Dispersionless and multicomponent BKP hierarchies with quantum torus symmetries

In this article, we will construct the additional perturbative quantum torus symmetry of the dispersionless BKP hierarchy based on the W∞ infinite dimensional Lie symmetry. These results show that the complete quantum torus symmetry is broken from the BKP hierarchy to its dispersionless hierarchies. Further a series of additional flows of the multicomponent BKP hierarchy will be defined and these flows constitute an N-folds direct product of the positive half of the quantum torus symmetries.