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

TL;DR

This paper establishes a connection between the reduced Ostrovsky equation and the $A_2^{(2)}$ Toda system, constructing explicit $N$-soliton solutions using pfaffians derived from bilinear equations and tau-functions.

Contribution

It introduces a novel pfaffian-based $N$-soliton solution for the reduced Ostrovsky equation via its link to the $A_2^{(2)}$ Toda system and related bilinear equations.

Findings

Constructed $N$-soliton solutions as pfaffians.

Linked the tau-functions to period 3-reduction of Toda systems.

Derived solutions from bilinear equations of extended BKP hierarchy.

Abstract

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 is 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.