A coupled Alice–Bob modified Korteweg de-Vries (mKdV) system is established from the mKdV equation in this paper, which is nonlocal and suitable to model two-place entangled events. The Lax integrability of the coupled Alice–Bob mKdV system is proved by demonstrating three types of Lax pairs. By means of the truncated Painlevé expansion, auto-Bäcklund transformation of the coupled Alice–Bob mKdV system and Bäcklund transformation between the coupled Alice–Bob mKdV system and the Schwarzian mKdV equation are demonstrated. Nonlocal residual symmetries of the coupled Alice–Bob mKdV system are researched. To obtain localized Lie point symmetries of residual symmetries, the coupled Alice–Bob mKdV system is extended to a system consisting six equations. Calculation on the prolonged system shows that it is invariant under the scaling transformations, space-time translations, phase translations and Galilean translations. One-parameter group transformation and one-parameter subgroup invariant solutions are obtained. The consistent Riccati expansion (CRE) solvability of the coupled Alice–Bob mKdV system is proved and some interaction structures between soliton–cnoidal waves are obtained by CRE. Moreover, Jacobi periodic wave solutions, solitary wave solutions and singular solutions are obtained by elliptic function expansion and exponential function expansion.

本文根据mKdV方程建立了一个耦合的Alice–Bob修正的Korteweg de-Vries（mKdV）系统，该系统是非局部的，适合于模拟两地纠缠事件。通过演示三种Lax对，证明了Alice-Bob mKdV耦合系统的Lax可积性。通过截断的Painlevé展开，证明了耦合的Alice–Bob mKdV系统的自动Bäcklund变换以及耦合的Alice–Bob mKdV系统与Schwarzian mKdV方程之间的Bäcklund变换。研究了耦合的Alice–Bob mKdV系统的非局部残差对称性。为了获得剩余对称性的局部Lie点对称性，将耦合的Alice–Bob mKdV系统扩展到包含六个方程的系统。对延长系统的计算表明，在缩放变换下它是不变的，时空翻译，相位翻译和伽利略翻译。得到一参数群变换和一参数子群不变解。证明了耦合的Alice–Bob mKdV系统具有一致的Riccati扩展（CRE）可溶性，并通过CRE获得了孤子-长波之间的一些相互作用结构。此外，通过椭圆函数展开和指数函数展开获得雅可比周期波解，孤波解和奇异解。