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Investigated in this paper is the  AB  system, which is used to describe the baroclinic instability processes in the geophysical flows. By introducing an auxiliary function, we construct the quasi-periodic wave solutions of the  AB  system, and find that behaviors of the quasi-periodic waves are similar to those of the periodic waves of the  AB  system except for certain phase shift in each evolution period. Especially, when, the quasi-periodic waves develop into the periodic waves. Besides, it is found that under certain conditions the quasi-periodic waves degenerate into the irregular solitary waves, which are characterized by shape variations or appearances, splits, combinations and disappearances of certain bulges in the evolution.

Quasi-periodic waves and irregular solitary waves of the AB system