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In this paper, we investigate the inverse scattering transform(IST) for thefocusing and defocusing mKdV equation with fully asymmetric nonzero boundaryconditions. Our analysis focuses on the properties of the Jost function,allowing us to establish the associated Riemann-Hilbert (RH) problem for boththe focusing and defocusing cases. Furthermore, we recover the potential fromthe RH problem, which plays a crucial role in calculating the long-timeasymptotic behavior of the solution. In the symmetric case, the eigenvalues aregiven by $\lambda^2 = k^2 \pm q_0^2$. However, in the asymmetric case, theeigenvalues are represented as $\lambda_\pm^2 = k^2 \pm q_\pm^2$. As a result,we directly handle the branch cut instead of utilizing the four-sheeted Riemannsurface. Naturally, this approach leads to the discontinuity of the functionsacross their respective branch cuts, which significantly impacts the entiredevelopment of the IST.

Focusing and defocusing mKdV equation with fully asymmetric nonzero boundary conditions: the inverse scattering transform