The physical significance and applications of infinitely many symmetries of the potential modified Korteweg–de Vries equation

Abstract The potential modified Korteweg–de Vries (pmKdV) equation possesses an infinite number of symmetries; however, the physical implications of these symmetries remain unexplored. Recent improvements have successfully elucidated that the physical significance of the infinite symmetries are related to wave number translation and wave center translation invariance. Following the ideas of these results, this paper aims to investigate the physical meanings associated with the infinite symmetries linked to the n -soliton solutions of the pmKdV equation. The findings indicate that the K -symmetries are related to the translational invariance of the wave center, while the τ -symmetry is associated with combined translational symmetries of both the wave center and wave number. Furthermore, these infinitely many symmetries are not entirely independent, only some of the special symmetries are autonomous. These results offer a novel method for deriving n -wave solutions of the pmKdV equation. Using this method, some special exact solutions of the pmKdV equation, such as the soliton, complexiton, breather and double-pole solutions, are presented.