On the $τ$-functions of the reduced Ostrovsky equation and the $A_2^{(2)}$ two-dimensional Toda system

Details On the $τ$-functions of the reduced Ostrovsky equation and the $A_2^{(2)}$ two-dimensional Toda system On the $τ$-functions of the reduced Ostrovsky equation and the $A_2^{(2)}$ two-dimensional Toda system

2012-03-12

arxiv.org/abs/1203.2677v1

Nonlinear Sciences

Exactly Solvable and Integrable Systems

20 pages, 3 figures

Scientific paper

The reciprocal link between the reduced Ostrovsky equation and the $A_2^{(2)}$ two-dimensional Toda system is used to construct the $N$-soliton solution of the reduced Ostrovsky equation. The $N$-soliton solution of the reduced Ostrovsky equation are presented in the form of pfaffian through a hodograph (reciprocal) transformation. The bilinear equations and the $\tau$-function of the reduced Ostrovsky equation are obtained from the period 3-reduction of the $B_{\infty}$ or $C_{\infty}$ two-dimensional Toda system, i.e., the $A_2^{(2)}$ two-dimensional Toda system. One of $\tau$-functions of the $A_2^{(2)}$ two-dimensional Toda system becomes the square of a pfaffian which also become 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.

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