Quasi-Periodic Relativistic Strings in the Minkowski Space $$\mathbf{R }^{1+n}$$

Abstract

In this article, we consider the motion of relativistic strings in the Minkowski space $$\mathbf{R }^{1+n}$$ . Those surfaces are known as a timelike minimal surface, and described by a system with n nonlinear wave equations of Born-Infeld type. The one dimensional Born-Infeld equation $$ x_{tt}(1+x_\theta ^2)-x_{\theta \theta }(1-x_t^2)=2x_tx_\theta x_{t\theta } $$ admits an exact time quasi-periodic solution $$ x(t,\theta )=\sin \Big ((\omega \cdot l)t+\theta \Big )-\sin \Big ((\omega \cdot l)t-\theta \Big ), $$ where $$\omega \in \mathbf{R }^n$$ denotes the frequencies, and $$l\in \mathbf{Z }^n$$ . By constructing a suitable Nash–Moser iteration scheme, we prove that relativistic strings can admit a more generalized time quasi-periodic motion in $$\mathbf{R }^{1+n}$$ . Moreover, those time quasi-periodic solutions are also timelike solutions.