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We develop a Riemann-Hilbert approach to the inverse scattering transformmethod for the short pulse (SP) equation $u_{xt}=u+\frac{1}{6}(u^3)_{xx}$ withzero boundary conditions (as $|x|\to\infty$). This approach is directly appliedto the Lax pair for the SP equation. It allows us to give a parametricrepresentation of the solution to the Cauchy problem. This representation isthen used for studying the long-time behavior of the solution as well as forretrieving the soliton solutions. Finally, the analysis of the long-timebehavior allows us to formulate, in spectral terms, a sufficient condition forthe wave breaking.

The short pulse equation by a Riemann-Hilbert approach