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

Abstract

Abstract. We present a systematic way to construct solutions of the (n = 5)-reduction of the BKP and CKP hierarchies from the general τ function τ (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 iε . Furthermore, we show that the soliton solutions of the nreduction 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 iε . Based on this classification, we detail the existence of two-peak solitons of the n-reduction from the Grammian τ 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.

The applications of the gauge transformation for the BKP hierarchy

“…Now let's review some results about the gauge transformation for the BKP hierarchy [7][8][9][10]. Assume…”

“…The gauge transformation [1,2] provides a simple way to construct solutions for integrable hierarchies. By now, the gauge transformations of many integrable hierarchies have been constructed, for example, the KP hierarchy [1,2], the constrained KP hierarchy [2][3][4][5][6], the constrained BKP and CKP hierarchy [7][8][9][10] (cBKP and cCKP), the discrete KP hierarchy [11,12], the q-KP hierarchy [13,14] and so on. In this paper, we will mainly study the gauge transformation for the BKP hierarchy.…”

“…The BKP hierarchy [15] is one of the most important sub-hierarchies of the KP hierarchy defined by restricting the Lax operator 1 L * = −∂L∂ −1 . In order to keep the restrictions on the Lax operator of the BKP hierarchy, one cannot do gauge transformation for the BKP hierarchy by only using one kind of the basic gauge transformation operators: the differential type T D and the integral type T I developed by Chau et al in [1], and instead has to use the combination of T D and T I [7][8][9][10]. In particular, the generating functions of the combination of T D and T I are dependent on each other, which is also requested by the restriction of the Lax operator.…”

The “ghost” symmetry in the CKP hierarchy

“…where dλ ≡ ∞ dλ 2πi = Res λ=∞ . By now, the CKP hierarchy has attracted many researches [3][4][5][6][7][8][9][10][11][12][13]. In contrast to the KP and the BKP cases, there seems not a single tau function to describe the CKP hierarchy in the form of Hirota bilinear equations (that is, Hirota's equations are no longer of the type P (D)τ · τ = 0) [2].…”

The determinant representation of the gauge transformation for the discrete KP hierarchy

“…Furthermore, in Theorem 3 of [18] Oevel also considered n-fold iteration of T d and binary DT for the dKP hierarchy. 1) In addition, the determinant representation [12] of the gauge transformation operators provides a simple method to construct the transformed τ function [11,[13][14][15] for several special cases of the KP hierarchy and q-KP hierarchy. In this paper we extend these results to discrete KP hierarchy.…”