The Short Pulse equation on the half lines: a Riemann--Hilbert approach

Abstract

In this work, we analyze the initial boundary value (IBV) problems for the Short Pulse (SP) equation $u_{xt}=u+\frac{1}{6}(u^3)_{xx}$ (classical solutions) on the half-lines $x\ge 0$ and $x\le 0$ under the condition of sufficient decay as $x\to +\infty$ or as $x\to -\infty$. Using the Riemann--Hilbert (RH) problem approach, we show that these two IBV problems differ substantially regarding the boundary values required for problems to be well-posed. Namely, the solution of the ``right problem'' ($x\ge 0$) is uniquely determined by the initial data $u(x,0)$, $x\ge 0$ alone, whereas for the ``left problem'' ($x\le 0$), two boundary functions $u(0,t)$ and $u_x(0,t)$ have to be prescribed, together with the initial data, for a well-posed problem.