The Extended C-Type of KP Hierarchy: Non-Auto Darboux Transformation and Solutions

2.1 CKP Hierarchy

It is known that KP hierarchy is given by

where L=∂+u₁∂⁻¹+u₂∂⁻²+…, ∂ denotes ∂/∂x, u_i(i=1, 2, …) depend on t=(t₁, t₂, t₃, …) with t₁=x, and B_n=(Lⁿ)_≥0 stands for the differential part of Lⁿ. The KP hierarchy also can be rewritten as Zakharov–Shabat (ZS) form

It is associated that Lax pair and adjoint Lax pair are given by

where ϕ and ψ are called the eigenfunction and the adjoint eigenfunction, respectively. The symbol * denotes a formal adjoint operation defined by A∗=∑i = 1∞(−1)i∂iai for an arbitrary pseudo-differential operator A∗=∑i = 1∞ai∂i and (AB)*=B*A* for two operators A, B.

The CKP hierarchy is obtained from KP hierarchy (1) by imposing the following restricted condition on the KP Lax operator

which leads to that CKP hierarchy only depends on the odd time flow. In addition, we immediately get from (4),

2.2 The Extended CKP Hierarchy

The multicomponent generalisation of the CKP hierarchy was considered in [8]. The CKP hierarchy is extended as follows

where L is define as (1) and (4), n, k are odd. u_i, q_i, and r_i depend on two sets of real variables t=(t₁, t₃, t₅, …) and τ=(τ₁, τ₃, …).

Under (8) and (9), the commutativity between (6) and (7) gives rise to

Actually the system (10) together with (8) and (9) are nothing but the CKP hierarchy with self-consistent sources. The corresponding Lax representation for (10) is given by

When n=3, k=5, (10) yields the first type of N-th CKPESCS, which has been derived in [8]. When n=5, k=3, (10) becomes the second type of N-th CKPESCS

whose Lax representation is given by

where q_i, r_i are defined by (14) and (15).

3.1 1–Time Non-Auto DT for Extended CKP Hierarchy

Theorem 3.1Let L, q_i, r_i(i=1, …, N) be the solution of the extended CKP hierarchy (6–9), f and g are two independent eigenfunction solutions of (11 and 12), Denote θ=f+b(τ_k)g, Then the non-auto DT defined by

gives the solution L[1], q_i[1], r_i[1](i=1, …, N, N+1) for the extended CKP hierarchy (6–9) with N replaced by N+1, where the notation∫fstands for∫fdxwith zero integration constants,

andT_D(θ)=θ∂θ⁻¹, T_I(θ⁽¹⁾)=(θ⁽¹⁾)⁻¹∂⁻¹θ⁽¹⁾, θ(1)=−TI(θ)(θ)=−∫θ2θ,b(τ_k), β(τ_k), and η(τ_k) are the functions ofτ_ksuch thatb′(τ_k) =β(τ_k)η(τ_k).

Remark 3.1For G₁in (22) and T_min (40), expansions with respect to the last column are understood, in which all sub-determinants are collected on the left side of ∂^k(k=−1, 0). While forG1−1in (23) andTm−1in (41), expansions with respect to the first column are understood (collecting all minors on the right side of ∂⁰and θ_i∂⁻¹).

Proof. Note that L[1]=G1LG1−1, we can derive the relation of B_n and B_n[1] described by

The detailed derivation can be found in Appendix A.

Similarly, we also get

Firstly, we show that (7) holds for L[1] and B_n[1]. In fact

Owing to (24), (26) becomes

(L[1])tn=[Bn[1], L[1]].

Secondly, we show that (8) holds for B_n[1] and q_i[1](i=1, …, N, N+1). For i=1, …, N,

For i=N+1, with the same proof as (27), we can get (qN + 1[1])tn=Bn[1](qN + 1[1]).

Thirdly, we show that B_n[1] and r_i[1](i=1, …, N, N+1) satisfy (9).

For i=1, …, N,

(ri[1])tn=[(G1−1)∗(ri)]tn=[(G1−1)∗]tn(ri)+(G1−1)∗(ri,tn) =−(G1−1)∗G1,tn∗(G1−1)∗(ri)−(G1−1)∗Bn∗(ri) =−(G1,tnG1−1+G1BnG1−1)∗(G1−1)∗(ri) =−Bn∗[1](ri[1])=Bn[1](ri[1]).

For i=N+1,

Note that

If taking ∂⁻¹(0)=1, then we get from (29)

Substituting (30) into (28) leads to

Lastly, we show that L[1], q_i[1], r_i[1](i=1, …, N, N+1) satisfy (6) with N replaced by N+1, that is,

(L[1])τk=[Bk[1]+∑i = 1N + 1(qi[1]∂−1ri[1]+ri[1]∂−1qi[1]), L[1]].

For the sake of convenience, we set Δ_i≜q_i∂⁻¹r_i+r_i∂⁻¹q_i, Δ_i[1]≜q_i[1]∂⁻¹r_i[1]+r_i[1]∂⁻¹q_i[1].

Noting that B_k[1] satisfy (25) and

(L[1])τk=[G1,τkG1−1, L[1]]+G1[Bk+∑i = 1NΔi, L]G1−1=[G1,τkG1−1+G1BkG1−1+∑i = 1NG1ΔiG1−1, L[1]],

then we obtain

Since θ=f+b(τ_k)g and f, g satisfy (11 and 12), we get

θτk=Bk(θ)+∑i = 1NΔi(θ)+b′(τk)g,

hence we obtain

Noting that G1(g)=(1−θ∫θ2∂−1θ)(g)=g−θ∫θg∫θ2 and b′(τ_k)=β(τ_k)η(τ_k), by the tedious computation, we can show that

Note that the following identity holds true

θ∫θ2∂−1gθ∂−1θ∫θ2=θ∫θ2∂−1(∂∫gθ−∫gθ∂)∂−1θ∫θ2=θ∫θg∫θ2∂−1θ∫θ2−θ∫θ2∂−1θ∫θg∫θ2,

so we derive that (34) is equal to zero. In addition, by the direct but tedious calculation, we show for ∀ i=1, …, N,

G1ΔiG1−1−Δi[1]−Δi(θ)∂−1θ∫θ2−θ∫θ2∂−1Δi(θ) +2θ∫θ2∫θΔi(θ)∂−1θ∫θ2−θ∫θ2∂−1Δi(θ)θ∂−1θ∫θ2=2θ∫θqiΩ(ri, θ)dx∫θ2∂−1θ∫θ2+2θ∫θriΩ(qi, θ)dx∫θ2∂−1θ∫θ2 −2θΩ(ri, θ)Ω(qi, θ)∫θ2∂−1θ∫θ2=2θ∫θ2(∫θqiΩ(ri, θ)dx+∫θriΩ(qi, θ)dx)∂−1θ∫θ2 −2θΩ(ri, θ)Ω(qi, θ)∫θ2∂−1θ∫θ2=2θΩ(ri, θ)Ω(qi, θ)∫θ2∂−1θ∫θ2−2θΩ(ri, θ)Ω(qi, θ)∫θ2∂−1θ∫θ2=0,

where Ω(ri, θ)=∫riθ, Ω(qi, θ)=∫qiθ. Hence, we have

(L[1])τk−[Bk[1]+∑i = 1N + 1(qi[1]∂−1ri[1]+ri[1]∂−1qi[1]), L[1]]=0.

So far, we have completed the proof of Theorem 3.1.□

After m steps iterations of the non-auto DT defined in Theorem 3.1, we obtain the m times repeated non-auto DT for the extended CKP hierarchy.

3.2 m-Time Repeated Non-Auto DT for Extended CKP Hierarchy

Theorem 3.2Let L, q_i, r_i(i=1, …, N) be the solution of extended CKP hierarchy (6–9), f_i and g_j(j=1, …, m) are 2m independent eigenfunction solutions of (11 and 12). Denote θ_j=f_j+b_j(τ_k)g_j, m-time repeated non-auto DT is given by

then L[m], q_i[m], r_i[m](i=1, …, N, …, N+m) satisfy extended CKP hierarchy (6–9) with N replaced by N+m, where

and

Gm=TI(θm(2m − 1))TD(θm(2m − 2)), θm(2m − 1)=−TI(θm(2m − 2))(θm(2m − 2)),θm(2m − 2)=Tm − 1(θm), TI(θm(2m − 1))=(θm(2m − 1))−1∂−1θm(2m − 1),TD(θm(2m − 2))=θm(2m − 2)∂(θm(2m − 2))−1.

Here b_j(τ_k), β_j(τ_k), and η_j(τ_k) are functions of τ_ksuch thatb′j(τk)=βj(τk)ηj(τk),the expansions of determinants (40) and (41) follow the rules given by Remark 1.

Proof. In what follows, we will prove Theorem 3.2 by the mathematical induction method. Obviously, Theorem 3.1 has indicated that L[1], q_i[1], r_i[1](i=1, …, N, N+1) satisfy extended CKP hierarchy (6–9) with N replaced by N+1. Provided that when m=s, L[s], q_i[s], r_i[s](i=1, …, N+s) satisfy (6–9) with N replaced by N+s. When m=s+1, the direct calculations lead to the recursion relations as follows: