An integrable coupled Alice-Bob modified Korteweg de-Vries system: Lax pairs, Backlund transformations, residual symmetries and exact solutions

摘要

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 Painleve expansion, auto-Backlund transformation of the coupled Alice-Bob mKdV system and Backlund 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.