Riemann-Hilbert approach for the FQXL model: A generalized Camassa-Holm equation with cubic and quadratic nonlinearity

In this paper, the inverse scattering transform associated with a Riemann-Hilbert problem is formulated for the FQXL model: a generalized Camassa-Holm equation m_t = 1/2 k₁[m(u² - u² _x )]x + 1/2 k₂(2mu_x + m_xu), m = u - u_xx, which was originally included in the work of Fokas [Physica D 87, 145 (1995)] and was recently shown to be integrable in the sense of Lax pair, bi-Hamilton structure, and conservation laws by Qiao, Xia, and Li [e-print arXiv:1205.2028v2 (2012)]. We have discussed the following properties: direct scattering problems and Jost solutions, asymptotical and analytical behavior of Jost solutions, the scattering equations in a Riemann- Hilbert problem, and the multi-soliton solutions of the FQXL model. Then, one-soliton and two-soliton solutions are presented in a parametric form as a special case of multi-soliton solutions.