On quasiperiodic solutions of the modified Kadomtsev–Petviashvili hierarchy

Applied Mathematics Letters 97 (2019) 27–33

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Applied Mathematics Letters www.elsevier.com/locate/aml

On quasiperiodic solutions of the modified Kadomtsev–Petviashvili hierarchy Peng Zhao a,b,c , Engui Fan d ,∗ a

College of Arts and Sciences, Shanghai Maritime University, Shanghai 201306, PR China College of Mathematics and Systems Sciences, Shandong University of Science and Technology, Qingdao 266590, PR China c School of Mathematical Sciences, Qufu Normal University, Qufu 273165, PR China d School of Mathematical Sciences, Institute of Mathematics and Key Laboratory of Mathematics for Nonlinear Science, Fudan University, Shanghai 200433, PR China b

article

info

abstract In this paper, we present the construction of quasiperiodic solutions of the pth equation in the modified Kadomtsev–Petviashvili hierarchy by seeking compatible solutions of the 1th and the pth equations in the derivative nonlinear Schrödinger hierarchy. © 2019 Elsevier Ltd. All rights reserved.

Article history: Received 4 March 2019 Received in revised form 3 May 2019 Accepted 3 May 2019 Available online 11 May 2019 Keywords: Modified Kadomtsev–Petviashvili hierarchy Gesztesy–Holden method Baker–Akhiezer function Riemann theta function Quasiperiodic solutions

1. Introduction The modified Kadomtsev–Petviashvili (mKP) equation ut =

1 3 3 3 uxxx − u2 ux + ux ∂ −1 uy + ∂ −1 uyy 4 2 2 4

(1.1)

plays important roles in a variety of different fields including modern string theory and in connection with the solution of the Schottky problem of compact Riemann surfaces [1]. In Sato theory [2–5], the mKP equation and its higher order flows are consistently described by isospectral deformations of the eigenvalue problem Lψ = λψ, λ ∈ C, and its time evolution ψtm = Bm ψ, where L is a pseudodifferential operator ∑∞ ∂ defined by L = ∂ + j=0 uj+1 ∂ −j , ∂ = ∂x , uj = uj (x, t1 , t2 . . .), and Bm is the differential part of ∑m m m j L : Bm = (L )+ = b ∂ . There has been a few remarkable work that associated with the m,j j=1 ∗ Corresponding author. E-mail address: [email protected] (E. Fan).

https://doi.org/10.1016/j.aml.2019.05.006 0893-9659/© 2019 Elsevier Ltd. All rights reserved.

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P. Zhao and E. Fan / Applied Mathematics Letters 97 (2019) 27–33

study of the mKP equation (hierarchy). An important development of the algebro-geometric method was the passage from 1+1 systems to the integration of 2+1 KP-like systems, realized by Krichever in 1976 [6], who constructed algebro-geometric solutions of the KP equation on a basis of a purely algebraic formulation of algebro-geometric approach. Then a generalized Miura transformation between the KP equation and mKP equation was discussed in [7]. Later it has been shown that the mKP hierarchy can be constructed on the basis of the Sato approach by means of the gauge transformation of the KP hierarchy which results in modification of the pseudo-differential operator L [8]. New B¨acklund transformations for the KP hierarchy and the possibility of transferring classes of KP solutions into those of mKP solutions are derived in [9]. An explicit theta function solution of the mKP equation is derived by the technique of nonlinearization of Lax pairs and Abel inversion [10]. In this paper, we will give an explicit construction of quasiperiodic solutions of the entire mKP hierarchy by applying a variant of Gesztesy–Holden’s method [11]. In Section 2, we review some basic results that exist in literature, including the derivative nonlinear Sch¨odinger (DNLS) hierarchy, symmetry constraint results of mKP spectral problem, spectral curves and auxiliary spectral points {µj }nj=1 , {νj }nj=1 . In Section 3, we explicitly construct the Baker–Akhiezer function of the mKP equation based on the conservation relations and then gives its analytic and asymptotic properties. In Section 4, using Riemann vanishing theorem and Riemann–Roch theorem, we reconstruct the Baker–Akhiezer function in terms of Riemann theta function and then derive quasiperiodic solutions of the entire mKP hierarchy. 2. Preliminaries Let us briefly review the results of [12]. Throughout this paper we suppose that q, r are functions of x, y and tp . Define the sequences of differential polynomials {fℓ }ℓ∈N0 , {gℓ }ℓ∈N0 , and {hℓ }ℓ∈N0 , recursively by g2ℓ+1 = f2ℓ = h2ℓ = 0, ℓ ∈ N0 , f1 = 2q, g0 = 2, h1 = −2r,

(2.1)

2f2ℓ+1 = −f2ℓ−1,x + 2qg2ℓ + (1/2)qrf2ℓ−1 , ℓ ∈ N0 ,

(2.2)

2h2ℓ+1 = h2ℓ−1,x − 2rg2ℓ + (1/2)qrh2ℓ−1 , ℓ ∈ N0 ,

(2.3)

g2ℓ,x = rf2ℓ+1 + qh2ℓ+1 , ℓ ∈ N0 .

(2.4)

ˆ ℓ , defined by vanishing of the integration constants The corresponding homogeneous coefficients fˆℓ , gˆℓ , and h ck , for k = 1, . . . , ℓ. In terms of fℓ , gℓ , hℓ the nth GI equation can be written as ( ) qtn − 2f2n+3 + qg2n+2 DNLSn (q, r) = = 0, tn ∈ R, n ∈ N0 . (2.5) rtn − 2h2n+3 − rg2n+2 Next we introduce −z 2 + 14 qr rz

) qz , z ∈ C, (2.6) z 2 − 14 qr ( ) −G2n+2 (z) F2n+1 (z) V2n+2 (z) = , n ∈ N0 , (2.7) −H2n+2 (z) G2n+2 (z) ∑n ∑n where Fn , Hn , Gn+1 , are polynomials defined by F2n+1 (z) = ℓ=0 f2ℓ+1 z 2(n−ℓ)+1 , H2n+1 (z) = ℓ=0 h2ℓ+1 ∑ n z 2(n−ℓ)+1 , G2n+2 (z) = ℓ=0 g2ℓ z 2(n−ℓ)+2 + 21 g2n+2 . (

U (z) =

Theorem 2.1 (See [13,14]). Assume (q, r) is a compatible solution of the 1th and the pth homogeneous DNLS equations ˆ 1 (q, r) = 0, DNLS ˆ p (q, r) = 0, p ≥ 2, DNLS (2.8) Then u(x, y, tp ) = −2p−1 q(−2x, y, (−2)p−1 tp )r(−2x, y, (−2)p−1 tp ) gives a solution of the pth mKP equation.

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To construct quasiperiodic solutions of the pth equation in the mKP hierarchy, we have to solve the following auxiliary linear problem ψx (z) = U (z)ψ(z), ψy (z) = Vˆ4 (z)ψ(z), ψtp (z) = Vˆ2p+2 (z)ψ(z), p ≥ 2,

(2.9)

where ψ(z) = (ψ1 (z, x, y, tp ), ψ2 (z, x, y, tp ))T and Vˆ2k+2 (z), k = 2, p, are matrices defined by setting all the integration constants in V2k+2 (z) to be zero. Let ψ ± (z) = (ψ1± (z, x, y, tp ), ψ2± (z, ( x, y, tp )) be two) fundamental solutions of linear system (2.9) and define F = ψ1+ ψ1− , H = ψ2+ ψ2− , G = 12 ψ1+ ψ2− + +ψ1− ψ2+ . From (2.6), (2.7) and (2.9), it is not difficult to verify ˆ2p+1 F , ˆ3 F , Gtp = Fˆ2p+1 H − H Gx = rzF + qzH , Gy = Fˆ3 H − H ( 1 ) 2 ˆ4 F , Ftp = 2Fˆ2p+1 G − 2G ˆ2p+2 F , Fx = 2qzG + −2z + qr F , Fy = 2Fˆ3 G − 2G 2 ( 1 ) ˆ2p+2 H − 2H ˆ2p+1 G . ˆ3 G , Htp = 2G ˆ 4 H − 2H Hx = 2rzG + 2z 2 − qr H , Hy = 2G 2

(2.10)

Following a standard method in literature, we can prove that for each fixed p, solutions of linear system (2.10) have the following form. Theorem 2.2. Assume q, r ∈ C ∞ (R2+1 ). Moreover, if (q, r) solves (2.8), then E = span{(G , F , H )| (G2j+2 , F2j+1 , H2j+1 ), j ∈ N0 }

(2.11)

forms a vector space on C for solutions of the system (2.10). Next, to emphasize the differences between different solutions in (2.11), we denote (G2n+2 , F2n+1 , H2n+1 ) = (G2n+2 , F2n+1 , H2n+1 ), n ∈ N. Then using (2.10), we infer that G 2 − F H is independent of x, t and tp . Hence the hyperelliptic curve X associated with the pth mKP equation can be introduced as √ √ √ √ 2 y 2 − 14 R4n+4 ( η) = y 2 − 41 G2n+2 ( η) + 14 F2n+1 ( η)H2n+1 ( η) = 0, where R4n+4 (·) is a polynomial of degree 4n + 4 and η = z 2 . The curve X is compactified by joining two points P∞± at infinity and can be viewed as a double covering of the complex z-plane. The upper and lower sheets of X are denoted by X+ and X− respectively. √ √ In the following the zeros of F2n+1 ( η) and H2n+1 ( η) will play a special role. We denote them by {µj }nj=1 , {νj }nj=1 and hence write F2n+1 = 2q(x, y, tp )z

n ∏

(η − µj (x, y, tp )), H2n+1 = −2r(x, y, tp )z

j=1

n ∏

(η − νj (x, y, tp )).

j=1

3. The Baker–Akhiezer functions The Baker–Akhiezer functions play important roles in the finite gap integration of soliton equations and there are numerous articles that have been devoted to this subject [15–17]. In this section, we give an explicit construction of “Baker–Akhiezer function” that satisfies (2.9). To this end one has to use the following conservation relations, which can be derived by a straightforward computation. Lemma 3.1. Suppose q, r ∈ C ∞ (R2+1 ). Then we have the following relations (t1 = y) ( ) ( ) q(x, y, tp )z Fˆ2j+1 (z, x, y, tp ) = , F2n+1 (z, x, y, tp ) tj F2n+1 (z, x, y, tp ) x

(

) (H ˆ 2j+1 (z, x, y, tp ) ) r(x, y, tp )z =− , j = 1, p, H2n+1 (z, x, y, tp ) tj F2n+1 (z, x, y, tp ) x

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( Fˆ ) ( H (H ( Fˆ (z, x, y, t ) ) ˆ 3 (z, x, y, tp ) ) ˆ 2p+1 (z, x, y, tp ) ) 3 p 2p+1 (z, x, y, tp ) = , = . F2n+1 (z, x, y, tp ) tp F2n+1 (z, x, y, tp ) y F2n+1 (z, x, y, tp ) tp F2n+1 (z, x, y, tp ) y Based on Lemma 3.1, one may derive explicit forms of the Baker–Akhiezer functions ψj (P ), j = 1, 2. Theorem 3.2. Suppose q, r ∈ C ∞ (R2+1 ). Then the Baker–Akhiezer functions which satisfy (2.9) can be represented by √ √ ( ∫ x q(x′ , y, t )√ηy(P ) F2n+1 ( η, x, y, tp ) p ψ1 (P, x, y, tp ) = exp −2 dx′ √ √ ′ F2n+1 ( η, x0 , y0 , tp,0 ) x0 F2n+1 ( η, x , y, tp ) ∫ y ∫ tp √ √ y(P )Fˆ3 ( η, x0 , y ′ , tp ) ′ y(P )Fˆ2p+1 ( η, x0 , y0 , t′ ) ′ ) √ (3.1) −2 dy − 2 dt , √ ′ F2n+1 ( η, x0 , y0 , t′ ) y0 F2n+1 ( z, x0 , y , tp ) tp,0 √ √ ( ∫ x r(x′ , y, t )√ηy(P ) H2n+1 ( η, x, y, tp ) p ψ2 (P, x, y, tp ) = exp 2 dx′ √ √ ′ F2n+1 ( η, x0 , y0 , tp,0 ) x0 F2n+1 ( η, x , y, tp ) ∫ y ∫ tp ˆ 3 (√η, x0 , y ′ , tp ) ˆ 2p+1 (√η, x0 , y0 , t′ ) ) y(P )H y(P )H ′ −2 dy − 2 dt′ . (3.2) √ √ ′ F2n+1 ( η, x0 , y0 , t′ ) y0 F2n+1 ( η, x0 , y , tp ) tp,0 Proof . We only prove (3.1) since the proof for (3.2) is similar. Using (2.9) and (2.10), one obtains ) ( 1 ± ψ1,x (z) = − z 2 − qr ψ1± (z) + qzψ2± (z) 4 ) (( ) ψ ± (z) qzG2n+2 (z) 1 F2n+1,x (z) + + qz 2± = − ψ1± (z) F2n+1 (z) 2 F2n+1 (z) ψ1 (z) √ ∓qz R4n+4 (z) + 12 F2n+1,x (z) ± = ψ1 (z), F2n+1 (z) ± ˆ 2j+2 (z)ψ ± (z) + Fˆ2j+1 (z)ψ ± (z) ψ1,t (z) = − G 1 2 j (( ) ) Fˆ2j+1 (z)G2n+2 (z) 1 F2n+1,tj (z) ψ2± (z) ˆ = − + + F2j+1 (z) ± ψ1± (z) F2n+1 (z) 2 F2n+1 (z) ψ1 (z) √ ∓Fˆ2j+1 (z) R4n+4 (z) + 12 F2n+1,tj (z) ± = ψ1 (z), j = 2, p. F2n+1 (z)

(3.3)

(3.4)

Thus, we have ( d ln(ψ1± (z, x, y, tp ))

=

) R4n+4 (z) + 12 F2n+1,x (z) dx F2n+1 (z) ( ) √ ∓Fˆ3 (z) R4n+4 (z) + 21 F2n+1,y (z) + dy F2n+1 (z) ( ) √ ∓Fˆ2p+1 (z) R4n+4 (z) + 12 F2n+1,tp (z) + dtp . F2n+1 (z)

∓qz

√

According to the relations Lemma 3.1, the integrals

∫ (x,y,tp ) (x0 ,y0 ,tp,0

( ψj± (z,x,y,tp ) √ d ln )

F2n+1 (z,x,y,tp )

)

, j = 1, 2, are not

dependent on the integration path. Therefore taking into account the normalization condition ψ1 (P, x0 , y0 , tp,0 ) = 1, and choosing a special path (x0 , y0 , tp,0 ) → (x0 , y0 , tp ) → (x0 , y, tp ) → (x, y, tp ), we finally obtain (3.1). □

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In the following we introduce 2n points on the two sheets X± of X by √ ) ( µ ˆj (x, y, tp ) = µj (x, y, tp ), −2 R4n+4 (µj (x, y, tp )) ∈ X− , √ ( ) νˆj (x, y, tp ) = νj (x, y, tp ), 2 R4n+4 (µj (x, y, tp )) ∈ X+ , j = 1, . . . , n. and use the abbreviation µ ˆ = (ˆ µ1 , . . . , µ ˆn ), νˆ = (ˆ ν1 , . . . , νˆn ). Then it is not difficult to verify: (a) ψ1 (P )ψ2 (P ∗ ) is a meromorphic function on X with 2n + 1 zeros at νˆ∗ (x, y, tp ), µ ˆ(x, y, tp ), P∞+ and 2n + 1 poles at ∗ ∗ ˆ(x0 , y0 , t0 ), P∞− . (b) ψ1 (P )ψ2 (P ) has the asymptotic expansions (the local coordinate µ ˆ (x0 , y0 , t0 ), µ ζ = z −1 ) ⎧ ⎨ q(x ,y2 ,t ) ζ −1 + O(1), as P → P∞+ , ζ→0 0 0 p,0 ∗ ψ1 (P, x, y, tp )ψ2 (P , x, y, tp ) = ⎩− q(x,y,tp )r(x,y,tp ) ζ + O(ζ 2 ), as P → P∞− . 2q(x0 ,y0 ,tp,0 ) Further calculation shows ψj (P, x, y, tp ), j = 1, 2, have the following analytic properties: (I). ψ1 (P, x, y, tp ) and ψ2 (P, x, y, tp ) are meromorphic on P ∈ X\{P∞± }. Their divisor of poles coincides with Dµˆ(x0 ,y0 ,tp,0 ) and the divisor of zeros for ψ1 (P, x, y, tp ) and ψ2 (P, x, y, tp ) coincides with Dµˆ(x,y,tr+1 ) and Dνˆ(x,y,tp ) , respectively. (II). As P → P∞± , the asymptotic behavior of ψ is given by the equations, ⎞ ⎛ q(x,y,tp ) ( ) + O(ζ) ( q(x0 ,y0 ,tp,0 ) ψ1 (P, x, y, tp ) ⎠ × exp −(x − x0 )ζ −1 = ⎝ q(x,y,t )r(x,y,t ) ψ2 (P, x, y, tp ) − 2q(x0p,y0 ,tp,0 )p ζ + O(ζ 2 ) ) − (y − y0 )ζ −2 − (tp − tp,0 )ζ −(p+1) − δ(x, y, tp ) as P → P∞+ , (

ψ1 (P, x, y, tp ) ψ2 (P, x, y, tp )

(

) =

)

1 + O(ζ) 2 −1 q(x,y,tp ) ζ

+ O(1)

( × exp (x − x0 )ζ −1

) + (y − y0 )ζ −2 + (t − tp,0 )ζ −(p+1) + δ(x, y, tp ) as P → P∞− , with δ(x, y, tp ) =

1 4

∫x x0

q(x′ , y, tp )r(x′ , y, tp )dx′ −

1 2

∫y y0

g4 (x0 , y ′ , tp )dy ′ −

1 2

∫ tp tp,0

g2p+2 (x0 , y0 , t′ )dt′ .

4. Quasiperiodic solutions In this section, we derive the theta function representation of the Baker–Akhiezer functions ψ1 (P ), ψ2 (P ) and quasiperiodic solutions of the whole mKP hierarchy. For more details about Riemann surfaces and theta functions one may refer to [18]. Next, choosing a convenient base point Q0 ∈ X \ {P∞± }, one may define the Abel maps AQ0 (·), αQ0 (·), Riemann matrix Γ , and Riemann constants Ξ Q0 . Moreover, and canonical homology basis {aj , bj }nj=1 on X. For brevity, we also define z(P, Q) = Ξ Q0 − AQ0 (P ) + αQ0 (DQ ). The Riemann theta function associated (2)

with X and the homology basis {aj , bj }nj=1 is denoted by θ(z). Moreover, let ωP∞± ,q be the normalized differentials of the second kind with a unique pole at P∞± , whose principal parts (2)

ωP∞± ,q =

ζ→0

have vanishing a-periods:

∫

with unique poles at P∞±

(

) ζ −2−q + O(1) dζ, P → P∞± , ζ = η −1 , q ∈ N0

(4.1)

(2)

ωP∞± ,q = 0, j = 1, . . . , n. The normalized differential of third kind ωP∞+ P∞− ∫ ∫P satisfy a ωP∞+ P∞− = 0, j = 1, . . . , n, and Q ωP∞− P∞+ = ζ −1 + c∞+ + O(ζ),

aj

0

j

(2)

(2)

(2)

P → P∞+ with c∞+ ∈ C. Moreover, we define Ωr = 12 (r + 1)(ωP∞+ ,r − ωP∞− ,r ), r ∈ N0 . and ∫ P (2) (2) ωr = limP →P∞+ Q Ωr . Then theta function representations for ψ1 (P ), ψ2 (P ∗ ) are uniquely determined 0 by the properties (I), (II).

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Lemma 4.1. Assume q, r satisfy (2.8). Moreover, let P ∈ X\{P∞± } and suppose the divisors Dνˆ(x,y,tp ) , Dµˆ(x,y,tp ) are nonspecial. Then the function ψ1 , ψ2 have the following representations ˆ(x, y, tp ))) w(P,t,x,tp ) θ(z(P, µ e , θ(z(P, µ ˆ(x0 , y0 , tp,0 ))) θ(z(P, νˆ(x, y, tp ))) w(P,t,x,tp ) ψ2 (P, x, y, tp ) = C− (x, y, tp ) e , θ(z(P, µ ˆ(x0 , y0 , tp,0 ))) ψ1 (P, x, y, tp ) = C+ (x, y, tp )

where w(P, x, y, tp ) = −(x−x0 ) and C± (x, y, tp ) =

∫P Q0

(2)

Ω0 −(y−y0 )

∫P Q0

(2)

Ω1 −(tp −tp,0 )

∫P Q0

(2)

Ωp +δ2j

√∫

(4.2) (4.3) P Q0

ωP∞+ P∞− , j = 1, 2

( c ∫ x θ(z(P∞− , µ ˆ(x0 , y0 , tp,0 ))) ˆ(x′ , y, tp ))) θ(z(P∞± , µ 0 × exp ± θ(z(P∞± , µ ˆ(x, y, tp ))) 2 x0 θ(z(P∞+ , µ ˆ(x′ , y, tp ))) θ(z(P∞+ , νˆ(x′ , y, tp ))) ′ ) × dx , θ(z(P∞− , νˆ(x′ , y, tp )))

(4.4)

and the Abel map linearizes the auxiliary divisors Dµˆ (x,y,tp ) , Dνˆ(x,y,tp ) in the sense that αQ0 (Dµˆ(x,y,tp ) ) (2)

(2)

= αQ0 (Dµˆ(x0 ,y0 ,tp,0 ) ) + U 0 (x − x0 ) + U 1 (y − y0 ) + U (2) ˆ(x,y,tp ) ) = αQ0 (Dµ ˆ (x,y,tp ) ) + r (tp − tp,0 ) and αQ0 (Dν ( ) 1 2 AQ0 (P∞− ) − AQ0 (P∞+ ) . Expanding (4.2) at P∞± , and comparing with the first expansion in (II), we finally derive the following theta function representations for q, r and u. It is remarkable that the formula (4.5) is compatible with that derived in [10]. Theorem 4.2. Assume the spectral curves are nonsingular and q, r satisfy (2.8). Moreover, we suppose the divisors Dνˆ(x,y,tp ) , Dµˆ(x,y,tp ) are nonspecial. Then q, r have the following theta function representations θ(z(P∞+ , µ ˆ(x0 , y0 , tp,0 ))) θ(z(P∞− , µ ˆ(x, y, tp ))) θ(z(P∞+ , µ ˆ(x, y, tp ))) θ(z(P∞− , µ ˆ(x0 , y0 , tp,0 ))) ( ∫ x θ(z(P∞− , µ ˆ(x′ , y, tp ))) θ(z(P∞+ , νˆ(x′ , y, tp ))) ′ ) × exp c0 dx , ˆ(x′ , y, tp ))) θ(z(P∞− , νˆ(x′ , y, tp ))) x0 θ(z(P∞+ , µ θ(z(P∞− , µ ˆ(x0 , y0 , tp,0 ))) θ(z(P∞+ , νˆ(x, y, tp ))) r(x, y, tp ) = − q(x0 , y0 , tp,0 ) θ(z(P∞− , νˆ(x, y, tp ))) θ(z(P∞+ , µ ˆ(x0 , y0 , tp,0 ))) ∫ x ( θ(z(P∞− , µ ˆ(x′ , y, tp ))) θ(z(P∞+ , νˆ(x′ , y, tp ))) ′ ) × exp −c0 dx , ˆ(x′ , y, tp ))) θ(z(P∞− , νˆ(x′ , y, tp ))) x0 θ(z(P∞+ , µ

q(x, y, tp ) = q(x0 , y0 , tp,0 )

and moreover, solutions of pth mKP equation (p ≥ 2) have the form of u(x, y, tp ) = 2p c0

θ(z(P∞− , µ ˆ(2x, −y, −(−2)p−1 tp ))) θ(z(P∞+ , νˆ(2x, −y, (−2)p−1 tp ))) , θ(z(P∞+ , µ ˆ(2x, −y, −(−2)p tp ))) θ(z(P∞− , νˆ(2x, −y, −(−2)p tp )))

(4.5)

with a constant c0 ∈ C. Remark 4.3. In the case p = 2, solutions of the 2th mKP equation can be related to those of the Jimbo– Miwa equation (the second equation in the KP hierarchy) via the well-known Miura relation (see [8] and references therein). Hence, the formula (4.5) is useful to construct explicit solutions of the Jimbo–Miwa equation such as rogue wave solutions. However, this issue is beyond the scope of this paper and one may discuss this problem elsewhere.

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