广义ZK方程和广义ZK-BBM方程的行波解分支
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摘要
随着非线性科学的发展,许多物理、化学和生命科学模型都可以转化为非线性方程,如非线性常微分方程、偏微分方程和差分方程等.非线性方程的求解已经成为非线性科学领域的一个重要研究课题.
非线性Zakharov-Kuznetsov方程(简称ZK方程)于1974年由Zakharov和Kuznetsov提出.由于该方程是KdV方程在二维空间的典型推广形式之一,因此研究该方程具有广泛的理论意义和实践意义.
2005年, Wazwaz结合ZK方程和BBM方程构造了ZK-BBM方程,并运用tanh方法和sine-cosine方法获得了ZK-BBM方程的一些精确解.
本文利用动力系统分支理论研究了广义ZK方程和广义ZK-BBM方程,由于它们的行波系统具有奇性,因此本文借助微分方程定性理论研究了对应的正则系统,获得了正则系统有界轨道的定性性质,指出了奇异直线的存在是导致正则系统出现非光滑的周期尖波、孤立尖波、compacton和破缺波的原因,进而分析了广义ZK方程和广义ZK-BBM方程的光滑行波解和非光滑行波解产生的分支参数条件,获得了各种有界行波解存在的充分条件,并求出了上述部分解的精确参数表示.
In the view of nonlinear science, a lot of physics, chemistry, and life sciences modelscan be changed into nonlinear equations, such as nonlinear ODE, PDE and di?erenceequation. Solving nonlinear equations has become an important research topic in thefield of nonlinear science.
In 1974, Zakharov and Kuznetsov posed the nonlinear Zakharov-Kuznetsov equa-tion(ZK equations in short). This equation is one of the best known two-dimensionalgeneralizations of the KdV equation. Studying this equation is important not only intheory but also in practice.
In 2005, Wazwaz constructed the ZK-BBM equation by combining the BBM equa-tion with the ZK equation, and obtained some exact solutions for the ZK-BBM equationby means of the tanh and the sine-cosine methods.
In this paper, by using the bifurcation theory of planar dynamical systems to thegeneralized ZK equations and the generalized ZK-BBM equations, since their travellingwave systems have singularity, after applying qualitative theory of di?erential equationsto the corresponding regular systems, the qualitative properties of bounded orbits ofthe regular system are obtained. It is emphasized that the existence of singular straightline is the reason for the appearance of non-smooth periodic cusp wave solutions,solitary cusp wave solutions and breaking wave solutions. Various su?cient conditionsto guarantee the existence of smooth and non-smooth travelling wave solutions aregiven. Therefore the bifurcation parameter conditions which lead to smooth and non-smooth travelling wave solutions to the generalized ZK equations and the generalizedZK-BBM equations are analyzed and the various su?cient conditions to guarantee theexistence of the bounded travelling wave solutions are obtained. Some exact explicitparametric representations of the above waves are determined.
非线性Zakharov-Kuznetsov方程(简称ZK方程)于1974年由Zakharov和Kuznetsov提出.由于该方程是KdV方程在二维空间的典型推广形式之一,因此研究该方程具有广泛的理论意义和实践意义.
2005年, Wazwaz结合ZK方程和BBM方程构造了ZK-BBM方程,并运用tanh方法和sine-cosine方法获得了ZK-BBM方程的一些精确解.
本文利用动力系统分支理论研究了广义ZK方程和广义ZK-BBM方程,由于它们的行波系统具有奇性,因此本文借助微分方程定性理论研究了对应的正则系统,获得了正则系统有界轨道的定性性质,指出了奇异直线的存在是导致正则系统出现非光滑的周期尖波、孤立尖波、compacton和破缺波的原因,进而分析了广义ZK方程和广义ZK-BBM方程的光滑行波解和非光滑行波解产生的分支参数条件,获得了各种有界行波解存在的充分条件,并求出了上述部分解的精确参数表示.
In the view of nonlinear science, a lot of physics, chemistry, and life sciences modelscan be changed into nonlinear equations, such as nonlinear ODE, PDE and di?erenceequation. Solving nonlinear equations has become an important research topic in thefield of nonlinear science.
In 1974, Zakharov and Kuznetsov posed the nonlinear Zakharov-Kuznetsov equa-tion(ZK equations in short). This equation is one of the best known two-dimensionalgeneralizations of the KdV equation. Studying this equation is important not only intheory but also in practice.
In 2005, Wazwaz constructed the ZK-BBM equation by combining the BBM equa-tion with the ZK equation, and obtained some exact solutions for the ZK-BBM equationby means of the tanh and the sine-cosine methods.
In this paper, by using the bifurcation theory of planar dynamical systems to thegeneralized ZK equations and the generalized ZK-BBM equations, since their travellingwave systems have singularity, after applying qualitative theory of di?erential equationsto the corresponding regular systems, the qualitative properties of bounded orbits ofthe regular system are obtained. It is emphasized that the existence of singular straightline is the reason for the appearance of non-smooth periodic cusp wave solutions,solitary cusp wave solutions and breaking wave solutions. Various su?cient conditionsto guarantee the existence of smooth and non-smooth travelling wave solutions aregiven. Therefore the bifurcation parameter conditions which lead to smooth and non-smooth travelling wave solutions to the generalized ZK equations and the generalizedZK-BBM equations are analyzed and the various su?cient conditions to guarantee theexistence of the bounded travelling wave solutions are obtained. Some exact explicitparametric representations of the above waves are determined.
引文
[1] Russel J S. Report of the committee on waves[C]. Rep. Meet. Birt. Assoc. Adv. Sci.7th Liverpool, London, John Murray. 1837, 417.
[2] Russel J S. Report on waves[C]. Rep. Meet. Birt. Assoc. Adv. Sci. 14th York, London,John Murray. 1844, 311.
[3] Korteweg D J, de Vries G. On the change of form long waves advancing in a rectangularcanal, and on a new of type of long stationary waves[J]. Phil Mag, 1895, 39:422-443.
[4] Zabusky N J, Kruskal M D. Interaction of solitons in acollisions plasma and the recur-rence of initial states[J]. Phys Rev Lett, 1965, 15: 240-243.
[5] Camassa R, Holm D D. An integrable shallow water equation with peaked solitons[J].Phys Rev Lett, 1993, 71(11): 1661-1664.
[6] Rosenau P, Hyman J M. Compactons: Solitons with finite wavelength[J]. Phys RevLett, 1993, 70(5): 564-567.
[7] Rosenau P. On nonanlytic solitary waves formed by a nonliearly dispersion[J]. PhysLett A, 1997, 230: 305-318.
[8] Chow S N, Hale J K. Method of Bifurcation Theory[M]. New York: Springer-Verlag,1981.
[9] Lawrence P. Di?erential Equations and Dynamical Systems[M]. New York: Springer-Verlag, 1991.
[10] Li J, Liu Z. Smooth and non-smooth travelling waves in a non-linearly dispersive equa-tion[J]. Appl Math Model, 2000, 25: 41-56.
[11] Li J, Liu Z. Travelling wave solutions for a class of nonlinear dispersive equations[J].Chin Ann Math, 2002, 23B: 397-418.
[12] Li J, Dai H. On the Study of Singular Nonlinear Travelling Wave Equations: DynamicalSystem Approach[M]. Beijing: Science Press, 2007.
[13] Shen J, Li J, Xu W. Bifurcations of travelling wave solutions in a model of the hydrogen-bonded systems[J]. Appl Math Comput, 2005, 171(1): 242-271.
[14] Shen J, Xu W, Xu Y. Travelling wave solutions in the generalized Hirota-Satsumacoupled KdV system[J]. Appl Math Comput, 2005, 161(2): 365-383.
[15] Tang S, Li M. Bifurcations of travelling wave solutions in a class of generalized KdVequation[J]. Appl Math Comput, 2006, 177(2): 589-596.
[16] Tang S, Huang W. Bifurcations of travelling wave solutions for the generalized doublesinh-Gordon equation[J]. Appl Math Comput, 2007, 189(2): 1774-1781.
[17] Tang S, Huang W. Bifurcations of travelling wave solutions for the K(n,-n,2n) equa-tions[J]. Appl Math Comput, 2008, 203(1): 39-49.
[18]唐生强,张彦文.三阶非线性Schro¨dinger方程行波解的分支[J].广西科学院学报,2006, 22(3): 148-152
[19]冯大河.非线性波方程的精确解与分支问题研究[D].昆明理工大学, 2007.
[20]张锦炎,冯贝叶.常微分方程几何理论与分支问题(第二版)[M].北京:北京大学出版社, 1980.
[21]叶彦谦,极限环论(修定版)[M].上海科技出版社, 1984.
[22]张芷芬,丁同仁等.微分方程定性理论[M].北京:科学出版社, 1985.
[23]李继彬,李存富.非线性微分方程[M].成都科技大学出版社, 1987.
[24]李继彬.非线性微分方程[M].昆明:云南科技出版社, 1987.
[25]陆启韶.常微分方程定性方法和分叉[M].北京:北京航空航天大学出版社, 1989.
[26]李继彬.浑沌与Melnikov方法[M].重庆:重庆大学出版社, 1989.
[27]丁同仁,李承治.常微分方程教程[M].北京:高等教育出版社, 1992.
[28]韩茂安,朱德明.微分方程分支理论[M].煤碳工业出版社, 1994.
[29]王高雄,周之铭等.常微分方程[M].北京:高等教育出版社, 1996.
[30]张芷芬,李承治等.向量场分支理论基础[M].北京:高等教育出版社, 1997.
[31]谷超豪等.应用偏微分方程[M].北京:高等教育出版社, 1993.
[32]刘式达,刘式适,物理学中的非线性方程[M].北京:北京大学出版社, 1998.
[33]李诩神,孤子与可积系统[M].上海:上海科技教育出版社, 1999.
[34]罗定军等.动力系统的定性与分支理论[M].北京:科学出版社, 2001.
[35]张锦炎,冯贝叶.常微分方程几何理论与分支问题[M].北京:北京大学出版社,2001.
[36]张芷芬,丁同仁,黄文灶等.微分方程定性理论[M].北京:科学出版社, 2003.
[37]范恩贵,可积系统与计算机代数[M].北京:科学出版社, 2004.
[38]谷超豪,胡和生,周子翔,孤立子理论中的达布变换及其几何应用(第二版)[M].上海:上海科学技术出版社, 2005.
[39]黄念宁,陈世荣.完全可积非线性方程的哈密尔顿理论[M].北京:科学出版社,2005.
[40] Zakharov V E, Kuznetsov E A. On three-dimensional solitons[J]. Sov Phys, 1974, 39:285-288.
[41] Sliivamoggi B K. Nonlinear ion-acoustic waves in weak magnetied plasma andZakharov-Kuznetsov equation[J]. J Plasma Phys, 1989, 41: 83-88.
[42] Shivamoggi B K. The Painlev′e analysis of the Zakharov-Kuznetsov equation[J]. PhysScr, 1990, 42: 641-642.
[43] Hamza A M. A kinetic derivation of a generalized Zakharov-Kuznetsov equation for ionacoustic turbulence in a magnetized plasma[J]. Phys Lett A, 1994, 190(3-4): 309-316.
[44] Munro S, Parkes E J. The derivation of a modified Zakharov-Kuznetsov equation andthe stability of its solutions[J]. J Plasma Phys, 1999, 62(3): 305-317.
[45] Munro S, Parkes E J. Stability of solitary-wave solutions to a modified Zakharov-Kuznetsov equation[J]. J Plasma Phys, 2000, 64(4): 411-426.
[46] Li B, Chen Y, Zhang H. Exact travelling wave solutions for a generalized Zakharov-Kuznetsov equation[J]. Appl Math Comput, 2003, 146: 653-666.
[47] Wazwaz A M. Special types of the nonlinear dispersive Zakharov-Kuznetsov equa-tion with compactons, solitons and periodic solutions[J]. Int J Comput Math, 2004,81(9):1107-1119.
[48] Wazwaz A M. Nonlinear dispersive special type of the Zakharov-Kuznetsov equationZK(n, n) with compact and noncompact structures[J]. Appl Math Comput, 2005, 161:577-590.
[49] Wazwaz A M. Exact solutions with solitons and periodic structures for the Zakharov-Kuznetsov (ZK) equation and its modified form[J]. Commun Nonlinear Sci NumerSimul, 2005, 10: 597-606.
[50] Wazwaz A M. Explicit travelling wave solutions of variants of the K(n, n) and theZK(n, n) equations with compact and noncompact structures[J]. Appl Math Comput,2006, 177(1): 213-230.
[51] Wazwaz A M. The extended tanh method for the Zakharov-Kuznetsov (ZK) equation,the modified ZK equation, and its generalized forms[J]. Commun Nonlinear Sci NumerSimul, 2008, 13: 1039-1047.
[52] Huang W. A polynomial expansion method and its application in the coupled Zakharov-Kuznetsov equations[J]. Chaos Solitons Fractals, 2006, 29: 365-371.
[53] Inc M. Exact solutions with solitary patterns for the Zakharov-Kuznetsov equationswith fully nonlinear dispersion[J]. Chaos Solitons Fractals, 2007, 33(15): 1783-1790.
[54] Elhanbaly A, Abdou M A. Exact travelling wave solutions for two nonlinear evolutionequations using the improved F-expansion method[J]. Math Comput Model, 2007, 46:1265-1276.
[55]闰振亚,张鸿庆.组合Zakharov-Kuznetsov方程的显式孤波解[J].纯粹数学与应用数学, 2000, 16(2): 31-35.
[56]林麦麦,段文山,吕克璞. (3+1)维ZK方程的N孤子解[J].西北师范大学学报, 2006,42(1): 41-45.
[57]胡越,田长安. Zakharov-Kuznetsov型方程的一类周期行波解[J].河南师范大学学报, 2007, 35(4): 35-37.
[58]刘式适,陈华,付遵涛,刘式达. Lam函数和非线性演化方程的多级准确解的不变性[J].物理学报, 2003,52(8): 1842-1847.
[59]套格图桑,斯仁道尔吉. BBM方程和修正的BBM方程新的精确孤立波解[J].物理学报, 2004,53(12): 4052-4060
[60] Lei Z. Existence and stability of solitary-wave solutions of equations of Benjamin-Bona-Mahony type[J]. J.Di?erential Equations, 2003,188: 1-32.
[61] Wazwaz A M. New travelling wave solutions of di?erent physical structures to gener-alized BBM equation[J]. Phys Lett A, 2006, 355(4-5): 358-362.
[62] Wazwaz A M. The extended tanh method for new compact and noncompact solutionsfor the KP-BBM and the ZK-BBM equations[J]. Appl Math Comput, 2005, 169(1):713-725.
[63] Wazwaz A M. New travelling wave solutions of di?erent physical structures to gener-alized BBM equation[J]. Chaos, Solitons Fractals, 2008, 38(5): 1505-1516.
[64]徐桂芳.积分表[M].上海:上海科学技术出版社, 1962.
[65] Byrd P F, Friedman M D. Handbook of Elliptic Integrals for Engineers and Physi-cists[M]. Berlin: Springer-Verlag, 1954.
[2] Russel J S. Report on waves[C]. Rep. Meet. Birt. Assoc. Adv. Sci. 14th York, London,John Murray. 1844, 311.
[3] Korteweg D J, de Vries G. On the change of form long waves advancing in a rectangularcanal, and on a new of type of long stationary waves[J]. Phil Mag, 1895, 39:422-443.
[4] Zabusky N J, Kruskal M D. Interaction of solitons in acollisions plasma and the recur-rence of initial states[J]. Phys Rev Lett, 1965, 15: 240-243.
[5] Camassa R, Holm D D. An integrable shallow water equation with peaked solitons[J].Phys Rev Lett, 1993, 71(11): 1661-1664.
[6] Rosenau P, Hyman J M. Compactons: Solitons with finite wavelength[J]. Phys RevLett, 1993, 70(5): 564-567.
[7] Rosenau P. On nonanlytic solitary waves formed by a nonliearly dispersion[J]. PhysLett A, 1997, 230: 305-318.
[8] Chow S N, Hale J K. Method of Bifurcation Theory[M]. New York: Springer-Verlag,1981.
[9] Lawrence P. Di?erential Equations and Dynamical Systems[M]. New York: Springer-Verlag, 1991.
[10] Li J, Liu Z. Smooth and non-smooth travelling waves in a non-linearly dispersive equa-tion[J]. Appl Math Model, 2000, 25: 41-56.
[11] Li J, Liu Z. Travelling wave solutions for a class of nonlinear dispersive equations[J].Chin Ann Math, 2002, 23B: 397-418.
[12] Li J, Dai H. On the Study of Singular Nonlinear Travelling Wave Equations: DynamicalSystem Approach[M]. Beijing: Science Press, 2007.
[13] Shen J, Li J, Xu W. Bifurcations of travelling wave solutions in a model of the hydrogen-bonded systems[J]. Appl Math Comput, 2005, 171(1): 242-271.
[14] Shen J, Xu W, Xu Y. Travelling wave solutions in the generalized Hirota-Satsumacoupled KdV system[J]. Appl Math Comput, 2005, 161(2): 365-383.
[15] Tang S, Li M. Bifurcations of travelling wave solutions in a class of generalized KdVequation[J]. Appl Math Comput, 2006, 177(2): 589-596.
[16] Tang S, Huang W. Bifurcations of travelling wave solutions for the generalized doublesinh-Gordon equation[J]. Appl Math Comput, 2007, 189(2): 1774-1781.
[17] Tang S, Huang W. Bifurcations of travelling wave solutions for the K(n,-n,2n) equa-tions[J]. Appl Math Comput, 2008, 203(1): 39-49.
[18]唐生强,张彦文.三阶非线性Schro¨dinger方程行波解的分支[J].广西科学院学报,2006, 22(3): 148-152
[19]冯大河.非线性波方程的精确解与分支问题研究[D].昆明理工大学, 2007.
[20]张锦炎,冯贝叶.常微分方程几何理论与分支问题(第二版)[M].北京:北京大学出版社, 1980.
[21]叶彦谦,极限环论(修定版)[M].上海科技出版社, 1984.
[22]张芷芬,丁同仁等.微分方程定性理论[M].北京:科学出版社, 1985.
[23]李继彬,李存富.非线性微分方程[M].成都科技大学出版社, 1987.
[24]李继彬.非线性微分方程[M].昆明:云南科技出版社, 1987.
[25]陆启韶.常微分方程定性方法和分叉[M].北京:北京航空航天大学出版社, 1989.
[26]李继彬.浑沌与Melnikov方法[M].重庆:重庆大学出版社, 1989.
[27]丁同仁,李承治.常微分方程教程[M].北京:高等教育出版社, 1992.
[28]韩茂安,朱德明.微分方程分支理论[M].煤碳工业出版社, 1994.
[29]王高雄,周之铭等.常微分方程[M].北京:高等教育出版社, 1996.
[30]张芷芬,李承治等.向量场分支理论基础[M].北京:高等教育出版社, 1997.
[31]谷超豪等.应用偏微分方程[M].北京:高等教育出版社, 1993.
[32]刘式达,刘式适,物理学中的非线性方程[M].北京:北京大学出版社, 1998.
[33]李诩神,孤子与可积系统[M].上海:上海科技教育出版社, 1999.
[34]罗定军等.动力系统的定性与分支理论[M].北京:科学出版社, 2001.
[35]张锦炎,冯贝叶.常微分方程几何理论与分支问题[M].北京:北京大学出版社,2001.
[36]张芷芬,丁同仁,黄文灶等.微分方程定性理论[M].北京:科学出版社, 2003.
[37]范恩贵,可积系统与计算机代数[M].北京:科学出版社, 2004.
[38]谷超豪,胡和生,周子翔,孤立子理论中的达布变换及其几何应用(第二版)[M].上海:上海科学技术出版社, 2005.
[39]黄念宁,陈世荣.完全可积非线性方程的哈密尔顿理论[M].北京:科学出版社,2005.
[40] Zakharov V E, Kuznetsov E A. On three-dimensional solitons[J]. Sov Phys, 1974, 39:285-288.
[41] Sliivamoggi B K. Nonlinear ion-acoustic waves in weak magnetied plasma andZakharov-Kuznetsov equation[J]. J Plasma Phys, 1989, 41: 83-88.
[42] Shivamoggi B K. The Painlev′e analysis of the Zakharov-Kuznetsov equation[J]. PhysScr, 1990, 42: 641-642.
[43] Hamza A M. A kinetic derivation of a generalized Zakharov-Kuznetsov equation for ionacoustic turbulence in a magnetized plasma[J]. Phys Lett A, 1994, 190(3-4): 309-316.
[44] Munro S, Parkes E J. The derivation of a modified Zakharov-Kuznetsov equation andthe stability of its solutions[J]. J Plasma Phys, 1999, 62(3): 305-317.
[45] Munro S, Parkes E J. Stability of solitary-wave solutions to a modified Zakharov-Kuznetsov equation[J]. J Plasma Phys, 2000, 64(4): 411-426.
[46] Li B, Chen Y, Zhang H. Exact travelling wave solutions for a generalized Zakharov-Kuznetsov equation[J]. Appl Math Comput, 2003, 146: 653-666.
[47] Wazwaz A M. Special types of the nonlinear dispersive Zakharov-Kuznetsov equa-tion with compactons, solitons and periodic solutions[J]. Int J Comput Math, 2004,81(9):1107-1119.
[48] Wazwaz A M. Nonlinear dispersive special type of the Zakharov-Kuznetsov equationZK(n, n) with compact and noncompact structures[J]. Appl Math Comput, 2005, 161:577-590.
[49] Wazwaz A M. Exact solutions with solitons and periodic structures for the Zakharov-Kuznetsov (ZK) equation and its modified form[J]. Commun Nonlinear Sci NumerSimul, 2005, 10: 597-606.
[50] Wazwaz A M. Explicit travelling wave solutions of variants of the K(n, n) and theZK(n, n) equations with compact and noncompact structures[J]. Appl Math Comput,2006, 177(1): 213-230.
[51] Wazwaz A M. The extended tanh method for the Zakharov-Kuznetsov (ZK) equation,the modified ZK equation, and its generalized forms[J]. Commun Nonlinear Sci NumerSimul, 2008, 13: 1039-1047.
[52] Huang W. A polynomial expansion method and its application in the coupled Zakharov-Kuznetsov equations[J]. Chaos Solitons Fractals, 2006, 29: 365-371.
[53] Inc M. Exact solutions with solitary patterns for the Zakharov-Kuznetsov equationswith fully nonlinear dispersion[J]. Chaos Solitons Fractals, 2007, 33(15): 1783-1790.
[54] Elhanbaly A, Abdou M A. Exact travelling wave solutions for two nonlinear evolutionequations using the improved F-expansion method[J]. Math Comput Model, 2007, 46:1265-1276.
[55]闰振亚,张鸿庆.组合Zakharov-Kuznetsov方程的显式孤波解[J].纯粹数学与应用数学, 2000, 16(2): 31-35.
[56]林麦麦,段文山,吕克璞. (3+1)维ZK方程的N孤子解[J].西北师范大学学报, 2006,42(1): 41-45.
[57]胡越,田长安. Zakharov-Kuznetsov型方程的一类周期行波解[J].河南师范大学学报, 2007, 35(4): 35-37.
[58]刘式适,陈华,付遵涛,刘式达. Lam函数和非线性演化方程的多级准确解的不变性[J].物理学报, 2003,52(8): 1842-1847.
[59]套格图桑,斯仁道尔吉. BBM方程和修正的BBM方程新的精确孤立波解[J].物理学报, 2004,53(12): 4052-4060
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