University of Warwick institutional repository: This paper is made available online in accordance with publisher

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1 University of Warwic institutional repository: This paper is made available online in accordance with publisher policies Please scroll down to view the document itself Please refer to the repository record for this item and our policy information available from the repository home page for further information To see the final version of this paper please visit the publisher s website Access to the published version may require a subscription Authors: Jingsong He, Yi Cheng and Rudolf A Römer Article Title: Solving bi-directional soliton equations in the KP hierarchy by gauge transformation Year of publication: 006 Lin to published version: /6-6708/006/03/03 Publisher statement: None

2 Communications in Mathematical Physics manuscript No will be inserted by the editor Solving bi-directional soliton equations in the KP hierarchy by gauge transformation Jingsong He,, Yi Cheng, Rudolf A Römer Department of Mathematics, University of Science and Technology of China, Hefei, 3006 Anhui, P R China Department of Physics and Centre for Scientific Computing, University of Warwic, Coventry CV4 7AL, United Kingdom Revision :, compiled March, 005 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 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 and also the bi-directional Satsuma-Hirota 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 n-reduction of the BKP and CKP hierarchies with n = j, j =,, 3,, can propagate along j directions in the 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-pea solitons of the n-reduction from the Grammian τ function of the subhierarchies BKP and CKP If n is even, we again find two-pea solitons Last, we obtain the stationary soliton for the higher-order KP hierarchy Key words KP hierarchy BKP hierarchy CKP hierarchy τ-function gauge transformation bsk equation equation equation periodic solution bidirectional soliton Introduction The Kadomtsev-Petviashvili KP hierarchy is of central interest for integrable systems and includes several well-nown partial differential equations such as the Korteweg-de Vries KdV and the KP equation With pseudo-differential Lax operator L given as [ 3] the corresponding generalized Lax equation L = u u 3, L t n = [B n, L], n =,, 3,, gives rise to the infinite number of partial differential equations PDEs of the KP hierarchy with dynamical variables {u i t, t, t 3, } with i =, 3, 4, Here B n = n b n,i i L n denotes the differential part of L n and in following we will use L n L n B n to denote the integral part i=0

3 Jingsong He, Yi Cheng, Rudolf A Römer sub-hierarchy Lax operator example equation BKP [, 4] L = L SK [6, 7], bsk [0, ] CKP [4] L = L KK [8, 9], [0, ] n-thkdv [5] L n = 0 KdV [], Boussinesq-type [], SH [3] n =, 3, 4 constrained KPcKP [7, ] L = φ ψ YO [4], MKdV [5], NLS [6] Table Examples of sub-hierarchies of the KP hierarchy, Lax operators used to construct them and resulting equations The symbol indicates the conjugation, for example, = There are some abbreviations used in Table: Sawada-Kotera SK, bi-directional Sawada-Kotera bsk, Kaup-Kupershmidt KK, bi-directional Kaup-Kupershmidt, Satsuma- Hirota SH, Yajima-Oiawa YO, Modified KdV MKdV, Non-linear Schrödinger NLS The simplest nontrivial PDE constructed from is the KP equation given as x 4 u t 3 u u x 3 u x 3 3 u t = 0 3 In Table we show the Lax operator and corresponding -dimensional examples of sub-hierarchies of the KP hierarchy An alternative way to express the KP hierarchy is given by the Zaharov-Shabat ZS equation [], B n B m [B n, B m ] = 0, m, n =, 3, 4, 4 t m t n The eigenfunction φ and the adjoint eigenfunction ψ of the KP hierarchy associated with Eq 4 are defined by φ ψ = B n φ, = B t n t nψ, 5 n where φ = φλ; t and ψ = ψλ; t and t = t, t, The n-reduction of the KP hierarchy corresponds to the situation L n = 0 such that L n = B n = n v n n v v 0 Then the v i, i = 0,,, n, are independent of t n, t n, t 3n, In this way the Lax pair of the -dimensional integrable system can be found Well-nown examples of such n-reductions include the 4-reduction of the KP hierarchy [3] with Lax pair corresponding to the Satsuma-Hirota SH equation [3] 4 x 4u x 4u x x u xx 4u vφ = λφ, 6 t φ = 3 x 3u x 3 u xφ, t = x, t 3 = t, 7 4u t uu x u xxx 3v x = 0, v t 6uv x v xxx = 0 8 Furthermore, eliminating v in the above equations, we can obtain a 6th order equation u = z x 8z tt z xxxxxx z xxxt 8z x z xxxx 36z xx z xxx 7z xz xx = 0, 9 which has been called bi-directional Satsuma-Hirota equation [3] Naturally, there also exist n- reductions of the BKP and CKP hierarchies For example, the 5-reduction of the BKP hierarchy with u = u is given as [ 5x 5u 3x 5u x x 5u 03 u xx 53 z t x ] φ = λφ, 0 t φ = 3 x 3u x φ, u = z x, t 3 = t, t = x, which is the Lax pair corresponding to bi-directional Sawada-Kotera bsk equation [0, ] zxxxxx 5z x z xxx 5z 3 x 5z x z t 5z xxt x 5z tt = 0

4 Solving bi-directional soliton equations 3 The 5-reduction of the CKP hierarchy u = u with Lax pair [0, ] [ x 5 5u x 3 5 u x x 5u 35 6 u xx 5 3 z t x 5uu x 5 3 u xxx 5 ] 6 u t φ = λφ, 3 t φ = 3 x 3u x 3 u xφ, u = z x, t 3 = t, t = x, 4 gives the bi-directional Kaup-Kupershmidt equation z xxxxx 5z x z xxx 5zx 3 5z x z t 5z xxt 45 4 z xx 5z tt = 0 5 x An essential characteristic of the KP hierarchy is the existence of the τ-function and all dynamical variables {u i }, i =, 3,, can be constructed from it [, ], eg, u = log τ, 6 x u 3 = 3 x 3 log τ, 7 x t So it is a central tas to construct the τ-function in order to solve the nonlinear PDEs associated with the KP hierarchy In the following, we will show that φ and ψ play a ey role in this construction Gauge transformations [4, 5] offer an efficient route towards the construction of the τ function of the KP hierarchy In Ref [6] two inds of such a gauge transformation have been proposed, namely, T D φ = φ φ, T I ψ = ψ ψ 8 resulting in a very general and universal τ function see Eq 37 of [6] and also IW,n in [7] The determinant representation of the gauge transformation operators with n steps is given in Ref [7] In particular, the Grammian τ function [8] of the KP hierarchy can be generated by an iteration of the transformation [6, 9, 30] This is straightforwardly understood from Chau s τ function and the determinant representation [7] if we impose a restriction on the generating functions of the gauge transformation Grammian τ function have also been used to solve the reduction of the constrained BKP and CKP hierarchies [3 33, 36] There are two issues that arise when one wants to study the solutions of the -dimensional solitons equations given by the n-reduction of the BKP and CKP hierarchies The first is how it retain the restrictions, ie L = L for BKP and L = L for CKP, for the transformed Lax operators L = T LT In other words, the problem is how to obtain the τ-functions τ n n BKP and τ CKP from the general τ-function τ n = IW,n τ 0 with the gauge transformation T n of the KP hierarchy Here τ 0 is the initial value of the τ-function of the KP hierarchy Also, the generating functions φ i, ψ i of the gauge transformation will be complex-valued and related to the n-th roots of e iε The second issue therefore is how to choose generating functions φ i = φλ i ; x, t and ψ i = ψµ i ; x, t such that τ n BKP and n correspond to a physical τ-function ˆτ, which is real and positive on the full x, t plane τ n CKP Eq In fact, the and bsk equations have been introduced recently by Dye and Parer [0, ] when looing for the bidirectional soliton analogues of the Sawada-Kotera SK [6, 7] and Kaup-Kupershmidt KK [8, 9] equations The Lax pairs of and bsk related similarly as the Lax pairs of KdV and Boussineq equation, thus ensuring their integrability Both and bsk equation have a bidirectional soliton solution [0, ] which have been obtained by the Hirota bilinear method [37] The profile of the solitons depend on their direction of propagation The right-going solitons of are standard

5 4 Jingsong He, Yi Cheng, Rudolf A Römer one-pea solitons, but the left-going solitons have two peas Very recently, Verhoeven and Musette [3] have plotted the bi-directional solitons for the and equation based on the Grammian τ function In this paper, we want to study why the 5-reduction of the BKP and CKP hierarchies have bidirectional soliton solutions, whereas their 3-reduction does not As a first step, we will therefore exhibit the relationship between the periodic, left-going and right-going solitons of the 5-reduction and the 5-th roots of e iε In order to do so, we derive the τ functions of the BKP and CKP hierarchies in sections -4 The explicit formulas of the corresponding τ-functions for solitons as well as for the periodic solutions of bsk and are given and the two-pea soliton is discussed in detail In section 5, we will prove that no two-pea solitons exist for the equation The one-pea soliton has bi-directional motion and we also obtain the periodic and two-soliton solutions In section 6, we will discuss the lower and higher-order reductions of BKP and CKP hierarchies and also the n =even-reductions of the KP hierarchy We will show that the soliton of the j -reduction of BKP and CKP hierarchies can move along j directions j =,,, investigate the relationship with the symmetric distribution of the j -th roots of e iε In particular, we will obtain the stationary soliton for the higher reduction of the KP hierarchy For the higher-order equation and even-reduction of KP hierarchy, we can again find a two-pea soliton τ functions for BKP and CKP hierarchies Let us first define the generalized Wronsian determinant IW,n IW,n g 0, g0,, g0 ; f 0, f 0,, f n 0 0 g f 0 0 g f 0 0 g f g f 0 n 0 g f 0 0 g f 0 0 g f g f 0 n 0 = g f 0 0 g f 0 0 g f g f 0 n f 0 f 0 f 0 3 f n 0 f 0,x f 0,x f 0 3,x f n,x 0 f 0 n f 0 n f 0 3 n f n 0 n In particular, IW 0,n = W n f 0, f 0,, f n 0 with f 0 f 0 i = i x is the usual Wronsian determinant of functions {f 0, f 0,, f n 0 } We shall also use the abbreviation f = fdx with integration constant equal to zero Lemma [6, 7] The τ function of the KP hierarchy generated by the gauge transformation T n is given as τ n = IW,n ψ 0, ψ0,, ψ0 ; φ0, φ0,, φ0 n τ 0, where φ 0 i, ψ 0 j = φλ i ; t, ψµ j ; t are solutions of Eq 5 with initial value τ 0 for the τ-function and the initial values of the {u i } are {u 0 i } Let us now discuss how to reduce the τ n in to the τ function of the BKP hierarchy The ey problem is how to eep the restriction L n = L n under the gauge transformation T n [7] It should be noted that t = t, t 3, t 5, in BKP hierarchy Proposition [30, 39]

6 Solving bi-directional soliton equations 5 The Lax operator transforms as L n = T n LT n under the gauge transformation T n with n = and generating functions ψ 0 i = φ 0 i,x for i =,,, n The τ function τ nn BKP of the BKP hierarchy is τ nn n,x, φ 0 BKP = IW n,nφ 0 n,x,, φ0,x ; φ0, φ0,, φ0 n 0 φ n,x φ 0 0 φ n,x φ 0 0 φ n,x φ 0 3 φ 0 n,x φ 0 n 0 φ n,x φ0 0 φ n,x φ0 0 φ n,x φ0 3 φ0 n 0 φ = 0 φ,x φ0 φ0 0 φ,x φ0 3 φ 0,x φ0 0 n φ φ0 0 φ,x φ0 0 φ,x φ0 3 φ 0,x φ0 n φ0 n n,x φ0 n,x φ0 n 0 φ,x φ0 n τ 0 BKP 3 Proof It is clear that a single step of the gauge transformations T D or T I can not eep the restriction So we use T T = T I ψ T D φ 0 4 such that the lax operator is L = T LT Let us chec whether it satisfies the required restriction which means in terms of T that T D ψ T I L = L 5 φ 0 = T I ψ T D Based on the determinant representation of T [7] we see from 6 that φ 0 rhs = ψ 0 0 φ0 ψ0 0 φ φ lhs = ψ0 ψ 0 0 φ ψ0 x φ 0 φ 0 ψ0,x φ0 ψ0 φ 0 ψ0 6, 7 8 This implies ψ 0 = φ 0,x So we have seen that in order to eep the restriction of the Lax operator, we have to regard T = T as basic building bloc in iteration of the gauge transformations T n In particular, T = T I φ 3,x T D φ T I φ,x T D φ 0, T 33 = T I φ 5 T D T I T D T I T D, 3,x φ 4 3 φ 3,x and so on such that = n and ψ 0 i = φ 0 i,x for i =,,, n According to the determinant of T n [7] and τ n [6, 7] with = n and ψ 0 i = φ 0 i,x, i =,,, n, τ nn BKP can be obtained directly from τ n as in Lemma For the CKP hierarchy, we have again t = t, t 3, t 5, and the restriction is L n = L n Proposition [30, 39] The appropriate gauge transformation T n is given by n = and generating functions ψ 0 i = φ 0 i for i =,,, n φ φ,x φ 0

7 6 Jingsong He, Yi Cheng, Rudolf A Römer The τ function τ nn CKP τ nn of the CKP hierarchy has the form CKP = IW n,nφ 0 n, φ 0 n,, φ0 ; φ0, φ0,, φ0 n 0 φ n φ 0 0 φ n φ 0 0 φ n φ φ n φ 0 0 n φ n φ 0 n 0 φ n φ0 0 φ n φ0 0 φ n φ0 3 φ 0 n φ0 0 n φ n φ0 n = τ 0 CKP 9 0 φ φ 0 0 φ φ 0 0 φ φ φ φ 0 0 n φ φ 0 n 0 φ φ 0 0 φ φ 0 0 φ φ φ φ 0 0 n φ φ 0 n Proof Similar to the BKP hierarchy we have to try the two-step gauge transformation T T = T I T D ψ φ 0 With L = T LT, the restriction L = L then implies T D T I = T I T D ψ φ 0 ψ Based on the determinant representation of T [7], we find from that rhs = lhs = 0 φ 0 φ 0 ψ 0 0 φ ψ0, ψ 0 φ 0 ψ0 φ 0 3 Then ψ 0 = φ 0 Again, we have to regard T = T I φ T D φ 0 as basic building bloc such that T = T I φ 3 T D φ T I φ T D φ 0, T 33 = T I φ 5 T D T I T D T I T D, 3 φ 4 3 φ 3 so = n and ψ 0 i = φ 0 i for i =,,, n According to the determinant of T n [7] and τ n [6, 7] with = n and ψ 0 i = φ 0 i, i =,,, n, τ nn CKP is obtained directly from τ n in Lemma In fact, we can let ψ 0 i = c i φ 0 i,x or ψ0 i = c i φ 0 i associated with ψ 0 i = c i φ 0 i,x or ψ0 i = c i φ 0 φ φ φ 0 with constants c i However, the new τ nn BKP nn or τ CKP i is equivalent to the ones in Proposition or Proposition Although Refs [30, 39] have results similar to our Propositions and, our approach is more direct and simpler for the construction τ nn nn BKP and τ CKP If the initial values of dynamical variables {u i} of BKPCKP hierarchy are zero, then the equations in 5 of φ 0 i, ψ 0 j = φλ i ; t, ψµ j ; t become more simpler as φλ; t t n = n x φλ; t, t= t, t 3, t 5,, 4 ψµ; t t n = n n x ψµ; t, t= t, t 3, t 5,, 5 and τ 0 0 BKP = τ CKP = Last, we note that for the generalized KP gkp hierarchy with Lax operator ˆL = L n, n =, 4, 6, 8,, and ˆL = ˆL, the τ function τ n gkp generated by gauge transformations T n has the same form as for the CKP hierarchy This result will afford a simple way to construct the τ function of equation in Section 5

8 Solving bi-directional soliton equations 7 3 Soliton solutions of the bsk equation As pointed out in the introduction, there are two steps en route from a τ function τ n generated by the gauge transformations T n of the KP hierarchy to the τ function of equations as the n-reduction of BKP or CKP hierarchies The second step is to build physical τ functions from the complex-valued τ nn BKP and τ nn CKP constructed in the last section In the following Sections, we will illustrate our approach by computing the τ function for the 5-reduction of BKP and CKP, ie, for the bsk and equations The 5-reduction of the BKP hierarchy is the bsk equation Assume for the initial value u = 0 in Eqs 0 and, then φ 0 i = φλ i ; x, t are solutions of 5 xφλ i ; x, t = λ i φλ i ; x, t, φλ i ; x, t t = 3 xφλ i ; x, t 3 So proposition with τ 0 BKP = implies that the τ function of bsk is given as follows Proposition 3 The τ function of the bsk equation generated by T nn from initial value is = IW n,n φ 0 n,x, φ 0 n,x,, φ0,x ; φ0, φ0,, φ0 n τ nn bsk 3 and the solution of the bsk equation generated by T nn from initial value 0 is u = x log τ nn bsk 33 Here φ 0 i = φλ i ; x, t are solutions of Eq 3 In general, this τ function τ nn bsk for bsk is complex and related to 5-th roots of e iε We have to find the real and non-zero τ function from it such that u in Eq 33 is a real and smooth solution of bsk This is main tas of this section We start by analysing the solution φλ; x, t of Eq 3and mae the universal ansatz 5 φλ; x, t = A j e xpjtp3 j, with p 5 j = λ 34 j= Here p j = exp επj 5 i, 5 = λ, R, 0 ε < π and j = 0,,, 3, 4 There are two important ingredients which we can use to find the desired solution The first is that the 5-th roots ε j = exp επj 5 i of e iε are distributed uniformly on the unit circle in C So for a suitable value of ε there exist combinations of p j s which are symmetric upon reflection on the x-axes; similarly for the y-axes for other values of ε The second ingredient is that τ bsk and exp αx βt τ bsk will imply the same solution u since u = x log τ bsk Here, α and β are arbitrary, complex constants Therefore we can obtain the desired real and smooth solutions of the bsk if τ bsk can be expressed as τ bsk = e αxβtˆτ bsk ˆτ bsk, in which ˆτ bsk is a real and nonzero function although τ bsk is complex We call ˆτ bsk the physical τ function for the bsk equation Based on the above arguments, let us mae the refined ansatz or φλ ; x, t = e pxp3 t B e qxq3 t, p = e iε, q = e iε, 5 = λ, R, 35 φλ ; x, t = e pxp3 t B e qxq3 t, p = e iε, q = e iε, 5 = λ, R, 36 and in the next step we need to fix the ratio B We stress that the above analysis is also true for the derivation of the equation

9 8 Jingsong He, Yi Cheng, Rudolf A Römer Proposition 4 Define ξ = x cos ε t 3 cos 3ε Then the physical τ function of bsk generated by T is ˆτ bsk = B B eξ e ξ 37 and the corresponding one soliton u = x log ˆτ bsk u = is given as 6 B cos ε 38 e ξ B e ξ B Here B > 0 The velocity of the moving soliton is v = cos 3ε and can be both positive and negative cos ε depending on the choice of ε Specifically, we have v = v ε= π < 0 and v 0 = v ε= 3π > 0 0 Proof τ bsk = φ 0 = A e ix sin εt3 sin 3ε [ e ξ B e ξ B ] 39 and φ 0 = φλ ; x, t defined by Eq 35 Let us point out a relation between the distribution of the 5-th roots of e iε and the direction of movement for the soliton e iε ε=π/0, e iε ε=π/0 one distribution of 5-th roots of e iε p = e iε ε=π/0, q = e iε ε=π/0 in Eq 35 v ε=π/0 < 0, left-going soliton u in Eq 38 ; e iε ε=3π/0, e iε ε=3π/0 another distribution of 5-th roots of e iε p = e iε ε=3π/0, q = e iε ε=3π/0 in Eq 35 v ε=3π/0 > 0, right-going soliton u in Eq 38 We can see from Eq 38 that the one-soliton of bsk has only one pea in its profile The process of generating a two-soliton by T is more complicated Lemma With φλ ; x, t and φλ ; x, t as in Eq 35, and using Proposition 3, τ bsk τ bsk = A e i[x sin εεt3 sin 3ε3 sin 3ε] { 4z f e ξξ 4 [ cosε ε ] 4z f e ξξ B B 4 [ cosε ε ] A 4z 3 f 3 e ξ ξ B 4z f 3 e ξ ξ 4 [ cosε ε ] A 4 [ cosε ε ] iz 5 f 5 e ξ [ cosε ε ] [ cosε ε ] iz 5 f 5 e ξ [ cosε ε ] [ cosε ε ] iz 7 f 5 e ξ [ cosε ε ] [ cosε ε ] iz 7 f 5 e ξ [ cosε ε ] [ cosε ε ] B A B B B B A B A is given by B

10 Solving bi-directional soliton equations 9 [ cosε ε ] [ cosε ε ] B B } 30 Here the z i, i =, 3, 5, 7 are given in Appendix A and f = [ cosε ε ], f3 = [ cosε ε ] and f5 = f f 3, as well as ξ i = x i cos ε i ti 3 cos 3ε i for i = and We now need to find a suitable solution of B {} i, i =, such that the summation of terms in the A i bracet of Eq 30 is a positive function on the whole x,t plane The following two lemmas is useful Let z = 4z f, z 3 = 4z 3 f 3, z 5 = iz 5 f 5, z 7 = iz 7 f 5, and z i, i =, 3, 5, 7 given in Appendix A Lemma 3 For z i, i =, 3, 5, 7 there exist relations z 7 z 5 z z 3 = z 5 ; z z 3 = z 7 3 Lemma 4 Let B = z, B = z, and g = z A z 3, g 6 = g 8 = z z 5, g 9 = z 3 then hold z 3 z z z z 5 z B A B B =, z 3 B = g, A B = g 6, A z 7 z z B z B B A =, 3 B = g 9, 33 A B = g 8, 34 The B i, i =, in Lemma 4 are what we are looing for, and then the physical τ function ˆτ bsk is give A i by following proposition Proposition 5 ˆτ bsk = { e ξξ 4 cosε ε g e ξξ 4 cosε ε e ξ ξ 4 cosε ε e ξ ξ 4 cosε ε e ξ cosε ε cosε ε g 6 e ξ cosε ε cosε ε e ξ cosε ε cosε ε g 8 e ξ cosε ε cosε ε A

11 0 Jingsong He, Yi Cheng, Rudolf A Römer g } 9 cosε ε cosε ε, 35 the two soliton solution is u = x log ˆτ bsk In particular, ε = ε = π 0 solitons and moving in negative direction; ε = ε = 3π 0 positive direction; ε = π 0, ε = 3π 0 results in two overtaing produces two overtaing solitons and moving in results in two head-on solitons We have plotted the one solitons in Fig associated with parameters = B =, = of Eq 38 The two solitons in Proposition 5 are shown in Fig associated with parameters = and = 3 Fig Single left- and right-going solitons for the bsk equation : ε = π 0 left, ε = 3π 0 right 4 Periodic and soliton solutions of equation The 5-reduction of the CKP hierarchy yields the equation Eq 5 Let the initial value be u = 0 in Eqs 3 and 4, then φ 0 i = φλ i ; x, t are solutions of 5 xφλ i ; x, t = λ i φλ i ; x, t, φλ i ; x, t t So the Proposition implies the τ function of equation = 3 xφλ i ; x, t 4 Proposition 6 The τ function of the equation generated by T nn from initial value is τ nn = IW n,nφ 0 n, φ 0 n,, φ0 ; φ0, φ0,, φ0 n 0 φ n φ 0 0 φ n φ 0 0 φ n φ φ n φ 0 0 n φ n φ 0 n 0 φ n φ0 0 φ n φ0 0 φ n φ0 3 φ 0 n φ0 0 n φ n φ0 n = 0 φ φ 0 0 φ φ 0 0 φ φ φ φ 0 0 n φ φ 0 n 0 φ φ 0 0 φ φ 0 0 φ φ φ φ 0 0 n φ φ 0 n 4

12 Solving bi-directional soliton equations Fig Two left- and right-going as well as head-on colliding solitons for the bsk equation : ε = ε = π 0 left, ε = ε = 3π 0 right and ε = π 0, ε = 3π 0 collision and the solution u of the from initial value zero is u = x log τ nn 43 Here φ 0 i = φλ i ; x, t are solutions of Eq 4 As before, τ nn τ function ˆτ nn is complex and related to the 5-th roots of eiε and again we have to find a physical such that u in Eq 43 is real and smooth solution includes solitons and periodic solutions The case of n = and n = will be discussed in detail Similar to the bsk equation, we should assume the solutions of Eq 4 as or φλ ; x, t = e pxp3 t B e qxq3 t, p = e iε, q = e iε, 5 = λ, R, 44 φλ ; x, t = e pxp3 t B e qxq3 t, p = e iε, q = e iε, 5 = λ, R, 45

13 Jingsong He, Yi Cheng, Rudolf A Römer to extract physical τ function ˆτ nn nn from τ At first, we would lie to give the two simple cases which are generated by the gauge transformations T Proposition 7 Let ξ = x cos ε t 3 cos 3ε, B = ie iε and φ 0 = φλ ; x, t defined by Eq 44, then the physical τ function of extracted from τ nn is n= and the corresponding one soliton u = x log ˆτ is ˆτ = eξ e ξ sin ε 46 u = 4 cos ε cosh ξ sin ε cosh ξ 47 sin ε with ε = π 3π or 0 0 The velocity of the soliton is v = cos 3ε In particular, the left-going soliton cos ε have two peas in its profile and the negative speed v = v ε= ; the right-going soliton have only one π 0 pea and positive speed v = v ε= 3π 0 Proof Taing φ 0 = φλ ; x, t of Eq 44 and n = bac into Proposition 6, the straightforward calculation leads to τ = φ 0 A = e ix sin εt3 sin 3ε p [ e ξ e ξ ] 48 sin ε Here ξ = x cos ε t 3 cos 3ε If let φ 0 = φλ ; x, t defined by Eq 45, then we can get a periodic solution as following proposition Proposition 8 Let η = x sin ε t 3 sin 3ε, and φ 0 = φλ ; x, t defined by Eq 45, = B = in φ 0 nn, then the physical τ function of extracted from τ is n= and the corresponding solution ˆτ = cos ε cosη ε 49 4 u = x log ˆτ sin cosη ε ε cos ε = 40 cosη ε cos ε is periodic Here ε = π 4π or 0 0 The velocity for the solution is v = sin 3ε If ε = π sin ε 0, u in Eq 40 is a left-going periodic wave If ε = 4π 0, u in Eq 40 is a right-going periodic wave

14 Solving bi-directional soliton equations 3 Proof τ = φ 0 e x cos [ ] εt3 cos 3ε = cosη ε 4 4 cos ε Here η = x sin ε t 3 sin 3ε, and φ 0 = φλ ; x, t is defined by Eq 45 Based on Propositions 7 and 8, we can find following corresponding relationship between symmetrical distributions of 5-th roots of e iε and moving direction of solutions e iε, e iε ε= π 0 the first distribution of 5-th roots of eiε p = e iε, q = e iε ε= π 0 in Eq 44 left-going two-pea soliton in Eq 47; e iε, e iε ε= 3π the second distribution of 5-th roots of eiε p 0 = e iε, q = e iε ε= 3π 0 in Eq 44 right-going one-pea soliton in Eq 47; 3 e iε, e iε ε= π the third distribution of 5-th roots of eiε p 0 = e iε, q = e iε ε= π 0 in Eq 45 left-going periodic wave in Eq 40; 4 e iε, e iε ε= 4π the fourth distribution of 5-th roots of eiε p 0 = e iε, q = e iε ε= 4π 0 in Eq 45 right-going periodic wave in Eq 40 There are only four distributions of 5-th roots of e iε, which are symmetric respect with x-axes or y- axes However, there exist several other pairs of roots in the above four distributions which will result in divergent solutions of through the above procedure For example, p = e i 3π 3π 0 i, q = e 0 or p = e i 8π 8π 0 i, q = e 0 Let us now concentrate on the two-pea soliton solution in Eq 47 Lemma 5 Let x, constant a > 0 and function y = yx = x a x, 4 a then if a > /, y x < 0; if a = /, then y x x= = 0; 3 if > a > 0, then there exists one point x > such that y x point of y, and y x > 0 x= x=x = 0, x is one extreme maximum Proof We have y x = y x = a a a x x 3 43 a Firstly, y x < 0 if a > / Secondly, if a = /, y x = 0 when x = At last, if / > a > 0, there exist x > such that y x = 0 Note that y x > 0 if x, x, y x < 0 if x > x So x is one extreme maximum point of y Proposition 9 Let a, b, be positive constants, ξ = x ct, c R, for following ind of solution b cosh ξ u = a, 44 cosh ξ a

15 4 Jingsong He, Yi Cheng, Rudolf A Römer if a /, u has one pea in its profile defined by ξ = 0; if 0 < a <, then there exist two peas in profile; 3 There exist no more than two peas in a soliton give by Eq 44 Proof By calculation, we have u x = b sinh ξ a a cosh ξ a a cosh ξ 3 According to the Lemma 5, we have a > /, there exist ξ = 0 such that u x = 0 because sinh ξ ξ=0 = 0 Note a cosh ξ < 0 a ξ a = /, there exist ξ = 0 such that u x = 0 because sinh ξ ξ=0 = 0 and a ξ=0 a cosh = 0 However, let ξ be sufficiently small, we have u x < 0 if ξ > 0 and u x > 0 if ξ < 0 So ξ = x ct = 0 defines one extreme maximum line of ux, t on x,t plane 3 / > a > 0, there exist ξ = 0 and ξ > 0 and ξ = ξ < 0 such that u x = 0 But u x > 0 if ξ < ξ ; u x < 0 if ξ ξ, 0; u x > 0 if ξ 0, ξ ; u x < 0 if ξ > ξ So ξ = x ct = 0 defines one extreme minimal line on x,t plane; 0 < ξ = x ct and 0 > ξ = x ct define two extreme maximum lines on the x,t plane Using u 0 if ξ, conclusions are proven Comparing Eq 44 with Eq 47 we get a = sin ε, and then can understand why sin ε ε=π/0 will lead to two peas in one soliton of but sin ε ε=3π/0 will lead only to one pea in one soliton of On the other hand, one soliton solution of u in Eq 47 have one pea or two peasmaximum case in its profile According to analysis above, we can claim from the point of view of reduction in KP hierarchy that the existence of two peas in the soliton is traced to three facts: The Grammian τ function in Proposition 6 which determines the form of soliton in Eq 47; The order of n-reduction, ie n 5 can produce two peas soliton in KP hierarchy; 3 The phase ε of n-th rootn 5 of e iε, such that 0 < a = sin ε < / Now we turn to the more complicated τ from Proposition 6, which generates the two soliton and periodic solution with two spectral parameters of equation The first case is the two soliton solution Lemma 6 Let φ 0 i = φλ i ; x, t, i =,, defined by Eq44, ξ i = x i cos ε i ti 3 cos 3ε i, η i = x i sin ε i ti 3 sin 3ε i, i =,, then τ nn n= gives out τ = A e iηη { ze ξξ 4 cosε ε z e ξξ B B 4 cosε ε A z 3e ξ ξ B 4i cosε ε z 3 e ξ ξ A 4i cosε ε z e ξ i sin ε cosε ε cosε ε z e ξ i sin ε cosε ε cosε ε B A B B A B

16 Solving bi-directional soliton equations 5 z 4e ξ i sin ε cosε ε cosε ε z 4 e ξ i sin ε cosε ε cosε ε B B 4 cos ε cos ε sin ε sin ε sin ε sin ε cosε ε cosε ε B A B Here z i, i =,, 3, 4, are given in Appendix B z i means the complex conjugation of z i B } In order to extract physical τ function ˆτ, we need following two Lemmas for suitable B i, i =, A i Lemma 7 For z i, i =,, 3, 4, are given in Appendix B, the following identities hold Lemma 8 Let B = i z, B = i z A z 4 z A 45 z = z z 3, z 4 = z z 3 46, and g 5 = z 3, g 6 = g 8 = z z, g 9 = z 4 z 4, then hold z 3 B z A z B z z B iz =, z 3 B = g 9, A B = g 8, A z z 4 iz B B z B =, 47 B = g 5, 48 A B = g 6 49 A Taing B i, i =,, and relations in Lemma 8 bac into Lemma 6, the physical τ function ˆτ A i obtained is Proposition 0 ˆτ = { e ξξ 4 cosε ε g 9 e ξξ 4 cosε ε e ξ ξ 4 cosε ε e ξ ξ 4 cosε ε e ξ sin ε cosε ε cosε ε g 8 e ξ sin ε cosε ε cosε ε e ξ sin ε cosε ε cosε ε

17 6 Jingsong He, Yi Cheng, Rudolf A Römer g 6 e ξ sin ε cosε ε cosε ε g 5 4 cos ε cos ε sin ε sin ε sin ε sin ε cosε ε cosε ε } 40 The two solitons solution is u = x log ˆτ In particular, ε = ε = π 0 soltions moving in negative direction; ε = ε = 3π 0 direction; ε = π 0, ε = 3π 0 results in two overtaing results in two overtaing soltions moving in positive results in head-on colliding two soltions The second case is a periodic solution with two spectral parameters of equation from Proposition 6 Lemma 9 Let φ 0 i = φλ i ; x, t, i =,, defined by Eq 45, ξ i = x i cos ε i ti 3 cos 3ε i, η i = x i sin ε i ti 3 sin 3ε ii =,, then τ nn gives n= τ = A e iξξ { ze iηη 4 cosε ε z e iηη B B 4 cosε ε A z 3e iη η B 4 cosε ε z 3 e iη η A 4 cosε ε z eiη cos ε cosε ε cosε ε z e iη cos ε cosε ε cosε ε z 4e iη cos ε cosε ε cosε ε z 4 e iη cos ε cosε ε cosε ε 4 cos ε cos ε sin ε sin ε B A B B B cos ε cos ε cosε ε cosε ε B A B A B Here z i, i =,, 3, 4, are given in Appendix C z i indicates the complex conjugation of z i B B } 4 Similar to the two solitons solution of equation, we need following two Lemmas to find suitable B i, i =,, to extract physical ˆτ from Eq 4 for periodic solution A i Lemma 0 For z i, i =,, 3, 4, are given in Appendix C, the following identities hold z = z z 3, z 4 = z z 3 4 Lemma Let z = z e iθ, =,, 3, 4, are given in Appendix C, and B = e iθ, B = e iθ4, and A g = z z, g 3 = z 3 z, g 4 = z 4 z, g 5 = z, then z3 B = z 3 B = g 3, 43 A z A z

18 Solving bi-directional soliton equations 7 hold z B z z z B A = z z B =, A B z B B = g, A B = g 5, 44 z4 z A B = z 4 z B B = g 4 45 A We can get the physical τ function ˆτ Lemma 9 by taing B i A i, i =, and relations in Lemma bac into Proposition ˆτ = { cos η η 4 cosε ε g 3 cos η η 4 cosε ε g cos η cos ε cosε ε cosε ε g 4 cos η cos ε cosε ε cosε ε g 5 4 cos ε cos ε sin ε sin ε cos ε cos ε cosε ε cosε ε } 46 The periodic solution with two parameters and is u = x log ˆτ Furthermore, ε = ε = π 0 results in two overtaing waves moving in negative direction ; ε = ε = 4π 0 results in two overtaing waves moving in positive direction; ε = π 0, ε = 4π 0 results in two head-on colliding waves We have plotted soliton solutions of in Fig 3, and there periodic solutions with two spectral parameters in Fig 4 5 Periodic and soliton solutions of equation The τ function of the equation is still in the form of a Grammian although the equation does not belong to the CKP hierarchy, which is obtained in [38] through the Bäclund transformation Similar to the equation, its τ function is in the form of Grammian, we can find τ function τ and τ of from Grammian τ function Let the initial value be u = 0 in Eqs 6 and 7, then = φλ i ; x, t are solutions of φ 0 i 4 xφλ i ; x, t = λ i φλ i ; x, t, φλ i ; x, t t = 3 xφλ i ; x, t 5 Proposition [38] The τ function of equation generated by Bäclund transformation from initial value u = 0 is τ nn = IW n,n φ 0 n, φ 0 n,, φ0 ; φ0, φ0,, φ0 n

19 8 Jingsong He, Yi Cheng, Rudolf A Römer Fig 3 Soliton solutions of the equation 5 Top left: one left-going two-pea soliton when ε = π 0 and = Top right: two left-going soliton when =, = 3, ε = ε = π Bottom: Head-on collision of left- and right-going 0 solitons when = 8, = 3, ε = 3π 0, ε = π 0 0 φ n φ 0 0 φ = 0 φ φ 0 0 φ φ 0 n φ0 0 φ n φ 0 0 φ n φ0 0 φ φ 0 0 φ φ 0 and the solution u of from initial value zero is u = x log τ nn 0 φ n φ φ n φ 0 0 n φ n φ 0 n 0 φ n φ0 3 φ 0 n φ0 0 n φ n φ0 n 0 φ φ φ φ 0 0 n φ φ 0 n 0 φ φ φ φ 0 0 n φ φ 0 n 5 53 Here φ 0 i = φλ i ; x, t are solutions of Eq 5 In fact, τ nn can be generated by gauge transformation T n n= The Lax pair of is L = 4 x 4u x 4u x x u xx 4u v, M = 3 x 3u x 3 u x,

20 Solving bi-directional soliton equations 9 Fig 4 Periodic solutions with two spectral parameters of equation 5 Top left: left-going periodic solution with = 0, = 03, ε = ε = π 0 Top right: right-going periodic solution when = 04, = 05, ε = ε = 4π 0 Bottom: Collision of left- and right-going periodic solution when =, = 5, ε = π 0, ε = 4π 0 and satisfy L = L, M = M Similar to the CKP hierarchy, let T = T = T I ψ T Dφ 0, and do gauge transformation L = T L T So L = L requires T Dψ T Iφ 0 = T I ψ T Dφ 0 as we have seen in CKP hierarchy The remaining procedure is the same as the gauge transformation of the CKP hierarchy as well as the equation Of course, the generating functions φ 0 i, ψ 0 i = φλ i ; x, t, ψλ i ; x, t satisfy Eq 6 and Eq 7 if the initial values are u 0, v 0, or Eq 5 if the initial values are u = 0, v = 0 Remar We should note that L v=0 = 4 x 4u x 4u x x u xx 4u = x u = L KdV The Lax pair of the KdV equation is L KdV = x u, M KdV = 3 x 3u x 3 u x T D φ 0 generates a single soliton solution of the KdV from zero initial value Here φ0 = φλ ; x, t satisfy L KdV φλ ; x, t = λ φλ ; x, t and φλ ; x, t = M KdV φλ ; x, t simultaneously The left-going t multi-soliton can be produced by using repeated iteration of T D

21 0 Jingsong He, Yi Cheng, Rudolf A Römer In order to get real and smooth solutions, such as soliton and periodic solution, we should construct physical τ function ˆτ from τ nn which is complex and related to 4-th roots of e iε The case of n = and n = will be discussed in detail Let us start to discuss the single soliton with two directional propagation To do this, similar to the above two sections, we should assume the solution of Eq 5 as or φλ ; x, t = e pxp3 t B e qxq3 t, p = e iε, q = e iε, 4 = λ, R, 54 φλ ; x, t = e pxp3 t B e qxq3 t, p = e iε, q = e iε, 4 = λ, R 55 For φλ i ; x, t, i =,, the difference between here and above two sections is the i 4 = λ i, i =,, instead of i 5 = λ i, i =, From Proposition we can extract physical τ function ˆτ nn from τ n= Proposition 3 Let ξ = x cos ε t 3 cos 3ε, B = ie ε, and φ 0 = φλ ; x, t as defined by Eq 54, then the physical τ function of extracted from τ nn n= is ˆτ = e eξ ξ 56 sin ε and the corresponding single soliton u = x log ˆτ is u = Here ε = π 4 The velocity of the soliton is v = cos 3ε ε= cos ε π > 0 4 4cos ε cosh ξ sin ε cosh ξ 57 sin ε Proof τ = φ 0 = e 3 ix sin εt sin 3ε p e ξ e ξ sin ε 58 As we discussed in Remar, the left-going soliton can also be generated by T D Proposition 4 Let ξ = x cos ε t 3 cos 3ε ε=0, and φ 0 = φλ ; x, t ε=0 as defined by Eq 54, then the physcial τ function of generated by T D φ 0 is and the corresponding single soliton u = ˆτ = e ξ 59 B x log ˆτ is u = Here A B > 0 The velocity of the soliton is v = < 0 4 B e ξ 50 e B ξ

22 Solving bi-directional soliton equations Proof τ = φ0 = B e ξ A e ξ B 5 It can be clarified by u = 0, v = 0 TDφ0 u 0, v = 0, and then τ = φ0 On the other hand, if φ 0 = φλ ; x, t as defined by Eq 55, then we can get periodic solution from Proposition Proposition 5 Let η = x sin ε t 3 sin 3ε, = B = in φ 0, then the physical τ function of equation for periodic solution extracted from τ nn n= is ˆτ = cosη ε 5 cos ε and the corresponding peroidic solution u = x log ˆτ is 4 sin cosη ε ε cos ε u = 53 cosη ε cos ε Here ε = π 4 The velocity of the solution is v = sin 3ε ε= sin ε π < 0 4 Proof τ = φ 0 = 3 cos εt ex cos 3ε 4 cosη ε 54 cos ε There are some relationship between the distributions of 4-th roots of e iε and moving direction of solutions e iε, e iε ε=0 the first distribution of 4-th roots of e iε > p = e iε, q = e iε ε=0 in Eq 54 > left-going soliton in Eq 50; e iε, e iε ε= π 4 the second distribution of 4-th roots of eiε > p = e iε, q = e iε ε= π 4 in Eq 54 > right-going soliton in Eq 57; 3 e iε, e iε ε= π 4 the third distribution of 4-th roots of eiε > p = e iε, q = e iε ε= π 4 in Eq 55 > left-going periodic wave in Eq 53 In the above discussion, we now the right-going soliton and left-going periodic wave of of have the completely same form with the equation, except ε = π/4 instead of ε = π/0 and ε = 3π/0 The reason is that the τ function of two equations is in the same Grammian of generating functions φ 0 i, and generating functions φ 0 i for two equations satisfy analogous linear partial differential equations with constant coefficients, ie Eq 4 for equation, Eq 5 for equation These relations between and are still true for their two soliton and two parameters periodic solutions Proposition 6 The two right-going solitons are given u = x log ˆτ 55 in which ˆτ is ˆτ = ˆτ ε =ε =π/4 56 and ˆτ is given by Proposition 0

23 Jingsong He, Yi Cheng, Rudolf A Römer Proposition 7 The right-going periodic wave with two spectral parameters and is given by u = x log ˆτ 57 in which ˆτ is ˆτ = ˆτ ε =ε =π/4 58 and ˆτ is given by Proposition According to the analysis in Remar, two left-going solitons of equation can be generated by a chain of gauge transformations u = 0, v = 0 TDφ0 u 0, v = 0 T Dφ u 0, v = 0 using the notation of [7], φ 0 i = φλ i ; x, t ε=0, i =,, are defined by Eq 54 Their τ function of generated by T = T D φ T Dφ 0 is τ = φ 0 φ 0 φ 0,x φ0,x From τ we can obtain the physical τ function ˆτ and two soliton solution 59 Proposition 8 Let φ 0 i = φλ i ; x, t ε=0 are defined by Eq 54, ξ i = i x 3 i t, i =, If B > 0, B < 0, >, then the physical τ function ˆτ is given by A The two soliton solution is u = ˆτ = e ξξ B B A e ξξ B e ξ ξ B e ξ ξ 50 A x log ˆτ, which is left-going The collision of two soliton is generated by gauge transformation chain u = 0, v = 0 TDφ0 u 0, v = 0 T I ψ TDφ u 0, v 0, φ 0 = φλ ; x, t ε=0 is defined by Eq 54, ψ 0 φλ ; x, t is defined by Eq 54 The corresponding τ function of is = φ 0, φ0 = τ 0 = ψ φ0 ψ 0 φ0 ψ 0 0 φ 0 φ 0 φ === φ0 φ 0 φ0 φ 0 φ 0 5 Taing φ 0 i, i =,, bac into Eq 5, we have its explicit expression as following Lemma Lemma Let ξ = x t 3, ξ = x cos ε t 3 cos 3ε, η = x sin ε t 3 sin 3ε, z i = c i d i, i =, 3, 5, are given in Appendix D τ = eiη A { ze ξξ z e ξ ξ cos ε cos ε B A B

24 Solving bi-directional soliton equations 3 z 3 e ξξ cos ε B z 3 e ξ ξ cos ε z 5 e ξ i sin ε cos ε cos ε z 5e ξ i sin ε cos ε cos ε Lemma 3 For z i = z i e θi i =, 3, 5, the following identities are true Lemma 4 Let z i, i =, 3, 5, are given by Appendix D, if B = z z3 then g = z B B z, g 4 = z 3 B A z A hold B A B A B A B } 5 z z 5 = z 3 z 5, e iθ3 θ = e iθ3 53, g 6 = z 5 iz, B A = i z z 5, g = g 4 = z z 5, g 6 =, B A B, 54 With the help of Lemmata 3 and 4, we deduce the physical τ function of colliding two soliton of equation from Lemma Proposition 9 Let g, g 4 are given in Lemma 4, then ˆτ = { e ξξ cos ε g e ξ ξ cos ε e ξξ g 4 e ξ ξ cos ε cos ε e ξ sin ε cos ε cos ε e ξ sin ε cos ε cos ε } 55 We have plotted the two soliton solutions of equation in Fig 5, and periodic solutions with one spectral parameter and with two spectral parameters of the same equation in Fig 6 6 Lower and Higher order reductions In this section, we want to discuss the general character of soliton equation from lower order to higher order in one same sub-hierarchy The purpose is to show the relation between propagation of soliton on x,t plane and the order of Lax pair, and show the difference between the lower reduction and higher reduction Let Lax pair of soliton equation is L, M, which defines φλ; x, t by Lφλ; x, t = λφλ; x, t, φλ; x, t t = Mφλ; x, t 6

25 4 Jingsong He, Yi Cheng, Rudolf A Römer Fig 5 Two left- and right-going as well as head-on colliding solitons for the equation 9 Parameters are chosen as: = B, A =, B =, = 5, = left; = 5, = 3, ε = ε = π 4 right; = 08, = 09, ε = π 4 collision Fig 6 Left-going periodic solutions with one left and two right spectral parameters for the equation 9 Parameters: =, ε = π 4 left; = 0, = 03, ε = ε = π 4 right

26 Solving bi-directional soliton equations 5 There are some examples of n-reduction of the KP hierarchy For the BKP hierarchy, Lax pair 3-reduction 5-reduction 7-reduction 9-reduction L B 3 B 5 B 7 B 9 M B5 B 3 B 3 B 3 Equation SK bsk higher order higher order 6 Here B 5 = x 5 5u x 3 5u x x 5u 0 3 u xx x, 63 SK [6, 7] : 9u t 45u u x u xxxxx 5uu xxx 5u x u xx = 0 64 B 5 and B 3 are given by Eqs 0 and For the CKP hierarchy, Lax pair 3-reduction 5-reduction 7-reduction 9-reduction L B 3 B 5 B 7 B 9 M B5 B 3 B 3 B 3 Equation KK higher order higher order 65 Here B 5 = x 5 5u x 3 5 u x x 5u 35 6 u xx x 5uu x 5 3 u xxx, 66 KK [8, 9] : 9u t 45u u x u xxxxx 5uu xxx 75 u xu xx = 0, 67 B 5 and B 3 are given in Eqs 3 and 4 There are several even-reductions of the KP hierarchy as following, Lax pair -reduction 4-reduction 6-reduction 8-reduction L B B 4 B 6 B 8 M B 3 B 3 B 3 B 3 68 Equation KdV higher order higher order Now we start to discuss the BKP hierarchy Lemma 5 Let ξ = x cos ε t 5 cos 5ε, then ˆτ SK is expressed by ˆτ SK = ˆτ bsk ξ > ξ e 69 and the corresponding single soliton is u = x log ˆτ SK The velocity of soliton v = 4 4 > 0 Here ˆτ bsk is given by Proposition 4 cos 5ε cos ε Proof Because the SK equation and bsk equation belong to the same sub-hierarchy BKP, so the results of bsk are also hold by SK equation only if we replace ξ in bsk by ξ = x cos ε t 5 cos 5ε For SK equation, the generating functions φ 0 i = φλ i ; x, t of gauge transformation satisfy 3 xφλ; x, t = λφλ; x, t, φλ; x, t t ε= π 6 = 5 xφλ; x, t, 60 which are different with Eq 3 for bsk equation So 3 = λ This difference determines replacement in Eq 69 Of course, similar to the bsk, we also should assume the solutions of Eq 60 be the form of φλ ; x, t = e pxp5 t B e qxq5 t, p = e iε, q = e iε, 3 = λ, R, 6 =

27 6 Jingsong He, Yi Cheng, Rudolf A Römer or φλ ; x, t = e pxp5 t B e qxq5 t, p = e iε, q = e iε, 3 = λ, R 6 Taing the generating functions φ 0 i in Eq 6 bac into the Proposition, then we can extract ˆτ SK from τ SK The relation ˆτ SK = ˆτ bsk ξ > ξ e is given by comparison In particular, there are two distributions of roots of third-order of e iε on circle, which is symmetric with respect to y-axes However, they are corresponding to same single soliton solution e i π 6, e i π 6 one distribution of 3-order root of e iε on unit circle p = e i π 6, q = e i π 6 in Eq 6 a single soliton in Lemma 5 e i π 6, e i π 6 one distribution of 3-order root of e iε on unit circle p = e i π π 6 i, q = e 6 in Eq 6 one soliton as Lemma 6 The higher order equations of the BKP hierarchy are defined by Eq 6 For the n- reduction equation of the BKP hierarchy nbkp, n = j, j = 3, 4, 5,, and let ξ mp = x m cos ε p tm 3 cos 3ε p, m n = m j = λ m, then the physical τ function of the nbkp generated by T is ˆτ nbkp = ˆτ bsk ξ > ξ e p, 63 and the corresponding single soliton of the nbkp is u = x log ˆτ nbkp Here ε p = p n π = p 4j π, p =,, 3,, j, ˆτ bsk is given by Proposition 4 So the single soliton can move along j directions in x,t plane, which are given by ξ p = 0 associated with j-value of ε p given before Proof Comparing the nbkp with the bsk equation, the main change here is the Lax pair L,M The Lax pair of the nbkp defines the generating functions φ 0 i = φλ i ; x, t are slight different as and then we assume or n x φλ; x, t = λφλ; x, t, φλ; x, t t = 3 xφλ; x, t 64 φλ ; x, t = e pxp3 t B e qxq3 t, p = e iεp, q = e iεp, n = λ, R, 65 φλ ; x, t = e pxp3 t B e qxq3 t, p = e iεp, q = e iεp, n = λ, R 66 In order to avoid the divergence of u, we only tae 0 < ε p < π, and then ε p = p n π = p 4j π, p =,, 3,, j This change results to the emergence of ξ mp = x m cos ε p tm 3 cos 3ε p, m n = m j = λ m The ˆτ nbkp and single soliton solution u = x log ˆτ nbkp can be derived directly from the Proposition and the generating functions φ 0 in Eq 65 associated with λ for the gauge transformation Further, for a given p, ξ p = 0 determines one moving direction of the single soltion on x,t plane, then the single soliton solution have j directions for propagation because p =,,, j From Lemmata 5, 6 and the results of the bsk equation, we have Proposition 0

28 Solving bi-directional soliton equations 7 The single soliton u = x log ˆτ nbkp of the nbkp equation, n = j, j =, 3, 4,, can move along a direction defined by ξ p = 0 on x,t plane for a given p e iεp, e iεp one distribution of n-th order roots of e iε on circle p = e iεp, q = e iεp in Eq 65 The single soliton moves along a line ξ { π p = 0 on x, t plane Here ε p 4j, 3π 4j, 5π j π },, 4j 4j 3 For a given n = j, the single soliton of the nbkp have j directions to propagate on x,t plane, which are defined ξ p = 0, p =,, 3, j Note that the result of j = in above Proposition is given by Lemma 5 Now we turn to the lower and higher reductions of the CKP hierarchy Similar to the discussion of the BKP hierarchy in this section, we can obtain parallel results in the CKP hierarchy, so we write out the results without proof in the following to save space Lemma 7 Let ξ = x cos ε t 5 cos 5ε, then ˆτ KK can be expressed by ˆτ KK = ˆτ ξ > e ξ 67 and the corresponding single soliton is u = x log ˆτ KK The velocity of soliton is ˆv = 4 4 > 0 Here ˆτ is given by Proposition 7 cos 5ε cos ε Lemma 8 The higher order equation of CKP defined by Eq 65 For n-reduction of CKP hierarchy nckp, n = j, j = 3, 4, 5, Let ξ mp = x m cos ε p tm 3 cos 3ε p, m n = m j = λ m, then the τ function of the nckp generated by T is ˆτ nckp = ˆτ ξ > ξ e p, 68 and the corresponding single soliton of the nckp equation is u = x log ˆτ nckp Here ε p = p n π = p 4jπ, p =,, 3,, j, and ˆτ is given by Proposition 7 So the single soliton can move along j directions on x, t plane, which are given by ξ p = 0 associated with j-value of ε p given before Using the Lemmata 7, 8 and results for the equation, we get Proposition The single soliton u = x log ˆτ nckp of the nckp, n = j, j =, 3, 4,, can move along a direction defined by ξ p = 0 on x, t plane for a given p e iεp, e iεp one distribution of n-th order roots of e iε on circle p = e iεp, q = e iεp in Eq 65 the single soliton moves along a line ξ { π p = 0 on x, t plane Here ε p 4j, 3π 4j, 5π j π },, 4j 4j 3 For a given n = j, the single soliton of the nckp can move along j directions on x,t plane, which are defined by ξ p = 0, p =,, 3,, j 4 In particular, if 0 < ε p < π/6, u = x log ˆτ nckp is a two-pea soliton π 6 =

29 8 Jingsong He, Yi Cheng, Rudolf A Römer In above Proposition, the case of j = is given by Lemma 7 This Proposition shows there exist several single two-pea solitons for nckp if n Corollary There are two single two-pea solitons for -reduction of CKP hierarchy, i e CKP equation, u CKP = x log ˆτ, in which ε = π/ and ε = 3π/ respectively Here ˆτ is given in Eq 46 We have plotted it out in Fig 7 with = 08 Fig 7 Left-going two-pea soliton with dashed line ε = π/ is faster, left-going two-pea soliton with full line ε = 3π/ The left is plotted when t = 0, the right is plotted when t = 0 We have nown that a = / in Lemma 5 and Proposition 9 is one crucial point to exist one-pea soliton or two-pea soliton It is more interesting that a = / will lead to stationary soliton of higher reductions of the BKP and the CKP hierarchy, which is not moving on x,t plane When ξ ε=π/6 = x cos ε t 3 cos 3ε ε=π/6 = x cos ε, ξ is independent with t So u is independent with t by taing this ξ into Proposition 4 and Proposition 7 Corollary There exists stationary single soliton for the 9-reduction of BKP hierarchy, which is u 9BKP = x log ˆτ bsk ε=3π/8 Here ˆτ bsk is given by Proposition 4; There exists stationary single soliton for the 9-reduction of CKP hierarchy, which is u 9CKP = x log ˆτ ε=3π/8 Here ˆτ is given by Proposition 7 We have plotted out stationary soliton for the 9-reduction of CKP in Fig 8 when = Corollary 3 There is single two-pea soliton u = x log ˆτ ε=π/8 for 8-reduction of the KP hierarchy; there is stationary single one-pea soliton u = x log ˆτ ε=π/6 for the 6-reduction of the KP hierarchy Here ˆτ is given by Proposition 3

30 Solving bi-directional soliton equations 9 Fig 8 Left: stationary soliton for 9-reduction of CKP, Right: Single two-pea soliton for 8-reduction of KP The two-pea soliton for the 8-reduction of KP hierarchy is plotted in Fig 8 when = For our best nowledge, this is first time to report the even-reduction of the KP hierarchy also has two-pea soliton solution The possession of two-pea soliton solution is not sole property of CKP hierarchy 7 Conclusions and Discussions We have presented a systematic way in which to obtain the solution of the n-reduction n = 4, 5 from the general τ function of the KP hierarchy Our approach is based on the determinant representation of gauge transformations T n [7] and τ n [6] It may be summarized as follows: τ n constraints of generating functions and =n τ n BKP assume the form of φ iand find suitable B i A i n or τ CKP 5 reduction τ n =n Eq = bsk, n efficient τ function ˆτ Eq =n=, We have applied this approach to various equations The one soliton, two soliton and periodic solution are constructed for bsk, and We show the corresponding relation between the distribution of 5-th or 4th roots of e iε on the unit circle and several types of solutions left-going one soliton, right-going one soliton, left/right-going periodic solutions We also show the reason for the existence of the two-pea soliton Furthermore, the lower reduction and higher reduction of BKP, CKP, and the even-reductions are explored by this method Our results show that the soliton of the n-reduction with n = j, j =,, 3, of BKP and CKP can move alone j directions, which are defined by ξ p = 0 Each direction corresponds to one symmetry distribution of n-th roots of e iε on the unit circle This supplies a very natural explanation why the 5-reduction BKP or CKP has bi-directional solitons whereas the 3-reduction of BKP or CKP has only single-directional solitons At last, the two-pea soliton is not a monopolizing phenomena of only the CKP hierarchy Rather, we find that the higher-order even-reduction of KP also exhibits two-pea solitons and we elucidate the criterion for its existence from the Grammian τ function At the same time, we show there is not three and more pea soliton from Grammian τ function The stationary soliton for higher order reduction of KP hierarchy is also obtained We thin that it is possible to construct an N-soliton solution of the bsk, and equations by this approach Namely, there exist suitable B i A i i =,,, N such that we can find a physical τ function ˆτ NN Eq for these equations from a complex-valued τ NN Eq, which is symmetric because we have assumed generating functions φ i in Eq 35 and Eq 36 with symmetric form Here Eq = bsk,, Additionally, it is worthy to discuss the phase shift in the collision of one-pea soliton Eq