A novel time-dependent coefficient complex mKdV-type equation with nonzero boundaries: Riemann–Hilbert method and soliton solutions

The modified Korteweg–de Vries (mKdV) equation is an important internal solitary wave model in ocean engineering. In this paper, a new time-dependent coefficient complex mKdV-type (tdccmKdV-type) equation with nonzero boundaries is proposed and analytically solved by using the Riemann–Hilbert (RH) method. First, the related spectral problems of the tdccmKdV-type equation are given. Then, a Riemann surface is introduced to handle the multivalued problem of spectral parameter, and a singularly valorized variable is used to restore the infeasible operations on the Riemann surface to the complex plane. The analyticity, symmetry and asymptotic property of the Jost solutions and spectral matrix are further analyzed. Subsequently, a RH problem associated with nonzero asymptotic boundaries is presented. Finally, by analyzing the discrete spectra and residual conditions of the poles in the presented RH problem, and calculating the trace formula corresponding to the reflection coefficients and discrete spectra, N-soliton solution of the tdccmKdV-type equation is derived. By simulating specific single- and two-soliton solutions, it is indicated that the time-dependent coefficients have a regulatory effect on the velocity and spatiotemporal structure of solitons. The novelty of this work lies in proposing the tdccmKdV-type equation with nonzero boundaries, applying the RH method for the first time to such a third-order time-dependent coefficient model with nonzero boundaries, and reporting novel nonlinear dynamical characteristics of nonzero boundary soliton solutions. This not only enriches the content of nonzero boundary solitons and nonlinear integrable systems with potential practical applications, but also expands the applicability of the RH method.