Communications in Mathematical Physics

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

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



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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© 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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