Inverse scattering transform for the complex short-pulse equation by a Riemann–Hilbert approach

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In this paper, we develop the inverse scattering transform (IST) for the complex short-pulse equation (CSP) on the line with zero boundary conditions at space infinity. The work extends to the complex case the Riemann–Hilbert approach to the IST for the real short-pulse equation proposed by A. Boutet de Monvel, D. Shepelsky and L. Zielinski in 2017. As a byproduct of the IST, soliton solutions are also obtained. Unlike the real SPE, in the complex case discrete eigenvalues are not necessarily restricted to the imaginary axis, and, as consequence, smooth 1-soliton solutions exist for any choice of discrete eigenvalue $$k_1\in {\mathbb {C}}$$ with $$\mathop {{\mathrm{Im}}}\nolimits k_1< |\mathop {{\mathrm{Re}}}\nolimits k_1|$$ . The 2-soliton solution is obtained for arbitrary eigenvalues $$k_1,k_2$$ , providing also the breather solution of the real SPE in the special case $$k_2=-k_1^*$$ .