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Solving bi-directional soliton equations in the KP hierarchy by gauge transformation

By J.S. He, Y. Cheng and Rudolf A. Roemer


We present a systematic way to construct solutions of the (n = 5)-reduction of the BKP and CKP hierarchies from the general tau function tau((n+k)) of the KP hierarchy. We obtain the one-soliton, two-soliton, and periodic solution for the bi-directional Sawada-Kotera (bSK), the bi-directional Kaup-Kupershmidt (bKK) and also the bi-directional Satsuma-Hirota (bSH) equation. Different solutions such as left- and right-going solitons are classified according to the symmetries of the 5th roots of e(i epsilon). Furthermore, we show that the soliton solutions of the n-reduction of the BKP and CKP hierarchies with n = 2j + 1, j = 1, 2, 3,..., can propagate along j directions in the 1 + 1 space-time domain. Each such direction corresponds to one symmetric distribution of the nth roots of e(i epsilon). Based on this classification, we detail the existence of two-peak solitons of the n-reduction from the Grammian tau function of the sub-hierarchies BKP and CKP. If n is even, we again find two- peak solitons. Last, we obtain the "stationary" soliton for the higher-order KP hierarchy

Topics: QC
Publisher: Institute of Physics Publishing Ltd.
Year: 2006
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