Communications in Mathematical Physics

, Volume 269, Issue 1, pp 153–174 | Cite as

The Dynamics of Relativistic Strings Moving in the Minkowski Space \(\mathbb{R}^{1+n}\)

Article

Abstract

In this paper we investigate the dynamics of relativistic (in particular, closed) strings moving in the Minkowski space \(\mathbb{R}^{1+n}\;(n\ge 2)\). We first derive a system with n nonlinear wave equations of Born-Infeld type which governs the motion of the string. This system can also be used to describe the extremal surfaces in \(\mathbb{R}^{1+n}\). We then show that this system enjoys some interesting geometric properties. Based on this, we give a sufficient and necessary condition for the global existence of extremal surfaces without space-like point in \(\mathbb{R}^{1+n}\) with given initial data. This result corresponds to the global propagation of nonlinear waves for the system describing the motion of the string in \(\mathbb{R}^{1+n}\). We also present an explicit exact representation of the general solution for such a system. Moreover, a great deal of numerical analyses are investigated, and the numerical results show that, in phase space, various topological singularities develop in finite time in the motion of the string. Finally, some important discussions related to the theory of extremal surfaces of mixed type in \(\mathbb{R}^{1+n}\) are given.

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References

  1. 1.
    Barbashov B.M., Nesterenko V.V., Chervyakov A.M. (1982). General solutions of nonlinear equations in the geometric theory of the relativistic string. Commun. Math. Phys. 84:471–481MATHMathSciNetCrossRefADSGoogle Scholar
  2. 2.
    Boillat G. (1972). Chocs caractéristiques. C. R. Acad. Sci., Paris, Sér. A 274:1018–1021MATHMathSciNetGoogle Scholar
  3. 3.
    Bordemann M., Hoppe J. (1994). The dynamics of relativistic membranes II: Nonlinear waves and covariantly reduced membrane equations. Phys. Lett. B 325:359–365MathSciNetCrossRefADSGoogle Scholar
  4. 4.
    Born M., Infeld L. (1934). Foundation of the new field theory. Proc. Roy. Soc. London A144:425–451MATHADSGoogle Scholar
  5. 5.
    Brenier Y. (2002). Some Geometric PDEs Related to Hydrodynamics and Electrodynamics. Proceedings of ICM 2002 3:761–772MATHGoogle Scholar
  6. 6.
    Calabi, E.: Examples of Bernstein problems for some nonlinear equations. In: Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), Providence, RI: Amer. Math. Soc., 1970 pp. 223–230.Google Scholar
  7. 7.
    Cheng S.Y., Yau S.T. (1976). Maximal space-like hypersurfaces in the Lorentz-Minkowski spaces. Ann. of Math. 104:407–419MATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Chae D., Huh H. (2003). Global existence for small initial data in the Born-Infeld equations. J. Math. Phys. 44:6132–6139MATHMathSciNetCrossRefADSGoogle Scholar
  9. 9.
    Christodoulou D. (1986). Global solutions of nonlinear hyperbolic equations for small initial data. Comm. Pure Appl. Math. 39:267–282MATHMathSciNetGoogle Scholar
  10. 10.
    Freistühler H. (1991). Linear degeneracy and shock waves. Math. Zeit. 207:583–596MATHGoogle Scholar
  11. 11.
    Gibbons G.W. (1998). Born-Infeld particles and Dirichlet p-branes. Nucl. Phys. B 514:603–639MATHMathSciNetCrossRefADSGoogle Scholar
  12. 12.
    Green, M.B., Schwarz, J.H., Witten, E.: Superstring theory: 1. Introduction. Second edition, Cambridge Monographs on Mathematical Physics, Cambridge: Cambridge University Press, 1988Google Scholar
  13. 13.
    Gu, C.H.: Extremal surfaces of mixed type in Minkowski space \(\mathbb{R}^{n+1}\). In: Variational methods (Paris, 1988), Progr. Nonlinear Differential Equations Appl. 4, Boston, MA: Birkhäuser Boston, 1990, pp. 283–296Google Scholar
  14. 14.
    Gu C.H. (1994). Complete extremal surfaces of mixed type in 3-dimensional Minkowski space. Chinese Ann. Math. 15B:385–400MATHGoogle Scholar
  15. 15.
    Hoppe J. (1994). Some classical solutions of relativistic membrane equations in 4-space-time dimensions. Phys. Lett. B 329:10–14MathSciNetCrossRefADSGoogle Scholar
  16. 16.
    Klainerman, S.: The null condition and global existence to nonlinear wave equations. In Lectures in Appl. Math. 23, Providence, RI: Amer. Math. Soc., 1986, pp. 293–326.Google Scholar
  17. 17.
    Kong D.X. (2004). A nonlinear geometric equation related to electrodynamics. Europhys. Lett. 66:617–623MathSciNetCrossRefADSGoogle Scholar
  18. 18.
    Kong, D.X., Sun, Q.Y., Zhou, Y.: The equation for time-like extremal surfaces in Minkowski space \(\mathbb{R}^{1+n}\). J. Math. Phys. 47, 013503 (2006)Google Scholar
  19. 19.
    Kong D.X., Tsuji M. (1999). Global solutions for 2 × 2 hyperbolic systems with linearly degenerate characteristics. Funkcialaj Ekvacioj 42:129–155MATHMathSciNetGoogle Scholar
  20. 20.
    Lax P.D. (1957). Hyperbolic systems of conservation laws II. Commun. Pure Appl. Math. 10:537–556MATHMathSciNetGoogle Scholar
  21. 21.
    Lindblad H. (2004). A remark on global existence for small initial data of the minimal surface equation in Minkowskian space time. Proc. Amer. Math. Soc. 132:1095–1102MATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    Milnor T. (1990). Entire timelike minimal surfaces in E 3,1. Michigan Math. J. 37:163–177MATHMathSciNetCrossRefGoogle Scholar
  23. 23.
    Serre D. (2000). Systems of Conservation Laws 2: Geometric Structures, Oscillations, and Initial-Boundary Value Problems. Cambridge University Press, CambridgeGoogle Scholar

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Department of MathematicsShanghai Jiao Tong UniversityShanghaiChina
  2. 2.Department of MathematicsCity University of Hong KongHong KongChina

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