Soliton solution, breather solution and rational wave solution for the coupled macroscopic fluctuation theory equation in the optimal path of the process

The solution of the macroscopic fluctuation theory (MFT) equation can describe the optimal path of the process, and the Darboux transformation (DT) method can solve soliton solution of some integrable equations. In this paper, we obtained the exact solutions of the coupled macroscopic fluctuation theory (CMFT) equations using the DT method. By constructing a novel type of Lax pairs with $\sqrt{ik}$, we derive some expressions for the 1-soliton, 2-soliton, and $n$-soliton solutions of the CMFT equations, including some soliton solutions, breather solutions and rational wave solutions. Based on these solutions, we consider the elastic interactions and dynamics between two solitons in CMFT equations. These results can present some novel phenomena in the optimal path of the process.