THE SHORT PULSE EQUATION BY A RIEMANN-HILBERT APPROACH
Abstract
We develop a Riemann-Hilbert approach to the inverse scattering transform method for the short pulse (SP) equation u_xt = u + 1/6 (u^3)_xx with zero boundary conditions (as |x| → ∞). This approach is directly applied to a Lax pair for the SP equation. It allows us to give a parametric representation of the solution to the Cauchy problem. This representation is then used for studying the long-time behavior of the solution as well as for retrieving the soliton solutions. Finally, the analysis of the long-time behavior allows us to formulate, in spectral terms, a sufficient condition for the wave breaking.
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