The dynamics of relativistic strings moving in the minkowski space ℝ¹⁺ⁿ

In this paper we investigate the dynamics of relativistic (in particular, closed) strings moving in the Minkowski space ℝ¹⁺ⁿ(n ≥ 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 ℝ¹⁺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 ℝ¹⁺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 ℝ¹⁺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 ℝ¹⁺ⁿ are given.