On the τ-functions of the reduced Ostrovsky equation and the A ⁽²⁾ ₂ two-dimensional Toda system

The reciprocal link between the reduced Ostrovsky equation and the A ⁽²⁾ ₂ two-dimensional Toda (2D-Toda) system is used to construct the N-soliton solution of the reduced Ostrovsky equation. The N-soliton solution of the reduced Ostrovsky equation is presented in the form of pfaffian through a hodograph (reciprocal) transformation. The bilinear equations and the τ-function of the reduced Ostrovsky equation are obtained from the period 3-reduction of the B or C 2D-Toda system, i.e. the A ⁽²⁾ ₂ 2D-Toda system. One of the τ-functions of the A ⁽²⁾ ₂ 2D-Toda system becomes the square of a pfaffian which also becomes a solution of the reduced Ostrovsky equation. There is another bilinear equation which is a member of the 3-reduced extended BKP hierarchy. Using this bilinear equation, we can also construct the same pfaffian solution.