commun Nonlinear Sci Numer Sim,2009,14(2):443-450.
[32]XIE Y X,TANG J S.New solitary wave solutions to the Kd V-Burgers equation[J].Internat J Theoret Phys,2005,44(3):293-301.
[33]XIA T C,ZHANG H Q,YAN Z Y.New explicit and exact traveling wave solutions for a compound Kd V-Burgers equation[J].Chin Phys,2001,10(8):694-699.