广义双约化理论运用于BBM方程的约化和精确解
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摘要
研究了Lie对称、守恒律、约化和Benjamin-Bona-Mahony(BBM)方程的精确解.运用乘子方法可以得到BBM方程的三个守恒律,根据这个方法我们发现守恒向量的Lie对称有两种不同形式.运用广义双约化理论将三阶BBM方程约化成二阶常微分方程,运用Sine-Cosine方法求出约化后的常微分方程的新的精确解.
We study the Lie symmetries,conservation laws,reductions,and new exact solutions of Benjamin-Bona-Mahony(BBM). The multiplier approach gets three conservation laws for BBM equation. We find the Lie symmetries is associated with the conserved vectors,and has two different cases arise. In this paper,the generalized double reduction theorem is then applied to reduce the third-order BBM equation to a second-order ordinary differential equation(ODE)and the implicit solutions are established. We use the Sine-Cosine method for the reduced second-order ODE to obtain the new explicit solutions of BBM equation.
We study the Lie symmetries,conservation laws,reductions,and new exact solutions of Benjamin-Bona-Mahony(BBM). The multiplier approach gets three conservation laws for BBM equation. We find the Lie symmetries is associated with the conserved vectors,and has two different cases arise. In this paper,the generalized double reduction theorem is then applied to reduce the third-order BBM equation to a second-order ordinary differential equation(ODE)and the implicit solutions are established. We use the Sine-Cosine method for the reduced second-order ODE to obtain the new explicit solutions of BBM equation.
引文
[1]NOETHER E,TAVEL M A.Invariant variation problems[J].Transport Theory and Statistical Physics,2005,1(3):186-207.
[2]BOKHARI A H,AL-DWEIK A Y,KARA A H,et al.Double reduction of a nonlinear(2+1)wave equation via conservation laws[J].Communications in Nonlinear Science and Numerical Simulation,2011,16(3):1244-1253.
[3]SJ?BERG A,MAHOMED F M.Non-local symmetries and conservation laws for one-dimensional gas dynamics equations[J].Applied Mathematics and Computation,2004,150(2):379-397.
[4]SJ?BERG A,MAHOMED F M.The association of non-local symmetries with conservation laws:applications to the heat and Burger’s equations[J].Applied Mathematics and Computation,2005,168(2):1098-1108.
[5]KHALIQUE C M,NTSIME P.On the Lagrangian formulation of a Lane-Emden-type equation and double reduction[J].Pamm,2007,7(1):2030049-2030050.
[6]Olver P J.Applications of Lie groups to differential equations[M].New York:Springer,1989.
[7]SJ?BERG A.Double reduction of PDEs from the association of symmetries with conservation laws with applications[J].Applied Mathematics and Computation,2007,184(2):608-616.
[8]BERG A.On double reductions from symmetries and conservation laws[J].Nonlinear Analysis Real World Applications,2009,10(6):3472-3477.
[9]BOKHARI A H,AL-DWEIK A Y,ZAMAN F D,et al.Generalization of the double reduction theory[J].Nonlinear Analysis Real World Applications,2010,11(5):3763-3769.
[10]NAZ R,KHAN M D,NAEEM I.Conservation laws and exact solutions of a class of non linear regularized long wave equations via double reduction theory and Lie symmetries[J].Communications in Nonlinear Science and Numerical Simulation,2013,18(4):826-834.
[11]NAZ R,MAHOMED F M,MASON D P.Comparison of different approaches to conservation laws for some partial differential equations in fluid mechanics[J].Applied Mathematics and Computation,2008,205(1):212-230.
[12]STEUDEL H.über die Zuordnung zwischen Invarianzeigenschaften und Erhaltungss?tzen[J].Zeitschrift Für Naturforschung A,1962,17(2):129-132.
[13]邢秀芝,程小强,卜春霞.BBM及(2+1)维BBM方程一些新的显示精确解[J].数学的实践与认识,2010,40(16):188-192.
[14]Al-MDALLAL Q M,SYAM M I.Sine-Cosine method for finding the soliton solutions of the generalized fifth-order nonlinear equation[J].Chaos Solitons and Fractals,2007,33(5):1610-1617.
[15]WAZWAZ A M.A sine-cosine method for handlingnonlinear wave equations[J].Mathematical and Computer Modelling,2004,40(5):499-508.
[16]WAZWAZ A M.The tanh and the sine-cosine methods for compact and noncompact solutions of the nonlinear Klein-Gordon equation[J].Applied Mathematics and Computation,2005,167(2):1179-1195.
[17]KARA A H,MAHOMED F M.Relationship between symmetries and conservation laws[J].International Journal of Theoretical Physics,2000,39(1):23-40.
[18]HKARA A,MMAHOMED F.A basis of conservation laws for partial differential equations[J].Journal of Nonlinear Mathematical Physics,2002,9(2):60-72.
[2]BOKHARI A H,AL-DWEIK A Y,KARA A H,et al.Double reduction of a nonlinear(2+1)wave equation via conservation laws[J].Communications in Nonlinear Science and Numerical Simulation,2011,16(3):1244-1253.
[3]SJ?BERG A,MAHOMED F M.Non-local symmetries and conservation laws for one-dimensional gas dynamics equations[J].Applied Mathematics and Computation,2004,150(2):379-397.
[4]SJ?BERG A,MAHOMED F M.The association of non-local symmetries with conservation laws:applications to the heat and Burger’s equations[J].Applied Mathematics and Computation,2005,168(2):1098-1108.
[5]KHALIQUE C M,NTSIME P.On the Lagrangian formulation of a Lane-Emden-type equation and double reduction[J].Pamm,2007,7(1):2030049-2030050.
[6]Olver P J.Applications of Lie groups to differential equations[M].New York:Springer,1989.
[7]SJ?BERG A.Double reduction of PDEs from the association of symmetries with conservation laws with applications[J].Applied Mathematics and Computation,2007,184(2):608-616.
[8]BERG A.On double reductions from symmetries and conservation laws[J].Nonlinear Analysis Real World Applications,2009,10(6):3472-3477.
[9]BOKHARI A H,AL-DWEIK A Y,ZAMAN F D,et al.Generalization of the double reduction theory[J].Nonlinear Analysis Real World Applications,2010,11(5):3763-3769.
[10]NAZ R,KHAN M D,NAEEM I.Conservation laws and exact solutions of a class of non linear regularized long wave equations via double reduction theory and Lie symmetries[J].Communications in Nonlinear Science and Numerical Simulation,2013,18(4):826-834.
[11]NAZ R,MAHOMED F M,MASON D P.Comparison of different approaches to conservation laws for some partial differential equations in fluid mechanics[J].Applied Mathematics and Computation,2008,205(1):212-230.
[12]STEUDEL H.über die Zuordnung zwischen Invarianzeigenschaften und Erhaltungss?tzen[J].Zeitschrift Für Naturforschung A,1962,17(2):129-132.
[13]邢秀芝,程小强,卜春霞.BBM及(2+1)维BBM方程一些新的显示精确解[J].数学的实践与认识,2010,40(16):188-192.
[14]Al-MDALLAL Q M,SYAM M I.Sine-Cosine method for finding the soliton solutions of the generalized fifth-order nonlinear equation[J].Chaos Solitons and Fractals,2007,33(5):1610-1617.
[15]WAZWAZ A M.A sine-cosine method for handlingnonlinear wave equations[J].Mathematical and Computer Modelling,2004,40(5):499-508.
[16]WAZWAZ A M.The tanh and the sine-cosine methods for compact and noncompact solutions of the nonlinear Klein-Gordon equation[J].Applied Mathematics and Computation,2005,167(2):1179-1195.
[17]KARA A H,MAHOMED F M.Relationship between symmetries and conservation laws[J].International Journal of Theoretical Physics,2000,39(1):23-40.
[18]HKARA A,MMAHOMED F.A basis of conservation laws for partial differential equations[J].Journal of Nonlinear Mathematical Physics,2002,9(2):60-72.