Riemann–Hilbert approach and nonlinear dynamics of the coupled higher-order nonlinear Schrödinger equation in the birefringent or two-mode fiber

The multi-soliton solutions and breathers to the coupled higher-order nonlinear Schrödinger (CH-NLS) equation are derived in this work via the Riemann–Hilbert approach. Firstly, the spectral structure of the CH-NLS equation is investigated and then a matrix Riemann–Hilbert problem on the real axis is strictly formulated. Secondly, by solving the special Riemann–Hilbert problem with no reflection where a jump matrix is taken to be the identity matrix, the formula of N-soliton solutions can be computed. Thirdly, we prove that the higher-order linear and nonlinear term r has important impact on the velocity, phase, period and wavewidth of wave dynamics. Besides, the localized waves characteristics together with collision dynamic behaviors of these explicit soliton solutions and breathers are shown graphically and discussed in detail. Interestingly, three solitons display different dynamics which demonstrate amplitudes of the right-direction waves gradually become larger during the propagation process.