关于三维空间中的Klein-Gordon-Zakharov方程
|
摘要
非线性波动系统是数学物理中最具吸引力的研究领域之一。一方面,它揭示着现代物理学中一些最深刻的规律和运动规则;另一方面,作为最重要的一类偏微分方程,它一直是核心数学的重要部分。经典的非线性波动系统主要是KdV方程,非线性Schr(?)dinger方程,非线性Klein-Gordon方程等。KdV方程是一种典型的孤立子波模型,而非线性Schr(?)dinger方程及Klein-Gordon方程则是量子力学中的重要模型。
近20年来,围绕上述三类模型的数学研究取得了一系列重要进展。尤其是在其典型性质如初值问题局部解的适定性、解在有限时间内的爆破性质及其动力学行为、整体解的存在性及其渐近行为、驻波解的存在性及其稳定性的研究上取得了丰硕的成果。作为这些成果取得的代表人物,W.A.Strauss,J.Ginibre,T.Cazenave,H.A.Levine,F.Merle,Y.Tsutsumi,李大潜、郭柏灵等数学家在偏微分方程及核心数学的现代进展中起着标志性的作用。
Klein-Gordon-Zakharov方程是近十年来引起关注的一个重要的非线性模型。它是一个耦合的数学物理方程组,描述了等离子区域中朗谬尔波与离子声波的相互作用等物理现象。该系统是由一个Klein-Gordon方程与一个经典的双曲波方程按Zakharov系统的耦合形式形成的一个非线性耦合方程组。除了它在物理背景上所表示的明确意义外,在数学上亦具有典型
Nonlinear wave system is one of the most attracting research fields in mathematical physics. On the one hand it delineates some abstruse laws and rules of motion in modern physics, and on the other hand, as one of the most important partial differential equations, it is one important part of the core mathematics. Classical nonlinear wave equations mainly include the Korteweg-de Vries equation, the nonlinear Schrodinger equation and the nonlinear Klein-Gordon equation etc. Korteweg-de Vries equation is a typical solitary wave model, and nonlinear Schrodinger equations as well as nonlinear Klein-Gordon equations are major models in quantum mechanics.In the recent twenty years, a series of important advances are achieved on the mathematical studies around the above three models. Especially for their typical properties such as the local well-posedness of the initial value problems, blowingup properties of the solutions in a finite time and their dynamical behavior, existence of the global solutions and their asymptotic behavior, existence of the standing waves and their stability, plentiful and substantial results are got. As the representative persons who accomplished these studies, mathematicians such as W.A.Strauss, J.Ginibre, T.Cazenave, F.Merle, Y.Tsutsumi, T.S.Li and B.L.Guo play a symbolic role in the modern advance of PDE and the core mathematics.The Klein-Gordon-Zakharov equations are an important nonlinear wave
model that is noticed in the recent five years. It is a coupled mathematical physics system which describes the interaction of the Langmuir wave and the ion acoustic wave in a plasma and so on. This system is a nonlinear coupled equations which are formed by a Klein-Gordon equation and a classical hyperbolic wave equation according to a coupled means of the Za-kharov equations. Besides the explicit meaning in physical background, it has typical feature in Mathematics. In [23], Ozawa, Tsutaya and Tsutsumi studied the Cauchy problem of the Klein-Gordon-Zakharov equations with different propagation speeds in three space dimensions. In terms of the correlated theory of the homogeneous Sobolev space of negative index and the discrepancy between the propagation speeds in the Klein-Gordon-Zakharov equations, they obtained the local well-posedness for the Cauchy problem in the energy space by using a harmonic analysis method and a contraction method. In addition, by combining this result and the energy conservation law, they obtained the unique global solutions for the Cauchy problem of the Klein-Gordon-Zakharov equations in the energy space for small initial data. Moreover, in [2,24], they got the global existence of small amplitude solution for the Cauchy problem of the Klein-Gordon-Zakharov equations in the case of c = 1.In the present paper, combining the local well-posedness for the Cauchy problem of the Klein-Gordon-Zakharov equations and the related theory of the homogeneous Sobolev space of negative index, we study the Klein-Gordon-Zakharov equations with different propagation speeds in three space dimensions under the framework of variational calculus, we first obtain the existence of the standing waves for the Klein-Gordon-Zakharov equations under the case of non-radial symmetry and radial symmetry by constructing two kinds of different constrained variational problems. We next get that the
solution for the Cauchy problem of the Klein-Gordon-Zakharov equations blows up in a finite time when the initial energy is negative by applying the potential well argument and the concavity method. We then derive out the sharp condition of global existence for the Cauchy problem of the Klein-Gordon-Zakharov equations by using scaling argument. Finally, in the light of the harmonic analysis method and the lower semi-continuity of weak limit, we prove the instability of the standing waves for the Klein-Gordon-Zakharov equations under the case of radial symmetry and non-radial symmetry.In introduction, first of all, we present the physical background and some known results on the Klein-Gordon-Zakharov equations. Secondly, we recall some known results on the classical nonlinear Klein-Gordon equation and Schrodinger equation as well as the Zakharov system. Finally, we present the main results of the present paper.In chapter 2, we first study the local well-posedness for the Cauchy problem of the Klein-Gordon-Zakharov equations with different propagation speeds in three space dimensions in the energy space, and obtain the unique existence of the solutions for the Cauchy problem on a maximal time interval in the light of [23]. We next introduce the expression form of the definition and the norm about the homogeneous Sobolev space of negative index and radially symmetric Sobolev space. At last, we construct proper functionals and manifolds in view of the distinctive feature of the Klein-Gordon-Zakharov equations.In chapter 3, we study the standing waves of the Klein-Gordon-Zakharov equations under the case of non-radial symmetry and radial symmetry, respectively. First of all, we construct two proper constrained variational problems. Next, according to the characteristics of the ground state and
the local theory, we prove the existence of the non-radial symmetric standing waves by variational calculus. Then, by solving a constrained varia-tional problem, which is equivalent to the constrained variational problem correlated to the radially symmetric standing waves, we obtain the solution of the constrained variational problem. Finally, by Lagrange multiplier method, we prove that the solution of the constrained variational problem is the solution of the nonlinear elliptic equations corresponding to the Klein-Gordon-Zakharov equations. Thus the existence of the radially symmetric standing waves is established.In chapter 4, we study the blowup of the solution for the Cauchy problem of the Klein-Gordon-Zakharov equations. Based on the characteristics of the ground state and the existence of the non-radially symmetric standing waves, we obtain the solution of the Cauchy problem blows up in a finite time when initial energy is negative by using the potential well argument and the concavity method.In chapter 5, we study the global solutions for the Cauchy problem of the Klein-Gordon-Zakharov equations. According to the existence of the standing waves with ground state under the case of non-radial symmetry, using the result in chapter 4, we derive out a sharp condition of global existence and blow up for the Cauchy problem. Moreover, by using scaling argument, we answer the question of how small the initial data are, the global solutions of the Cauchy problem exist.In chapter 6, we prove the instability of the standing waves for the Klein-Gordon-Zakharov equations under the case of non-radial symmetry and radial symmetry, respectively. Firstly, we obtain the strong instability of the standing waves under the case of non-radial symmetry by applying potential well argument and the concavity method. Secondly, by using
the original definition of stability, we show that the standing waves for the Klein-Gordon-Zakharov equations under the case of radial symmetry are always instability, regardless of the solutions for the Cauchy problem of the Klein-Gordon-Zakharov equations exist locally or globally. In the end of this chapter, we compare the two type of standing waves and their instabilities under the case of non-radial symmetry and radial symmetry for the Klein-Gordon-Zakharov equations and obtain that the standing waves under the two cases are different. The instability of the standing waves under the case of non-radial symmetry is initiated by blow up in a finite time, yet, the instability of the standing waves under the case of radial symmetry has nothing to do with the solutions blow up in a finite time or exist globally.
近20年来,围绕上述三类模型的数学研究取得了一系列重要进展。尤其是在其典型性质如初值问题局部解的适定性、解在有限时间内的爆破性质及其动力学行为、整体解的存在性及其渐近行为、驻波解的存在性及其稳定性的研究上取得了丰硕的成果。作为这些成果取得的代表人物,W.A.Strauss,J.Ginibre,T.Cazenave,H.A.Levine,F.Merle,Y.Tsutsumi,李大潜、郭柏灵等数学家在偏微分方程及核心数学的现代进展中起着标志性的作用。
Klein-Gordon-Zakharov方程是近十年来引起关注的一个重要的非线性模型。它是一个耦合的数学物理方程组,描述了等离子区域中朗谬尔波与离子声波的相互作用等物理现象。该系统是由一个Klein-Gordon方程与一个经典的双曲波方程按Zakharov系统的耦合形式形成的一个非线性耦合方程组。除了它在物理背景上所表示的明确意义外,在数学上亦具有典型
Nonlinear wave system is one of the most attracting research fields in mathematical physics. On the one hand it delineates some abstruse laws and rules of motion in modern physics, and on the other hand, as one of the most important partial differential equations, it is one important part of the core mathematics. Classical nonlinear wave equations mainly include the Korteweg-de Vries equation, the nonlinear Schrodinger equation and the nonlinear Klein-Gordon equation etc. Korteweg-de Vries equation is a typical solitary wave model, and nonlinear Schrodinger equations as well as nonlinear Klein-Gordon equations are major models in quantum mechanics.In the recent twenty years, a series of important advances are achieved on the mathematical studies around the above three models. Especially for their typical properties such as the local well-posedness of the initial value problems, blowingup properties of the solutions in a finite time and their dynamical behavior, existence of the global solutions and their asymptotic behavior, existence of the standing waves and their stability, plentiful and substantial results are got. As the representative persons who accomplished these studies, mathematicians such as W.A.Strauss, J.Ginibre, T.Cazenave, F.Merle, Y.Tsutsumi, T.S.Li and B.L.Guo play a symbolic role in the modern advance of PDE and the core mathematics.The Klein-Gordon-Zakharov equations are an important nonlinear wave
model that is noticed in the recent five years. It is a coupled mathematical physics system which describes the interaction of the Langmuir wave and the ion acoustic wave in a plasma and so on. This system is a nonlinear coupled equations which are formed by a Klein-Gordon equation and a classical hyperbolic wave equation according to a coupled means of the Za-kharov equations. Besides the explicit meaning in physical background, it has typical feature in Mathematics. In [23], Ozawa, Tsutaya and Tsutsumi studied the Cauchy problem of the Klein-Gordon-Zakharov equations with different propagation speeds in three space dimensions. In terms of the correlated theory of the homogeneous Sobolev space of negative index and the discrepancy between the propagation speeds in the Klein-Gordon-Zakharov equations, they obtained the local well-posedness for the Cauchy problem in the energy space by using a harmonic analysis method and a contraction method. In addition, by combining this result and the energy conservation law, they obtained the unique global solutions for the Cauchy problem of the Klein-Gordon-Zakharov equations in the energy space for small initial data. Moreover, in [2,24], they got the global existence of small amplitude solution for the Cauchy problem of the Klein-Gordon-Zakharov equations in the case of c = 1.In the present paper, combining the local well-posedness for the Cauchy problem of the Klein-Gordon-Zakharov equations and the related theory of the homogeneous Sobolev space of negative index, we study the Klein-Gordon-Zakharov equations with different propagation speeds in three space dimensions under the framework of variational calculus, we first obtain the existence of the standing waves for the Klein-Gordon-Zakharov equations under the case of non-radial symmetry and radial symmetry by constructing two kinds of different constrained variational problems. We next get that the
solution for the Cauchy problem of the Klein-Gordon-Zakharov equations blows up in a finite time when the initial energy is negative by applying the potential well argument and the concavity method. We then derive out the sharp condition of global existence for the Cauchy problem of the Klein-Gordon-Zakharov equations by using scaling argument. Finally, in the light of the harmonic analysis method and the lower semi-continuity of weak limit, we prove the instability of the standing waves for the Klein-Gordon-Zakharov equations under the case of radial symmetry and non-radial symmetry.In introduction, first of all, we present the physical background and some known results on the Klein-Gordon-Zakharov equations. Secondly, we recall some known results on the classical nonlinear Klein-Gordon equation and Schrodinger equation as well as the Zakharov system. Finally, we present the main results of the present paper.In chapter 2, we first study the local well-posedness for the Cauchy problem of the Klein-Gordon-Zakharov equations with different propagation speeds in three space dimensions in the energy space, and obtain the unique existence of the solutions for the Cauchy problem on a maximal time interval in the light of [23]. We next introduce the expression form of the definition and the norm about the homogeneous Sobolev space of negative index and radially symmetric Sobolev space. At last, we construct proper functionals and manifolds in view of the distinctive feature of the Klein-Gordon-Zakharov equations.In chapter 3, we study the standing waves of the Klein-Gordon-Zakharov equations under the case of non-radial symmetry and radial symmetry, respectively. First of all, we construct two proper constrained variational problems. Next, according to the characteristics of the ground state and
the local theory, we prove the existence of the non-radial symmetric standing waves by variational calculus. Then, by solving a constrained varia-tional problem, which is equivalent to the constrained variational problem correlated to the radially symmetric standing waves, we obtain the solution of the constrained variational problem. Finally, by Lagrange multiplier method, we prove that the solution of the constrained variational problem is the solution of the nonlinear elliptic equations corresponding to the Klein-Gordon-Zakharov equations. Thus the existence of the radially symmetric standing waves is established.In chapter 4, we study the blowup of the solution for the Cauchy problem of the Klein-Gordon-Zakharov equations. Based on the characteristics of the ground state and the existence of the non-radially symmetric standing waves, we obtain the solution of the Cauchy problem blows up in a finite time when initial energy is negative by using the potential well argument and the concavity method.In chapter 5, we study the global solutions for the Cauchy problem of the Klein-Gordon-Zakharov equations. According to the existence of the standing waves with ground state under the case of non-radial symmetry, using the result in chapter 4, we derive out a sharp condition of global existence and blow up for the Cauchy problem. Moreover, by using scaling argument, we answer the question of how small the initial data are, the global solutions of the Cauchy problem exist.In chapter 6, we prove the instability of the standing waves for the Klein-Gordon-Zakharov equations under the case of non-radial symmetry and radial symmetry, respectively. Firstly, we obtain the strong instability of the standing waves under the case of non-radial symmetry by applying potential well argument and the concavity method. Secondly, by using
the original definition of stability, we show that the standing waves for the Klein-Gordon-Zakharov equations under the case of radial symmetry are always instability, regardless of the solutions for the Cauchy problem of the Klein-Gordon-Zakharov equations exist locally or globally. In the end of this chapter, we compare the two type of standing waves and their instabilities under the case of non-radial symmetry and radial symmetry for the Klein-Gordon-Zakharov equations and obtain that the standing waves under the two cases are different. The instability of the standing waves under the case of non-radial symmetry is initiated by blow up in a finite time, yet, the instability of the standing waves under the case of radial symmetry has nothing to do with the solutions blow up in a finite time or exist globally.
引文
[1] V. Georgiev, Global solution of the system of wave and Klein-Gordon equations, Mathematische Zeitschrift, 203(1990), 683-698.
[2] T. Ozawa, K. Tsutaya, Y. Tsutsumi, Normal form and global solutions for the Klein-Gordon-Zakharov equations, Annales de I'Institut Henri Poincaré, 12(4)(1995), 459-503.
[3] R. Melrose, R. Ritter, Interaction of nonlinear progressing waves, Annals of Mathematics, 121(2)(1985), 187-213.
[4] J. Rauch, M. Reed, Singularities produced by the nonlinear interaction of three progressing waves: examples, Comm. PDE., 7(1982), 1117-1133.
[5] JCH. Simon, E. Taflin, Wave operators and analytic solutions for systems of nonlinear Klein-Gordon equations and of nonlinear Schrodinger equations, Comm. Math. Phys., 99(1985), 541-562.
[6] Y. Tsutsumi, J. Zhang, Instability of optical solitons for two-wave interaction model in cubic nonlinear media, Advances in Mathematical Sciences and Applications, 8(1999), 691-713.
[7] R. O. Dendy, Plasma Dynamics, Oxford University Press, Oxford, 1990.
[8] V. E. Zakharov, Collapse of Langmuir waves, Sov. Phys. JETP, 35(1972), 908-914.
[9] H. A. Levine, Instability and nonexistence of global solutions to nonlinear wave equations of the form Pu_(tt)=-Au-F(u), Transactions of the American Mathematical Society, 92(1974), 1-21.
[10] L. E. Pagne, D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel Journal of Mathematics, 22(3-4)(1975), 273-303.
[11] W. A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys., (55)(1977), 149-162.
[12] H. Berestycki, T. Cazenave, Instabilité des états stationnaires dans les équations de Schrodinger et de Klein-Gordon non linéairees, C. R. Acad. Sci. Paris, 293(1981), 489-492.
[13] J. Shatah, W. Strauss, Instability of nonlinear bound states, Comm. Math. Phys., 100(1985), 173-190.
[14] W. Strauss, On weak solutions of semi-linear hyperbolic equations, Ann. Acad. Brasil. Cienc., 42(1970), 645-651.
[15] W. Strauss, Stability of solitary waves, Contemporary Math., 107(1990), 123-129.
[16] P. E. Souganidis, W. A. Strauss, Instability of a class of dissipersive solitary waves, Proc. Roy. Soc. Edin., 114A(1990), 195-212.
[17] M. Weinstein, Lyapunov stability of ground states of nonlinear dispersive evolution equations, Comm. Pure Appl. Math., 39(1986), 51-68.
[18] J. Shatah, Stable standing waves of nonlinear Klein-Gordon equations, Comm. Math, Phys., 91(1983), 313-327.
[19] J. Shatah, Unstable ground state of nonlinear Klein-Gordon equations, Transactions of the American Mathematical Society, 290(2)(1985), 701-710.
[20] M, Grillakis, J, Shatah, W. Strauss, Stability theory of solitary waves in the presence of symmetry, Part Ⅰ, J, Funct. Anal., 74(1987), 160-197.
[21] M. Grillakis, J. Shatah, W. Strauss, Stability theory of solitary waves in the presence of symmetry, Part Ⅱ, J. Funct. Anal., 94(1990), 308-348.
[22] J. Zhang, Sharp conditions of global existence for nonlinear Schrodinger and KleinGordon equations, Nonlinear Anal. TMA, 48(2002), 191-207.
[23] T. Ozawa, K. Tsutaya, Y. Tsutsumi, Well-posedness in energy space for the Cauchy problem of the Klein-Gordon-Zakharov equations with different propagation speeds in three space dimensions, Math. Ann., 313(1999), 127-140.
[24] K, Tsutaya, Global existence of small amplitude solutions for the Klein-GordonZakharov equations, Nonlinear Anal, TMA., 27(1996), 1373-1380.
[25] T. B. Benjamin, The stability of solitary waves, Proceedings of the Royal Society London Series A, 328(1972), 153-183.
[26] H. Berestycki, P. L, Lions, Nonlinear scalar field equations, Ⅰ, Ⅱ, Archive for Rational Mechanics and Analysis, 82(1983), 313-375.
[27] E. Infeld, G. Rowlands, Nonlinear Waves, Solitons and Chaos, Cambridge University Press: Cambridge, 1990.
[28] Y. Liu, Instability of solitary waves for generalized Boussinesq equations, Journal of Dynamics and Differential Equations, 1993, 537-558.
[29] M. Ohta, Instability of standing waves for the generalized Davey-Stewartson system, Annales de I'Institut Henri Poincaré, 62(1)(1995), 69-80.
[30] W. Strauss, Nonlinear Wave Equations, C. B. M. S., Vol. 73. American Mathemat- ical Society: Providence, RI, 1989.
[31] E. Zeidler, Applied Functional Analysis, Springer: Berlin, 1995.
[32] J. Ginibre, G. Velo, On a class of nonlinear Schrodinger equations, J. Funct. Anal., 32(1979), 1-71.
[33] R. T. Glassey, On the blowing up of solutions to the Cauchy problem for nonlinear Schrodinger equations, J. Math. Phys., 18(9)(1977), 1794-1797.
[34] M. Tsutsumi, Nonexistence of global solutions to the Cauchy problem for nonlinear Schrodinger equations, SIAM J. Math. Anal., 15(1984), 357-366.
[35] T. Ogawa, Y. Tsutsumi, Blow-up of H~1 solution for the nonlinear Schrodinger equation, J. Differential Equations, 92(2)(1991), 317-330.
[36] T. Ogawa, Y. Tsutsumi, Blow-up of H~1 solution for the nonlinear Schrodinger equation with critical power nonlinearity, Proc. Amer. Math. Soc., 111(2)(1991), 487-496.
[37] J. Ginibre, T. Ozawa, Long range scattering for nonlinear Schrodinger and Hartree equations in space dimensions n≥2, Comm. Math. Phys., 15(1993), 619-645.
[38] J. Ginibre, G. Velo, The global Cauchy problem for the nonlinear Schrodinger equation, revisited, Ann. Inst. H. Poincaré, Anal. Non Linéaire, 2(1985), 309-327.
[39] N. Hayashi, Y. Tsutsumi, Scattering theory for Hartree type equations, Ann. Inst. H. Poincaré Phys. Théor., 46(1987), 187-213.
[40] N. Hayashi, K. Nakamitsu, M. Tsutsumi, On solutions of the initial value problem for the nonlinear Schrodinger equations, J. Funct. Anal., 71(1987), 218-245.
[41] C. Kenig, G. Ponce, L. Vega, Small solution to nonlinear Schrodinger equations, Ann. Inst. H. Poincaré, Anal. Non Linéaire, 10(1993), 255-288.
[42] T. Cazenave, An introduction to nonlinear Schrodinger equations, Textos de Metodos Mathematicos, Vol. 22, Rio de Janeiro, 1989.
[43] Z. Gan, J. Zhang, Sharp criterions of global existence and collapse for coupled nonlinear Schrodinger equations, J. Partial Diff. Eqs., 17(2004), 207-220.
[44] J. Zhang, On the standing wave in coupled nonlinear Klein-Gordon equations, Math. Meth. Appl. Sci., 26(2003), 11-25.
[45] J. Zhang, Cross-Constrained variational problem and nonlinear Schrodinger equation, Proceedings of International Conference on Foundations of Computational Mathematics in honor of Professor Steve Smale's 70th Birthday, HongKong, 2001.
[46] M. I. Weinstein, Nonlinear Schrodinger equations and sharp interpolations estimates, Comm. Math. Phys., 87(1983), 567-576.
[47] C. C. Bradley, C. A. Sackett and R. G. Hulet, Bose-Einstein condensation of Lithium observation of limited condensate number, Phys. Rev. Lett., 78(1997), 985-989.
[48] B. M. Caradoc-Davies, R. J. Ballagh and K. Burnett, Coherent dynamics of vortex formation in trapped Bose-Einstein consenstates, Phys. Rev. Lett., 83(5)(1999), 895-898.
[49] F. Dalfovo, S. Giorgini and L. P. Pitaevskii, et al., Theory of Bose-Einstein condensation in trapped gases, Reviews of Modern Physics, 71(3)(1999), 463-512.
[50] J. Garcia-Ripoll and V. M. Pérez-Garcia, Extended parametric resonances in nonlinear Schrodinger systems, Phys. Rev. Lett., 83(9)(1999), 1715-1718.
[51] T. Tsutsumi and M. Wadati, Collapses of wave functions in multidimensional nonlinear Schrodinger equations under harmonic potential, J. Phys. Soc. Jpn., 66(1997), 3031-3034.
[52] J. Zhang, A variationa approach to nonlinear Schrodinger equations under a harmonic potential, Preprint(2000).
[53] H. Pecher, L~P-Abschatzungen and klassiche Losungen für nichtlineare wellengeichungen, I, Math. Z., 150(1976), 159-183.
[54] J. M. Ball, Finite time blow-up in nonlinear problems, in: M. G. Grandall(Ed.), Nonlinear Evolution Equations, Academic Press, New York, 1978, 189-205.
[55] A. Softer, M. I. Weinstein, Resonance, radiation damping and instability in Hamiltonian nonlinear wave equations, Invent. Math., 136(1999), 9-74.
[56] Z. Gan, J. Zhang, Standing waves for nonlinear Klein-Gordon equations with nonnegative potentials, Applied Mathematics and Computation(accepted).
[57] H. Added and S. Added, Existence globale de solutions fortes pour les équations de la turbulence de Langmuir en dimension 2, C. R. Acad. Sci. Paris, 299(1984), 551-554.
[58] H. Added and S. Added, Equations of Langmuir turbulence and nonlinear Schrodinger equation: smoothness and approximation, J. Funct. Anal., 79(1988), 183-210.
[59] J. Bourgain, On the Cauchy and invariant measure problem for the periodic Za- kharov system, Duke Math. J., 76(1994), 175-203.
[60] J. Bourgain and J. Colliander, On well-posedness of the Zakharov system, Int. Math. Res. Not., 11(1996), 515-546.
[61] L. Glangetas and F. Merle, Existence of self-similar blow up solutions for Zakharov equation in dimension 2, Comm. Math. Phys., 160(1994), 173-215 and 349-389.
[62] C. Kenig, G. Ponce, and L. Vega, On the Zakharov and Zakharov-Schulman system, J. Funct. Anal., 127(1995), 204-234.
[63] T. Ozawa, Y. Tsutsumi, Existence and smoothing effect of solutions for the Zakharov equations, RIMS Kyoto Univ., 28(1992), 329-361.
[64] T. Ozawa, Y. Tsutsumi, The nonlinear Schrodinger limit and the initial layer of the Zakharov equations, Proc. Jap. Acad., A 67(1991), 113-116.
[65] T. Ozawa, Y. Tsutsumi, The nonlinear Schrodinger limit and the initial layer of the Zakharov equations, Diff. Int. Eq., 5(1992), 721-745.
[66] S. Schochet and M. Weinstein, The nonlinear Schrodinger limit of the Zakharov equations governing Langmuir turbulence, Commun. Math. Phys., 106(1986), 569-580.
[67] C. Sulem and P. L. Sulem, Quelques résultats de régularité pour les équations de la turbulence de Langmuir, C. R. Acad. Sci. Paris, 289(1979), 173-176.
[68] C. Kenig, G. Ponvr, and L. Vega, The initial value problem for a class of nonlinear dispersive equations, Lect Notes Math., 1450(1990), 141-156.
[69] J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and application to nonlinear evolution equations I. Schrodinger equations, GAFA, 3(1993), 107-156.
[70] J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and application to nonlinear evolution equations Ⅱ. The KdV equation, GAFA, 3(1993), 209-262.
[71] J. Ginibre and Y. Tsutsumi, On the Cauchy problem for the Zakharov system, J. Funct. Anal., 151(2) (1997), 384-436.
[72] 李大潜,陈韵梅,非线性发展方程,科学出版社,北京,1997.
[2] T. Ozawa, K. Tsutaya, Y. Tsutsumi, Normal form and global solutions for the Klein-Gordon-Zakharov equations, Annales de I'Institut Henri Poincaré, 12(4)(1995), 459-503.
[3] R. Melrose, R. Ritter, Interaction of nonlinear progressing waves, Annals of Mathematics, 121(2)(1985), 187-213.
[4] J. Rauch, M. Reed, Singularities produced by the nonlinear interaction of three progressing waves: examples, Comm. PDE., 7(1982), 1117-1133.
[5] JCH. Simon, E. Taflin, Wave operators and analytic solutions for systems of nonlinear Klein-Gordon equations and of nonlinear Schrodinger equations, Comm. Math. Phys., 99(1985), 541-562.
[6] Y. Tsutsumi, J. Zhang, Instability of optical solitons for two-wave interaction model in cubic nonlinear media, Advances in Mathematical Sciences and Applications, 8(1999), 691-713.
[7] R. O. Dendy, Plasma Dynamics, Oxford University Press, Oxford, 1990.
[8] V. E. Zakharov, Collapse of Langmuir waves, Sov. Phys. JETP, 35(1972), 908-914.
[9] H. A. Levine, Instability and nonexistence of global solutions to nonlinear wave equations of the form Pu_(tt)=-Au-F(u), Transactions of the American Mathematical Society, 92(1974), 1-21.
[10] L. E. Pagne, D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel Journal of Mathematics, 22(3-4)(1975), 273-303.
[11] W. A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys., (55)(1977), 149-162.
[12] H. Berestycki, T. Cazenave, Instabilité des états stationnaires dans les équations de Schrodinger et de Klein-Gordon non linéairees, C. R. Acad. Sci. Paris, 293(1981), 489-492.
[13] J. Shatah, W. Strauss, Instability of nonlinear bound states, Comm. Math. Phys., 100(1985), 173-190.
[14] W. Strauss, On weak solutions of semi-linear hyperbolic equations, Ann. Acad. Brasil. Cienc., 42(1970), 645-651.
[15] W. Strauss, Stability of solitary waves, Contemporary Math., 107(1990), 123-129.
[16] P. E. Souganidis, W. A. Strauss, Instability of a class of dissipersive solitary waves, Proc. Roy. Soc. Edin., 114A(1990), 195-212.
[17] M. Weinstein, Lyapunov stability of ground states of nonlinear dispersive evolution equations, Comm. Pure Appl. Math., 39(1986), 51-68.
[18] J. Shatah, Stable standing waves of nonlinear Klein-Gordon equations, Comm. Math, Phys., 91(1983), 313-327.
[19] J. Shatah, Unstable ground state of nonlinear Klein-Gordon equations, Transactions of the American Mathematical Society, 290(2)(1985), 701-710.
[20] M, Grillakis, J, Shatah, W. Strauss, Stability theory of solitary waves in the presence of symmetry, Part Ⅰ, J, Funct. Anal., 74(1987), 160-197.
[21] M. Grillakis, J. Shatah, W. Strauss, Stability theory of solitary waves in the presence of symmetry, Part Ⅱ, J. Funct. Anal., 94(1990), 308-348.
[22] J. Zhang, Sharp conditions of global existence for nonlinear Schrodinger and KleinGordon equations, Nonlinear Anal. TMA, 48(2002), 191-207.
[23] T. Ozawa, K. Tsutaya, Y. Tsutsumi, Well-posedness in energy space for the Cauchy problem of the Klein-Gordon-Zakharov equations with different propagation speeds in three space dimensions, Math. Ann., 313(1999), 127-140.
[24] K, Tsutaya, Global existence of small amplitude solutions for the Klein-GordonZakharov equations, Nonlinear Anal, TMA., 27(1996), 1373-1380.
[25] T. B. Benjamin, The stability of solitary waves, Proceedings of the Royal Society London Series A, 328(1972), 153-183.
[26] H. Berestycki, P. L, Lions, Nonlinear scalar field equations, Ⅰ, Ⅱ, Archive for Rational Mechanics and Analysis, 82(1983), 313-375.
[27] E. Infeld, G. Rowlands, Nonlinear Waves, Solitons and Chaos, Cambridge University Press: Cambridge, 1990.
[28] Y. Liu, Instability of solitary waves for generalized Boussinesq equations, Journal of Dynamics and Differential Equations, 1993, 537-558.
[29] M. Ohta, Instability of standing waves for the generalized Davey-Stewartson system, Annales de I'Institut Henri Poincaré, 62(1)(1995), 69-80.
[30] W. Strauss, Nonlinear Wave Equations, C. B. M. S., Vol. 73. American Mathemat- ical Society: Providence, RI, 1989.
[31] E. Zeidler, Applied Functional Analysis, Springer: Berlin, 1995.
[32] J. Ginibre, G. Velo, On a class of nonlinear Schrodinger equations, J. Funct. Anal., 32(1979), 1-71.
[33] R. T. Glassey, On the blowing up of solutions to the Cauchy problem for nonlinear Schrodinger equations, J. Math. Phys., 18(9)(1977), 1794-1797.
[34] M. Tsutsumi, Nonexistence of global solutions to the Cauchy problem for nonlinear Schrodinger equations, SIAM J. Math. Anal., 15(1984), 357-366.
[35] T. Ogawa, Y. Tsutsumi, Blow-up of H~1 solution for the nonlinear Schrodinger equation, J. Differential Equations, 92(2)(1991), 317-330.
[36] T. Ogawa, Y. Tsutsumi, Blow-up of H~1 solution for the nonlinear Schrodinger equation with critical power nonlinearity, Proc. Amer. Math. Soc., 111(2)(1991), 487-496.
[37] J. Ginibre, T. Ozawa, Long range scattering for nonlinear Schrodinger and Hartree equations in space dimensions n≥2, Comm. Math. Phys., 15(1993), 619-645.
[38] J. Ginibre, G. Velo, The global Cauchy problem for the nonlinear Schrodinger equation, revisited, Ann. Inst. H. Poincaré, Anal. Non Linéaire, 2(1985), 309-327.
[39] N. Hayashi, Y. Tsutsumi, Scattering theory for Hartree type equations, Ann. Inst. H. Poincaré Phys. Théor., 46(1987), 187-213.
[40] N. Hayashi, K. Nakamitsu, M. Tsutsumi, On solutions of the initial value problem for the nonlinear Schrodinger equations, J. Funct. Anal., 71(1987), 218-245.
[41] C. Kenig, G. Ponce, L. Vega, Small solution to nonlinear Schrodinger equations, Ann. Inst. H. Poincaré, Anal. Non Linéaire, 10(1993), 255-288.
[42] T. Cazenave, An introduction to nonlinear Schrodinger equations, Textos de Metodos Mathematicos, Vol. 22, Rio de Janeiro, 1989.
[43] Z. Gan, J. Zhang, Sharp criterions of global existence and collapse for coupled nonlinear Schrodinger equations, J. Partial Diff. Eqs., 17(2004), 207-220.
[44] J. Zhang, On the standing wave in coupled nonlinear Klein-Gordon equations, Math. Meth. Appl. Sci., 26(2003), 11-25.
[45] J. Zhang, Cross-Constrained variational problem and nonlinear Schrodinger equation, Proceedings of International Conference on Foundations of Computational Mathematics in honor of Professor Steve Smale's 70th Birthday, HongKong, 2001.
[46] M. I. Weinstein, Nonlinear Schrodinger equations and sharp interpolations estimates, Comm. Math. Phys., 87(1983), 567-576.
[47] C. C. Bradley, C. A. Sackett and R. G. Hulet, Bose-Einstein condensation of Lithium observation of limited condensate number, Phys. Rev. Lett., 78(1997), 985-989.
[48] B. M. Caradoc-Davies, R. J. Ballagh and K. Burnett, Coherent dynamics of vortex formation in trapped Bose-Einstein consenstates, Phys. Rev. Lett., 83(5)(1999), 895-898.
[49] F. Dalfovo, S. Giorgini and L. P. Pitaevskii, et al., Theory of Bose-Einstein condensation in trapped gases, Reviews of Modern Physics, 71(3)(1999), 463-512.
[50] J. Garcia-Ripoll and V. M. Pérez-Garcia, Extended parametric resonances in nonlinear Schrodinger systems, Phys. Rev. Lett., 83(9)(1999), 1715-1718.
[51] T. Tsutsumi and M. Wadati, Collapses of wave functions in multidimensional nonlinear Schrodinger equations under harmonic potential, J. Phys. Soc. Jpn., 66(1997), 3031-3034.
[52] J. Zhang, A variationa approach to nonlinear Schrodinger equations under a harmonic potential, Preprint(2000).
[53] H. Pecher, L~P-Abschatzungen and klassiche Losungen für nichtlineare wellengeichungen, I, Math. Z., 150(1976), 159-183.
[54] J. M. Ball, Finite time blow-up in nonlinear problems, in: M. G. Grandall(Ed.), Nonlinear Evolution Equations, Academic Press, New York, 1978, 189-205.
[55] A. Softer, M. I. Weinstein, Resonance, radiation damping and instability in Hamiltonian nonlinear wave equations, Invent. Math., 136(1999), 9-74.
[56] Z. Gan, J. Zhang, Standing waves for nonlinear Klein-Gordon equations with nonnegative potentials, Applied Mathematics and Computation(accepted).
[57] H. Added and S. Added, Existence globale de solutions fortes pour les équations de la turbulence de Langmuir en dimension 2, C. R. Acad. Sci. Paris, 299(1984), 551-554.
[58] H. Added and S. Added, Equations of Langmuir turbulence and nonlinear Schrodinger equation: smoothness and approximation, J. Funct. Anal., 79(1988), 183-210.
[59] J. Bourgain, On the Cauchy and invariant measure problem for the periodic Za- kharov system, Duke Math. J., 76(1994), 175-203.
[60] J. Bourgain and J. Colliander, On well-posedness of the Zakharov system, Int. Math. Res. Not., 11(1996), 515-546.
[61] L. Glangetas and F. Merle, Existence of self-similar blow up solutions for Zakharov equation in dimension 2, Comm. Math. Phys., 160(1994), 173-215 and 349-389.
[62] C. Kenig, G. Ponce, and L. Vega, On the Zakharov and Zakharov-Schulman system, J. Funct. Anal., 127(1995), 204-234.
[63] T. Ozawa, Y. Tsutsumi, Existence and smoothing effect of solutions for the Zakharov equations, RIMS Kyoto Univ., 28(1992), 329-361.
[64] T. Ozawa, Y. Tsutsumi, The nonlinear Schrodinger limit and the initial layer of the Zakharov equations, Proc. Jap. Acad., A 67(1991), 113-116.
[65] T. Ozawa, Y. Tsutsumi, The nonlinear Schrodinger limit and the initial layer of the Zakharov equations, Diff. Int. Eq., 5(1992), 721-745.
[66] S. Schochet and M. Weinstein, The nonlinear Schrodinger limit of the Zakharov equations governing Langmuir turbulence, Commun. Math. Phys., 106(1986), 569-580.
[67] C. Sulem and P. L. Sulem, Quelques résultats de régularité pour les équations de la turbulence de Langmuir, C. R. Acad. Sci. Paris, 289(1979), 173-176.
[68] C. Kenig, G. Ponvr, and L. Vega, The initial value problem for a class of nonlinear dispersive equations, Lect Notes Math., 1450(1990), 141-156.
[69] J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and application to nonlinear evolution equations I. Schrodinger equations, GAFA, 3(1993), 107-156.
[70] J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and application to nonlinear evolution equations Ⅱ. The KdV equation, GAFA, 3(1993), 209-262.
[71] J. Ginibre and Y. Tsutsumi, On the Cauchy problem for the Zakharov system, J. Funct. Anal., 151(2) (1997), 384-436.
[72] 李大潜,陈韵梅,非线性发展方程,科学出版社,北京,1997.