Optical solitons and their propagations in a time-varying coefficient modified nonlinear Schrödinger equation with non-zero boundary conditions via Riemann–Hilbert method

• A time-varying coefficient modified nonlinear Schrödinger equation with non-zero boundary conditions at infinity, phase term of which contains time-varying coefficients, is proposed. • The paper extends the celebrated Riemann–Hilbert method for the first time to solve the time-varying coefficient soliton equations with non-zero boundary conditions. • This study shows that the selection of time-varying coefficients, distribution of discrete spectral points, and constraints from non-zero boundary conditions all have significant impacts on the properties of soliton solutions. In this article, a time-varying coefficient modified nonlinear Schrödinger equation (tvcmNLSE) with non-zero boundary conditions (NZBCs) at infinity, phase term of which contains time-varying coefficients, is proposed to describe the nonlinear propagation behavior of optical pulses in non-uniform optical fibre. Meanwhile, the Riemann–Hilbert method (RHM) for the tvcmNLSE with NZBCs is presented for the first time. Specifically, a special transformation is constructed to convert the tvcmNLSE with NZBCs into the case with constant NZBCs, and the associated asymptotic spectral problem (ASP) is obtained. By introducing a suitable two-piece Riemann surface to map the original spectral parameter into a single-valued one, and analyzing the ASP, we construct the corresponding Jost solutions and spectral matrix, and sequentially analyze their analytical, symmetric, and asymptotic properties. Then, a generalized Riemann–Hilbert problem (RHP) is established that relates to the converted tvcmNLSE with constant NZBCs. Using the reconstruction formula, we further derive N -soliton solution with NZBCs in the absence of reflection potential, and analyze the dynamic characteristics of single and double solitons. Most notably, by modulating discrete spectral points (DSPs) and time-varying coefficients, we discover some previously unreported new waveforms, such as parabolic and periodic solitons with NZBCs. These waveforms have important implications in both theory and practice. In addition, the influences of different time-varying coefficients, NZBCs, and DSPs on soliton solutions are also discussed and analyzed. This study shows that the selection of time-varying coefficients, distribution of DSPs, and constraints from NZBCs all have significant impacts on the properties of soliton solutions.