用户名: 密码: 验证码:
两类数学物理方程的整体解
详细信息    本馆镜像全文|  推荐本文 |  |   获取CNKI官网全文
摘要
本文主要研究两类数学物理方程整体解的性态,即Minkowski时空中相对论膜的非线性动力学和水波理论中Nwogu型Boussinesq方程模型的孤立波。主要内容由以下几章组成。
     第一章对所考虑的两类数学物理方程的研究现状做一个简单介绍,并阐述本文要研究的问题,叙述我们得到的主要结果。
     第二章研究Minkowski空间中相对论膜的非线性动力学。通过变分法和几何方法,推导出Minkowski空间R1+n(n≥3)中相对论膜的运动方程。它是(1+2)维拟线性双曲型方程组,具有很多重要的性质,如非严格双曲性、常重特征、线性退化性和强零条件等;研究还发现,方程的平面波解都是类光极值子流形;反之,除了一类特殊的类光极值子流形外,其余所有的类光极值子流形都是方程的平面波解。
     第三章进一步研究相对论膜的非线性动力学。主要研究Minkowski空间R1+n(n≥3)中,我们所推导的相对论膜的运动方程,与以往所给出的经典方程之间的区别和联系。我们证明它们是等价的,并且从Noether定理角度重新认识此方程。同时,对于时空中相对论弦的情形给出类似相应的讨论。
     第四章研究水波理论中的Nwogu型Boussinesq方程模型的孤立波和周期波。此模型包含一个独立参数,这个参数与在不同水深时相应的水平速度有关。本章利用平面动力系统的分支方法,定性地研究此方程模型孤立波和周期波的存在条件。我们发现,在这个模型中,会出现一类新的尖峰波解—尖峰型周期波。
     第五章进一步研究上述Nwogu型Boussinesq方程模型,考察相向而行的孤立波的对撞问题。通过摄动方法,首先得到了方程的近似解。其次分析孤立波对撞的力学特征。由于模型包含独立参数,着重分析这个独立参数对于对撞的相移和最大波幅的影响;并将所得结果与经典可积的Boussinesq方程进行了比较。
The thesis concerns with the properties of global solutions for two kinds of mathemat-ical physical equations, precisely speaking, studies the nonlinear dynamics of relativistic membrane in the Minkowski space and solitary waves of the Nwogu's Boussinesq equation in water wave theory. It is organized as follows.
     Chapter 1 briefly recalls the present situation of the study on the two kinds of mathe-matical physical equations. The central problems under consideration and the main results obtained are stated.
     Chapter 2 concentrates on the nonlinear dynamics of relativistic membrane. By variational method and geometrical method, the system of equations for the relativistic membrane in the Minkowski space R1+n (n≥3) is derived. It can also be reduced to a (1+2)-dimensional quasilinear hyperbolic system and possesses many important properties such as non-strict hyperbolicity, constant multiplicity of eigenvalues, linear degeneracy of all characteristic fields, strong null condition, etc. An interesting phenomenon is found and proved, that is all plane wave solutions to this system are light-like extremal sub-manifolds and vice versa except for a type of special solution.
     Chapter 3 furthermore studies the nonlinear dynamics of relativistic membrane and mainly focus on the relationships between the equations for the relativistic membrane moving in the Minkowski space R1+n (n≥3) derived by us and that in canonical form in literature. The equivalence between them is proved and another geometric explanation about it through Noether's second theorem is also given. Moreover, for the motion of relativistic string in the Minkowski space, the similar argument about the equivalence is also presented.
     Chapter 4 investigates the solitary waves and periodic waves of Nwogu's Boussinesq equation in water wave theory. This model contains one independent parameter which is related to that the horizontal velocities at what level are chosen as the horizontal velocity variables. We employ the bifurcation method to qualitatively analyze the existence conditions of solitary waves and periodic waves for this model. Meanwhile, we find that the cusp periodic waves appear.
     Chapter 5 pays attention to the head-on collsions between two solitary waves of the Nwogu's Boussinesq equation. We first apply the perturbation method to this model and derive an approximate solution. Then the mechanics of the head-on collision, especially the impacts of the independent parameter on the phase shifts and the maximum run-up amplitude of two colliding waves is investigated. Comparison between our results and that of the integrable classical Boussinesq equation is also given
引文
[1]Dirac P. A. M., An extensible model for the electron, Proc. R. Soc. A 268 (1962),57-67.
    [2]Barbashov B. M., Nesterenk V. V., Introduction to the relativistic string theory, World Scientific Publishing Co., Inc., Teaneck, NJ, (1990).
    [3]Gilbarg D. and Trudinger N.S., Elliptic partial differential equations of second order (2nd Edition), Springer-Verlag, Berlin-Heidelberg-New York, (1984).
    [4]Plateau J.A.F., Statique experimentale et theorique des liquides soumis aux seules forces moleculaires, Paris, Gauthier-Villars, (1873).
    [5]Douglas Jesse, Solution of the problem of Plateau, Trans. Amer. Math. Soc.33 263-321, (1931).
    [6]Rado Tibor, On Plateau's problem, Ann. of Math.31457-469, (1930).
    [7]Osserman R. A., A survey of minimal surfaces, Second edition, Dover Publications, Inc. New York, (1986).
    [8]Calabi E., Examples of Bernstein problems for some nonlinear equations, in 1970 Global Analysis (Proc. Sympos. Pure Math., Vol. XV, Berkeley, Calif.,1968), Amer. Math. Soc. Providence, R.I.,223-230.
    [9]Cheng S. Y. and Yau S. T., Maximal space-like hypersurfaces in the Lorentz-Minkowski spaces, Ann. of Math.104, (1976),407-419.
    [10]E. Bombieri, E. De Giorgi and E. Giusti, Minimal cones and the Bernstein problem, Invent. Math.7, (1969),243-268.
    [11]Milnor T., Entire timelike minimal surfaces in E3+1, Michigan Math. J.37 (1990),163-177.
    [12]Barbashov B. M., Nesterenko V. V. and Chervyakov A. M., General solutions of nonlin-ear equations in the geometric theory of the relativistic string, Commun. Math. Phys.84, (1982),471-481.
    [13]Gu Chaohao, The extremal surfaces in the 3-dimensional Minkowski space, Acta Math. Sinica (N.S.) 1 (1985),173-180.
    [14]Gu Chaohao, A global study of extremal surfaces in 3-dimensional Minkowski space, in Differential geometry and differential equations (Shanghai,1985),26-33, Lecture Notes in Math.1255, Springer, Berlin, (1987).
    [15]Gu Chaohao, A class of boundary problems for extremal surfaces of mixed type in Minkowski 3-space, J. Reine Angew. Math.385 (1988),195-202.
    [16]Gu Chaohao, Extremal surfaces of mixed type in Minkowski space Rn+1, in Varia-tional methods (Paris,1988),283-296, Progr. Nonlinear Differential Equations Appl.4, Birkhauser Boston, Boston, MA, (1990).
    [17]Gu Chaohao, Complete extremal surfaces of mixed type in 3-dimensional Minkowski space, Chinese Ann. Math.15B (1994),385-400.
    [18]Kong De-Xing, Sun Qing-You and Zhou Yi, The equation for time-like extremal surfaces in Minkowski space R1+n, J. Math. Phys.,47, (2006),013503.
    [19]Kong De-Xing, Zhang Qiang and Zhou Qing, The dynamics of relativistic strings moving in the Minkowski space R1+n, Commun. Math. Phys.,269 (2007),153-174.
    [20]Kong De-Xing and Zhang Qiang, Solution formular and time periodicity for the motion of relativistic strings in the Minkowski space R1+n, to appear.
    [21]Kong De-Xing, A nonlinear geometric equation related to electrodynamics, Europhys. Lett., Vol.66, Iss.5, (2004),617-623.
    [22]Kong De-Xing, Lectures on dynamics of relativistic strings and membranes moving in phys-ical space-time, The Chinese University of Hong Kong (2006).
    [23]Benzoni-Gavage S., Serre D., Multidimensional hyperbolic partial differential equations. First-order systems and applications, Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford (2007).
    [24]Lax P. D., Hyperbolic systems of conservation laws II, Comm. Pure Appl. Math.10,537-566 (1957).
    [25]Lax P. D., Hyperbolic systems of conservation laws in several space variables, Current topics in partial differential equations,327-341, Kinokuniya, Tokyo (1986).
    [26]Kong De-Xing, Cauchy Problem for Quasilinear Hyperbolic Systems, MSJ Memoirs 6, the Mathematical Society of Japan, Tokyo (2000).
    [27]Kong De-Xing, Tsuji M., Global solutions for 2×2 hyperbolic systems with linearly degen-erate characteristics, Funkcialaj Ekvacioj 42, (1999),129-155.
    [28]Bordeman Martin, Hoppe Jens, The dynamics of relativistic membranes:Reduction to 2-dimensional fluid dynamics, Phys. Lett. B 317 (1993),315-320.
    [29]Bordeman Martin, Hoppe Jens, The dynamics of relativistic membranes II:Nonlinear waves and covariantly reduced membrane equations, Phys. Lett. B 325 (1994),359-365.
    [30]Hoppe Jens, Quantum theory of a massless relativistic surface and a two-dimensional bound state problem, MIT PhD. Thesis, (1982).
    [31]Hoppe Jens, Relativistic minimal surface, Phys. Lett. B 196 (1987),451-455.
    [32]Hoppe Jens, Some classical solutions of relativistic membrane equations in 4 space-time dimensions, Phys. Lett. B 329 (1994),10-14.
    [33]Born M. and Infeld L., Foundation of the new field theory, Proc. Roy. Soc. London A 144 (1934),425-451.
    [34]Lindblad H., A remark on global existence for small initial data of minimal surface equation in Minkowski space time, Proc. Amer. Math. Soc.132, (2004),1095-1102.
    [35]Christodoulou D., Global solutions of nonlinear hyperbolic equations for small initial data, Comm. Pure Appl. Math.39 (1986),267-282.
    [36]Klainerman S., The null condition and global existence to nonlinear wave equations, in Lectures in Appl. Math.23, Amer. Math. Soc., Providence, RI, (1986),293-326.
    [37]Christodoulou D., Klainerman S., The global nonlinear stability of the Minkowski space, Princeton Math. Ser.41, Princeton University Press, (1993).
    [38]Lindblad H., Global solutions of nonlinear wave equations, Comm. Pure Appl. Math.45, (1992),1063-1096.
    [39]Lindblad H., Rodnianski Igor, The weak null condition for Einstein's equations, C. R. Acad. Sci. Paris, Ser. Ⅰ 336, (2003),901-906.
    [40]Chae Dongho, Huh Hyungjin, Global exsitence for small initial data in the Born-Infeld equations, J. Math. Phys.44, (2003),6132-6139.
    [41]Whitham G. B., Linear and nonlinear waves, Wiley, New York, (1999), reprint of the 1974 original, a Wiley-Interscience publication.
    [42]吴耀祖,水波动力学研究进展,力学进展,第31卷,第3期,(2001),327-343.
    [43]Zhang Jin E., Water Wave Equations, Advances in Engineering Mechanics:Reflections and Outlooks, In honor of Theodore Y.-T. Wu, World Scientific Publishing Co. Ltd., (2005), 48-59.
    [44]李大潜,秦铁虎,物理学与偏微分方程(上册),高等教育出版社,(2005).
    [45]李大潜,秦铁虎,物理学与偏微分方程(下册),高等教育出版社,(2005).
    [46]Courant R., Friedrichs K. O., Supersolic flow and shock waves, Springer-Verlag, (1976).
    [47]Su C.H. and Mirie Ridam., On head-on collisions between two solitary waves, Journal of Fluid Mechanics,98, (1980),509-525.
    [48]Stokes G G. On the theory of oscillatary waves Trans. Cambrige Phil Soc.8, (1847),441-455.
    [49]Airy G B. Tides and waves, Encyc Metrop, (1845), Art 192.
    [50]Boussinesq J., Theorie des ondes et des remous qui se propagent le long d'un canal rect-angulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sen-siblement pareilles de la surface au fond, Journal de Mathematique Pures et Appliquees, Deuxieme Serie 17, (1872),55-108.
    [51]Korteweg D.J.and Vries G.de, On the Change of Form of Long Waves Advancing in a Rectangular Canal, and on a New Type of Long Stationary Waves, Philosophical Magazine, 39, (1895),422-443.
    [52]Ursell F. The long wave paradox in the theory of gravity waves, Proc Cambrige Phil Soc, 49, (1953),685-694.
    [53]Hirota R., Exact N-soliton solution of the wave equation of long wave in shallow water and in nonlinear lattices, J. Math. Phys.14, (1973),810-814.
    [54]Zhang Jin E. and Li Yi-Shen, Bidirectional solitons on water, Physical Review E 67, (2003), 016306.
    [55]李翊神,孤子与可积系统,上海科学教育出版社,(1999).
    [56]Zabusky N. J. and Kruskal M. D., Interaction of "solitons" in a collisionless plasma and the recurrence of initial states,Phys. Rev. Lett.15, (1965),240-243.
    [57]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,上海科学技术出版社,(2002).
    [58]楼森岳,唐晓艳,非线性数学物理方法,科学出版社,(2006).
    [59]Gardner C. S., J. M. Greene, Kruskal M. D., and Miura R. M., Method for solving the Korteweg-de Vries equation Phys. Rev. Lett.19, (1967),1095-1097.
    [60]Peregrine D. H., Long waves on a beach, J. Fluid Mech.27, (1967),815-827.
    [61]Wu Theodore Yao-tsu, Long waves in ocean and costal waters, J.Engeneering Mechanics, 107, (1981),501-522.
    [62]Nwogu O., An alternative form of the Boussinesq equations for nearshore wave propagation, J. Waterw., Port, Coastal, Ocean Eng.119, (1993),618-638.
    [63]Wei G., Kirby J. T., Grilli S. T. and Subramanya R., A fully nonlinear Boussinesq model for surface waves Part 1. Highly nonlinear unsteady waves, J. Fluid Mech.294, (1995), 71-92.
    [64]Wei G. and Kirby J. T., Waterw J., A time-dependent numerical code for extended Boussi-nesq equations, Port, Coastal, Ocean Eng.121, (1995),251-261.
    [65]Wu Theodore Yao-tsu, Zhang Jin E., On modelling nonlinear long waves, In:Cook LP, Roytburd V, Tulin M, editors. Mathematics is for solving problem:a volume in honor of Julian Cole on his 70th birthday. SIMA, (1996),233-49.
    [66]Chen Chun-Li, Tang Xiao-Yan and Lou Sen-Yue, Solutions of a (2+1) -dimensional dis-persive long wave equation, Physical Review E 66, (2002),036605.
    [67]Ji Xiao-Da, Chen Chun-Li, Zhang Jin E. and Li Yi-Shen, Lie symmetry analysis and some new exact solutions of the Wu-Zhang equation, J.Math.Phys.45, (2004),448-460.
    [68]Li Yi-Shen, Some Water Wave Equations and Integrability, Journal of Nonlinear Mathe-matical Physics,12, Supplement 1, (2005),466-481.
    [69]Chen Chun-Li and Lou Sen-Yue, Soliton excitations and periodic waves without dispersion relation in shallow water system, Chaos, Solitons and Fractals 16, (2003),27-35.
    [70]Weiss J., Tabor M., and Carnevale G., The Painleve property for partial differential equa-tions, J. Math. Phys.24,(1983),522-526.
    [71]Conte R., Invariant Painleve analysis of partial differential equations, Phys. Lett. A 140, (1989),383.
    [72]Pickering A., A new truncation in Painleve analysis, J. Phys. A 26, (1993),4395-4405.
    [73]Lou Sen-Yue, Extended Painleve Expansion, Nonstandard Truncation and Special Reduc tions of Nonlinear Evolution Equations, Z. Naturforsch. A 53, (1998),251-258.
    [74]Chen Chun-Li, Li Yi-Shen, Lou Sen-Yue, Solitary wave solutions for a general Boussinesq type fluid model, Communications in Nonlinear Science and Numerical Simulation,9 (2004), 583-601.
    [75]Zhang Jin E., Chen Chun-Li and Li Yi-Shen, On Boussinesq models of constant depth, Physics of fluids 16 No.5, (2004),1287-1296.
    [76]Kaup DJ. A higher order water-wave equation and the method for solving it Prog. Theor. Phys.,54 (1975),396-408.
    [77]Kupershmidt BA., Mathematics of dispersive water-waves, Comm. Math. Phys.,99 (1985), 51-73.
    [78]Chen Chun-Li and Lou Sen-Yue,Soliton excitations and periodic waves without dispersion relation in shallow water system Chaos, Solitons and Fractals,16 2003,27-35.
    [79]Conte R., Musette M. and Pickering A., Factorization of the 'classical Boussinesq'system, J. Phys. A:Math. Gen.27 (1994),2831-2836.
    [80]Pickering A., The singular manifold method revisited, J. Math. Phys.,37 (1996),1894-1927.
    [81]Musette M. and Conte R. The two-singular-manifold method:I. Modified Korteweg-de Vries and sine-Gordon equations, J. Phys. A:Math. Gen.,27, (1994),3895-3915.
    [82]Musette M., Conte R. and Pickering A. The two-singular manifold method:II Classical Boussinesq system., J. Phys. A:Math. Gen.,28 (1995),179-188.
    [83]Chen Chun-Li, Huang Shou-Jun and Zhang Jin E., On Head-on Collisions Between Two Solitary Waves of Nwogu's Boussinesq Model, Journal of the Physical Society of Japan 77, (2008),014003.
    [84]Hawking S., The large scale structure of space-time, Cambrige University Press, (1973).
    [85]陈维桓,黎曼几何引论(上册),北京大学出版社,(2002).
    [86]Majda A., Compressible Fluid Flow and System of Conservation Laws in Several Space Variables, Applied Mathematical Sciences 53, Springer-Verlag (1984).
    [87]Kong De-Xing, A necessary and sufficient condition for the diagonalization of multi-dimensional quasilinear systems, Electron. J. Diff. Eqns., Vol.1999, No.25, (1999),1-14.
    [88]John Fritz, Nonlinear wave equations, Formation of Singularities, Pitcher Lectures in the Mathematical Sciences, American mathematical society, (1989).
    [89]Collins P. A., Tucker R. W., Classical and quantum mechanics of free relativistic mem-branes, Nucl. Phys. B 112 (1976),150-176.
    [90]Huang Shou-Jun, Kong De-Xing, Equations of relativistic torus in the Monkowski space R1+n, J. Math. Phys.,48 (2007),083510.
    [91]Huang Shou-Jun, He Chun-Lei and Shi Jiang-Li, Remarks on the equations for the motion of relativistic membranes in the Minkowski space R1+n, submitted.
    [92]Larsen A.L., Lousto C.O., On the stability of spherical membranes in curved space-times, Nuclear Physics B 472, (1996),361-376.
    [93]Noether E., Invariante Variationsprobleme, Nachr. Konig. Gesell. Wissen. Gottingen, Math. Phys. Kl. (1918),235-257 (see Transport Theory and Stat. Phys.1 (1971) 186-207 for an English translation).
    [94]Olver P. J., Applications of Lie Groups to Differential Equations, Springer-Verlag, New York, (1993).
    [95]Huang Fei, Tang Xiao-Yan and Lou Sen-Yue, Exact Solutions for a Higher-Order Nonlinear Schrodinger Equation in Atmospheric Dynamics, Commun. Theor. Phys.45, (2006),573-576.
    [96]Lou Sen-Yue, Self-Steepening and Third-Order Dispersion Induced Optical Solitons in Fiber , Commun. Theor. Phys.,35 (2001),589-592.
    [97]Li Yi-Shen, Zhang Jin E., Bidirectional soliton solutions of the classical Boussinesq system and AKNS system, Chaos, Solitons and Fractals 16 (2003),271-277.
    [98]罗定军,张祥,董梅芳,动力系统的定性与分支理论,科学出版社,(2001).
    [99]李继彬,赵晓华,刘正荣,广义Hamilton系统的理论和应用,科学出版社,(1994).
    [100]韩茂安,动力系统的周期解与分支理论,科学出版社,(2002).
    [101]Perko L., Differential Equations and Dynamical Systems, Springer-Verlag, New York, (1991).
    [102]Chou SN, Hale JK, Method of Bifurcation Theory, Springer-Verlag, New York,1981.
    [103]Guckenheimer J., Holmes PJ, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer-Verlag, New York, (1983).
    [104]Huang Guo-Xiang, Lou Sen-Yue and Xu Zai-Xin, Head-on collisions between two solitary waves in a Rayleigh-Benard convecting fluid, Physical Review E,47, (1993),3830-3833.
    [105]Dai Hui-Hui, Dai Shi-Qiang and Huo Yi, Head-on collision between two solitary waves in a compressible Mooney-Rivlin elastic rod, Wave Motion,32, (2000),93-111.
    [106]Bona J.L., Chen M., and Saut J.C., Boussinesq Equations and Other Systems for Small-Amplitude Long Waves in Nonlinear Dispersive Media. I:Derivation and Linear Theory, J. Nonlinear Sci.12, (2002),283-318.
    [107]Bona J.L., Chen M., and Saut J.C., Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media:II. The nonlinear theory, Nonlinearity 17, (2004),925-952.
    [108]Huang Shou-Jun and Chen Chun-Li, Study on solitary waves of a general Boussinesq model, Communications in Theoretical Physics,48 (2007),773-780.
    [109]Xue Ju-Kui, Head-on collision of the blood solitary waves, Physics Letters A,331 (2004), 409-413.
    [110]Van Dyke, Perturbation methods in fluid mechanics, New York, Academic Press Inc. (1964).
    [111]Oikawa M. and Yajima N., Interactions of Solitary Waves:A Perturbation Approach to Nonlinear Systems, J. Phys. Soc. Jpn.34, (1973),1093-1099.
    [112]Oikawa M. and Yajima N., A Perturbation Approach to Nonlinear Systems. II. Interaction of Nonlinear Modulated Waves, J. Phys. Soc. Jpn.37, (1974),486-496.

© 2004-2018 中国地质图书馆版权所有 京ICP备05064691号 京公网安备11010802017129号

地址:北京市海淀区学院路29号 邮编:100083

电话:办公室:(+86 10)66554848;文献借阅、咨询服务、科技查新:66554700