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

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