Solitons and rogue waves of the quartic nonlinear Schrödinger equation by Riemann–Hilbert approach

We systematically develop a Riemann–Hilbert approach for the quartic nonlinear Schrödinger equation on the line with both zero boundary condition and nonzero boundary conditions at infinity. For zero boundary condition, the associated Riemann–Hilbert problem is related to two cases of scattering data: N simple poles and one Nth-order pole, which allows us to find the exact formulae of soliton solutions. In the case of nonzero boundary conditions and initial data that allow for the presence of discrete spectrum, the pure one-soliton solution and rogue waves are presented. The important advantage of this method is that one can study the long-time asymptotic behavior of the solutions and the infinite order rogue waves based on the associated Riemann–Hilbert problems.