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We extend the Riemann-Hilbert (RH) method to study the inverse scatteringtransformation and high-order pole solutions of the focusing and defocusingnonlocal (reverse-space-time) modified Korteweg-de Vries (mKdV) equations withnonzero boundary conditions (NZBCs) at infinity and successfully find itsmultiple soliton solutions with one high-order pole and multiple high-orderpoles. By introducing the generalized residue formula, we overcome thedifficulty caused by calculating the residue conditions corresponding to thehigher-order poles. In accordance with the Laurent series of reflectioncoefficient and oscillation term, the determinant formula of the high-orderpole solution with NZBCs is established. Finally, combined with specificparameters, the dynamic propagation behaviors of the high-order pole solutionsare further analyzed and some very interesting phenomena are obtained,including kink solution, anti kink solution, rational solution andbreathing-soliton solution.

Inverse scattering transform and multiple high-order pole solutions for  the nonlocal focusing and defocusing modified Korteweg-de Vries equation with  the nonzero boundary conditions