We explore the inverse scattering transforms with matrix Riemann–Hilbert problems for both focusing and defocusing modified Korteweg–de Vries (mKdV) equations with non-zero boundary conditions (NZBCs) at infinity systematically. Using a suitable uniformization variable, the direct and inverse scattering problems are proposed on a complex plane instead of a two-sheeted Riemann surface. For the direct scattering problem, the analyticities, symmetries and asymptotic behaviors of the Jost solutions and scattering matrix, and discrete spectra are established. The inverse problems are formulated and solved with the aid of the matrix Riemann-Hilbert problems, and the reconstruction formulas, trace formulas, and theta conditions are also posed. In particular, we present the general solutions for the focusing mKdV equation with NZBCs and both simple and double poles, and for the defocusing mKdV equation with NZBCs and simple poles. Finally, some representative reflectionless potentials are in detail studied to illustrate distinct nonlinear wave structures containing solitons and breathers for both focusing and defocusing mKdV equations with NZBCs.

我们探索了矩阵Riemann-Hilbert问题的逆散射变换，用于在无穷大的条件下系统地聚焦和散焦具有非零边界条件（NZBCs）的修正Korteweg-de Vries（mKdV）方程。使用合适的均匀化变量，可以在复杂平面上而不是在两层黎曼曲面上提出正向散射和逆向散射问题。对于直接散射问题，建立了Jost解和散射矩阵的解析性，对称性和渐近行为，以及离散谱。借助矩阵黎曼-希尔伯特（Riemann-Hilbert）问题来公式化和求解反问题，并提出了重构公式，迹线公式和θ条件。尤其是，我们给出了带有NZBCs和单极点和双极点的聚焦mKdV方程以及带有NZBCs和单极点的散焦mKdV方程的一般解。最后，对一些代表性的无反射电势进行了详细研究，以说明包含孤子和呼吸器的独特非线性波结构，用于带有NZBC的聚焦和散焦mKdV方程。