Journal of Nonlinear Mathematical Physics

Volume 28, Issue 1, March 2021, Pages 134 - 149

The Complex Hamiltonian Systems and Quasi-periodic Solutions in the Hirota Equation

Authors
Jinbing Chen*, Rong Tong
School of Mathematics, Southeast University, Nanjing, Jiangsu 210096, P. R. China
*Corresponding author. Email: cjb@seu.edu.cn
Corresponding Author
Jinbing Chen
Received 7 May 2020, Accepted 21 August 2020, Available Online 10 December 2020.
DOI
https://doi.org/10.2991/jnmp.k.200922.010How to use a DOI?
Keywords
Hirota equation, complex finite-dimensional Hamiltonian system, quasi-periodic solution
Abstract

The Hirota equation is reduced to a pair of complex Finite-dimensional Hamiltonian Systems (FDHSs) with real-valued Hamiltonians, which are proven to be completely integrable in the Liouville sense. It turns out that involutive solutions of the complex FDHSs yield finite parametric solutions of the Hirota equation. From a Lax matrix of the complex FDHSs, the Hirota flow is linearized to display its evolution behavior on the Jacobi variety of a Riemann surface. With the technique of Riemann–Jacobi inversion, the quasi-periodic solution of the Hirota equation is presented in the form of Riemann theta functions.

Copyright
© 2020 The Authors. Published by Atlantis Press B.V.
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. INTRODUCTION

The Hirota equation was introduced in 1973 as a generalization of the Nonlinear Schrödinger (NLS) equation and the Modified Korteweg–de Vries (mKdV) equation [23]

ivt+δv|v|2+αvxx+3iγvx|v|2+iβvxxx=0, (1.1)
where v is a scalar function depending on x and t ((x,t)𝕉2) , i is the imaginary unit, and α, β, δ, and γ are real positive constants satisfying αγ = βδ. Let δ be 2α and γ be 2β. The Hirota equation (1.1) can be rewritten as
ivt+α(2v|v|2+vxx)+iβ(6vx|v|2+vxxx)=0. (1.2)

Obviously, as α = 0, β = 1, and v being real, the Hirota equation becomes the focusing mKdV equation; whereas α = 1, β = 0, and v being complex it is reduced to the focusing NLS equation.

The NLS equation is a universal model with various physical applications ranging from nonlinear optics and hydrodynamics to Bose–Einstein condensates due to a simple balance between nonlinear and dispersive effects. Thanks to the significant complexity of ocean waves, the third-order dispersion vxxx and a time-delay correction to the cubic term vx|v|2 are added to the NLS equation for a more precise description [35], similar to those high-order equations related to water waves considered by Osborne [33]. Under the Hasimoto map, it has been shown the relevance of the Hirota equation (1.2) in the modelling of the vortex string motion for a three dimensional Euler incompressible fluid [16,25]. As for the wave propagation of picosecond pulses in optical fibers [29], one needs to bring in the high-order dispersion and some other nonlinear effects for the simulation. Therefore, such an integrable extension of the NLS equation is relevant to the physical contexts in the high-intensity and short pulse picosecond regime [20,28].

The Hirota equation is of also mathematical interests, since it can be identified as an integrable PT-symmetric extension of the NLS equation [7]. The N envelope-soliton solution has been derived by the Hirota’s bilinear method [23]. A more general soliton solution formula was obtained through the inverse scattering transformation, which includes the N-soliton solution, the breather solution, and a class of multipole soliton solutions [14]. With the nonlinear steepest descent method, the long-time asymptotic was analysed for the Hirota equation [24], as well as that of initial and boundary value problems on the half line [22]. Remarkably, by modifying the Darboux transformation method, it is found that the second-order rational solution of the Hirota equation (1.2) can be used to describe high-order rogue waves under random initial conditions with a given small amplitude of chaotic perturbations [2].

From the isospectral nature of Lax representations [26], a linear spectral problem usually results in a hierarchy of soliton equations, including both the positive and negative directions in view of bidirectional Lenard gradients [8]. It has been confirmed that the integrable couplings of arbitrary two commutable flows lying in the same soliton hierarchy are integrable in the sense of Lax compatibility [38]. Seen from the profile of equation (1.2), the Hirota equation can be regarded as an integrable coupling of NLS and mKdV flows in reference to the Ablowitz–Kaup–Newell–Segur (AKNS) spectral problem [1]. However, this kind of combination does not automatically give us explicit solutions to the integrable equation.

It is necessary to know not only soliton solutions, but also quasi-periodic (finite-gap, or algebro-geometric) solutions of integrable Nonlinear Evolution Equations (NLEEs) in a number of physical problems. The quasi-periodic solutions to the NLS and mKdV equations have been obtained using either by the algebro-geometric method or by the combination of commutation methods and Hirota’s τ-function approach in Belokolos et al. [4], Gesztesy [19] and some others, but the quasi-periodic solutions are still missing for the Hirota equation. Using the nonlinearization of Lax pair [5], the rogue periodic waves to the NLS and mKdV equations have been presented in Chen and Pelinovsky [9], Chen and Pelinovsky [10], Chen et al. [11], and the rogue waves on the periodic background have been given to the Hirota equation in Gao and Zhang [17], Peng et al. [34]. In the present work, the complex Finite-dimensional Hamiltonian Systems (FDHSs) with real-valued Hamiltonians are generalized to deduce some quasi-periodic solutions of the Hirota equation in view of finite-dimensional integrable reductions.

The real FDHSs have been used to derive soliton solutions, quasi-periodic solutions, and rogue periodic waves for NLEEs [6,811,13,17,18,34]. A natural issue is whether the complex FDHSs can be adapted to deduce solutions for complex NLEEs. To obtain solutions of integrable NLEEs, no matter N-solitons or quasi-periodic solutions, one key step is to specify a finite-dimensional invariant subspace associated with the phase flows [27,32]. It was known that the solution space of Novikov equation is a finite-dimensional invariant set of infinite-dimensional integrable systems. Recently, it is found that integration constants appearing in the Novikov equation are determined by eigenvalues and conserved quantities of FDHSs, from which the branch points of spectral bands are figured out in view of the symmetric constraint [5]. As a result, some interesting exact solution, such as the algebraically decaying solitons and the rogue periodic waves, are obtained by means of the Darboux transformation [911,17,34]. In this study, we reduce the Hirota equation to two complex FDHSs and construct its quasi-periodic solution.

The purpose of this work is to develop an alternative algorithm for getting quasi-periodic solutions of the Hirota equation by virtue of complex FDHSs. Subject to the finite-dimensional integrable reduction, the Hirota equation is decomposed into a pair of complex FDHSs with real-valued Hamiltonians by separating temporal and spatial variables. The relation between the Hirota equation and the complex FDHSs is established in view of the commutability of complex Hamiltonian flows, which simplifies the process of getting explicit solutions. Also, the finite-gap potential to the complex Novikov (high-order stationary) equation is presented, which cuts out a finite-dimensional invariant subspace for the Hirota flow via the symmetric constraint. Followed by a set of elliptic variables of complex FDHSs, a systematic way is given to elaborate Abel–Jacobi variables that straighten out the complex Hamiltonian and Hirota flows on the Jacobi variety of a Riemann surface. By using the technique of Riemann–Jacobi inversion [21,30], the Abel–Jacobi solution of the Hirota flow is transformed to the potential represented by Riemann theta functions. Although our computations are reported in the context of Hirota equation, the constructing scheme can also be applied to some other complex integrable NLEEs [12].

This paper is organized as follows. Section 2 is to decompose the Hirota equation into two complex FDHSs. The connection between the Hirota equation and the complex FDHSs is established in Section 3. Section 4 exhibits the evolution behavior of various flows on the Jacobi variety of a Riemann surface. Finally, in Section 5 the algebraic geometrical datum are processed to deduce quasi-periodic solutions for the Hirota equation.

2. REDUCTION TO THE HIROTA EQUATION

To reduce the Hirota equation, we first reformulate it into the Lenard scheme. Let us begin with the AKNS spectral problem [1]

φx=Uφ,U=iλσ1+v¯σ2vσ3,φ=(φ1,φ2)T, (2.1)
where v¯ is the complex conjugate of v, and
σ1=(1001),σ2=(0100),σ3=(0010).

Solve the stationary zero-curvature equation of the AKNS spectral problem (2.1)

Vx=[U,V],V=aσ1+bσ2+cσ3=j0(ajσ1+bjσ2+cjσ3)λj, (2.2)
which coincides with
ajx=vbj+v¯cj,bjx=2ibj+12v¯aj,cjx=2icj+12vaj,j0. (2.3)

Let a0 = 2i and b0 = c0 = 0 be the initial values. Up to constants of integration, aj, bj and cj can be uniquely determined by means of the recursive formula (2.3), for example

a1=0,b1=2v¯,c1=2v,a2=i|v|2,b2=iv¯x,c2=ivx,a3=12(v¯vxv¯xv),b3=12(v¯xx+2v¯|v|2),c3=12(vxx+2v|v|2),a4=i4(vv¯xx+v¯vxxvxv¯x+3|v|4),b4=i4(v¯xxx+6|v|2v¯x),c4=i4(vxxx+6|v|2vx). (2.4)

Based on the recurrence chain (2.3), we introduce the Lenard gradients {gj} and the Lenard operator pair K and J:

Kgj=Jgj+1,gj=(icj+1,ibj+1)T,j1, (2.5)
where
K=(2iv¯x1v¯i(x+2v¯x1v)i(x+2vx1v¯)2ivx1v),J=(0220), (2.6)
are two skew-symmetric operators, and x1 is to denote the inverse operator of x = /x under the condition xx1=x1x=1 . Recalling (2.4), it is clear to see that
g1=(00),g0=(2iv2iv¯),g1=(vxv¯x),g2=(i2vxx+iv|v|2i2v¯xx+iv¯|v|2),g3=(14(vxxx+6|v|2vx)14(v¯xxx+6|v|2v¯x)). (2.7)

It is assumed that φ satisfies a spectral problem determined by the Lenard gradients {gj}

φtn=V(n)φ,V(n)=x1(vg(2)v¯g(1))σ1+g(2)σ2g(1)σ3,n0, (2.8)
where
g=(g(1),g(2))T=ij=0ngj1λnj.

The zero-curvature equation of spectral problems (2.1) and (2.8), i.e. UtnVx(n)+[U,V(n)]=0, gives the focusing NLS hierarchy

(v¯tn,vtn)T=JgnXn,n0, (2.9)
together with a fundamental identity
Vx(n)[U,V(n)]=U*[i(KλJ)g], (2.10)
where
U*[Ξ]=ddɛ|ɛ=0U(v¯+ɛΞ1,v+ɛΞ2),Ξ=(Ξ1,Ξ2)T.

It is found that the Hirota equation (1.2) is the compatibility condition of Lax pair (2.1) and

φt=V(2,3)φ,V(2,3)=αV(2)+2βV(3), (2.11)
where
V(2)=(2iλ2i|v|2)σ1+(2λv¯iv¯x)σ2(2λv+ivx)σ3,V(3)=[2iλ3iλ|v|2+12(vxv¯vv¯x)]σ1+[2λ2v¯iλv¯x12(v¯xx+2|v|2v¯)]σ2+[2λ2viλvx+12(vxx+2|v|2v)]σ3. (2.12)

Let λ1, λ2, ⋯, λN be N arbitrary distinct nonzero complex eigenvalues, namely, (λiλ¯j,1i,jN) , (ψ1j, ψ2j)T be the vector eigenfunction pertinent to λj. Due to the symmetry of (2.1), (ψ¯2j,ψ¯1j)T corresponds to the eigenvalue λ¯j . Only for the convenience, we make the conventions Λ = diag(λ1, λ2, ⋯, λN), ψ1 = (ψ11, ψ12, ⋯, ψ1N)T, and ψ2 = (ψ21, ψ22, ⋯, ψ2N)T. The diamond bracket 〈. , .〉 stands for the vector product: ξ,η=j=1Nξjηj , where ξ = (ξ1, ξ2, ⋯, ξN)T and η = (η1, η2, ⋯, ηN)T. According to the nonlinearization of Lax pair [5], we consider N copies of spectral problem (2.1)

{ψ1jx=iλjψ1j+v¯ψ2j,ψ2jx=vψ1jiλjψ2j,ψ¯1jx=iλ¯jψ¯1j+vψ¯2j,ψ¯2jx=v¯ψ¯1j+iλ¯jψ¯2j. (2.13)

It follows from [37,39] that the functional gradients of λj and λ¯j with respect to v¯ and v are

λj=(δλjδv¯δλjδv)=(2iψ2j22iψ1j2),λ¯j=(δλ¯jδv¯δλ¯jδv)=(2iψ¯1j22iψ¯2j2), (2.14)
which satisfy the Lenard eigenvalue equations
(KλjJ)λj=0,(Kλ¯jJ)λ¯j=0. (2.15)

Recall the Bargmann (symmetric) constraint

g0=j=1N(λj+λ¯j), (2.16)
which gives a Bargmann map to connect the potential v with the eigenfunctions (ψ1, ψ2)
v=ψ¯1,ψ¯1+ψ2,ψ2. (2.17)

On 𝔺2N , we define the symplectic structure [3]

ω2=j=1N(dψ1jdψ2j+dψ¯1jdψ¯2j), (2.18)
and the Poisson bracket
{f,g}=j=1N(fψ2jgψ1j+fψ¯2jgψ¯1jfψ1jgψ2jfψ¯1jgψ¯2j). (2.19)

Substituting (2.17) back into (2.1) and (2.11), we arrive at two complex FDHSs with real-valued Hamiltonians

ψ1x={ψ1,H1},ψ2x={ψ2,H1},ψ¯1x={ψ¯1,H1},ψ¯2x={ψ¯2,H1}, (2.20)
where
H1=iΛψ1,ψ2+iΛ¯ψ¯1,ψ¯212|ψ1,ψ1+ψ¯2,ψ¯2|2, (2.21)
and
ψ1t={ψ1,H(2,3)},ψ2t={ψ2,H(2,3)},ψ¯1t={ψ¯1,H(2,3)},ψ¯2t={ψ¯2,H(2,3)}, (2.22)
where H(2,3) = α H2 + 2β H3 together with
H2=2i(Λ2ψ1,ψ2Λ¯2ψ¯1,ψ¯2)+i|ψ1,ψ1+ψ¯2,ψ¯2|2(ψ1,ψ2ψ¯1,ψ¯2)(ψ1,ψ1+ψ¯2,ψ¯2)(Λψ2,ψ2+Λ¯ψ¯1,ψ¯1)(ψ2,ψ2+ψ¯1,ψ¯1)(Λψ1,ψ1+Λ¯ψ¯2,ψ¯2). (2.23)
and
H3=2i(Λ3ψ1,ψ2Λ¯3ψ¯1,ψ¯2)+i|ψ1,ψ1+ψ¯2,ψ¯2|2(Λψ1,ψ2Λ¯ψ¯1,ψ¯2)+i(ψ1,ψ2ψ¯1,ψ¯2)×[(ψ1,ψ1+ψ¯2,ψ¯2)(Λψ2,ψ2+Λ¯ψ¯1,ψ¯1)+(ψ2,ψ2+ψ¯1,ψ¯1)(Λψ1,ψ1+Λ¯ψ¯2,ψ¯2)]+|ψ1,ψ1+ψ¯2,ψ¯2|2(ψ1,ψ2ψ¯1,ψ¯2)2(ψ1,ψ1+ψ¯2,ψ¯2)(Λ2ψ2,ψ2+Λ¯2ψ¯1,ψ¯1)(Λ2ψ1,ψ1+Λ¯2ψ¯2,ψ¯2)(ψ2,ψ2+ψ¯1,ψ¯1)|Λψ1,ψ1+Λ¯ψ¯2,ψ¯2|2+14|ψ1,ψ1+ψ¯2,ψ¯2|4. (2.24)

It is noted that the Hirota equation (1.2) can be represented as the compatibility condition of spectral problems (2.1) and (2.11). The Hirota equation (1.2) is indeed reduced to two complex FDHSs separating its temporal and spatial variables over (𝔺2N,ω2) .

3. RELATION BETWEEN THE HIROTA EQUATION AND THE COMPLEX FDHSS

In order to establish the relation between the Hirota equation (1.2) and the complex FDHSs (2.20) and (2.22), it is necessary for us to prove the Liouville integrability of the complex FDHSs. The Liouville’s definition of integrability is based on the notion of integrals of motion [3]. We need to construct a sufficient number of involutive integrals of motion for the complex FDHSs (2.20) and (2.22). Firstly, let us bring in a bilinear generating function

Gλ=i2j=1N(λjλλj+λ¯jλλ¯j)=(Qλ(ψ2,ψ2)+Qλ¯(ψ¯1,ψ¯1)Qλ(ψ1,ψ1)+Qλ¯(ψ¯2,ψ¯2)), (3.1)
where
Qλ(ξ,η)=j=1Nξjηjλλj,Qλ¯(ξ,η)=j=1Nξjηjλλ¯j.

It follows from (2.15) that

(KλJ)Gλ=0. (3.2)

Substituting Gλ back into the expression of V(n) gives rise to a Lax matrix

Vλ=(iQλ(ψ1,ψ2)+Qλ¯(ψ¯1,ψ¯2)Qλ(ψ1,ψ1)+Qλ¯(ψ¯2,ψ¯2)Qλ(ψ2,ψ2)Qλ¯(ψ¯1,ψ¯1)i+Qλ(ψ1,ψ2)Qλ¯(ψ¯1,ψ¯2)), (3.3)
which satisfies the Lax equation
(Vλ)x[U,Vλ]=0, (3.4)
in view of (2.10) and (3.2). It follows from (3.4) that detVλ is a generating function of integrals of motion for the complex FDHSs (2.20) [36]. With |λ| > max{|λ1|, |λ2|, ⋯, |λN|}, we come to
Fλ=detVλ=1+2iQλ(ψ1,ψ2)+Qλ(ψ1,ψ1)Qλ(ψ2,ψ2)Qλ2(ψ1,ψ2)2iQλ¯(ψ¯1,ψ¯2)+Qλ¯(ψ¯1,ψ¯1)Qλ¯(ψ¯2,ψ¯2)Qλ¯2(ψ¯1,ψ¯2)+Qλ(ψ1,ψ1)Qλ¯(ψ¯1,ψ¯1)+2Qλ(ψ1,ψ2)Qλ¯(ψ¯1,ψ¯2)+Qλ(ψ2,ψ2)Qλ¯(ψ¯2,ψ¯2)=1+j=1NEjλλj+j=1NE¯jλλ¯j=1+k=0Fkλk1, (3.5)
where
Ej=2iψ1jψ2j+k=1,kjN(ψ1jψ2kψ1kψ2j)2λjλk+k=1N(ψ1jψ¯1k+ψ2jψ¯2k)2λjλ¯k, (3.6)
F0=2i(ψ1,ψ2ψ¯1,ψ¯2), (3.7)
F1=2i(Λψ1,ψ2Λ¯ψ¯1,ψ¯2)+ψ1,ψ1ψ2,ψ2ψ1,ψ22+ψ¯1,ψ¯1ψ¯2,ψ¯2ψ¯1,ψ¯22+ψ1,ψ1ψ¯1,ψ¯1+2ψ1,ψ2ψ¯1,ψ¯2+ψ2,ψ2ψ¯2,ψ¯2, (3.8)
F2=2i(Λ2ψ1,ψ2Λ¯2ψ¯1,ψ¯2)+(Λψ1,ψ1+Λ¯ψ¯2,ψ¯2)(ψ2,ψ2+ψ¯1,ψ¯1)+(Λψ2,ψ2+Λ¯ψ¯1,ψ¯1)(ψ1,ψ1+ψ¯2,ψ¯2)2(Λψ1,ψ2Λ¯ψ¯1,ψ¯2)(ψ1,ψ2ψ¯1,ψ¯2), (3.9)
Fk=2i(Λkψ1,ψ2Λ¯kψ¯1,ψ¯2)+j=0k1|Λjψ1,ψ1+Λ¯jψ¯2,ψ¯2Λkj1ψ1,ψ2Λ¯kj1ψ¯1,ψ¯2Λjψ1,ψ2Λ¯jψ¯1,ψ¯2Λkj1ψ2,ψ2+Λ¯kj1ψ¯1,ψ¯1|,k3. (3.10)

Let Fλ be a real-valued Hamiltonian on (𝔺2N,ω2) , and τλ be the flow variable of Fλ. From the Poisson bracket, a direct calculation results in two canonical Hamiltonian equations

ddτλ(ψ1jψ2j)=({ψ1j,Fλ}{ψ2j,Fλ})=W(λ,λj)(ψ1jψ2j), (3.11)
ddτλ(ψ¯2jψ¯1j)=({ψ¯2j,Fλ}{ψ¯1j,Fλ})=W(λ,λ¯j)(ψ¯2jψ¯1j), (3.12)
where
W(λ,μ)=2λμVλ. (3.13)

Lemma 3.1.

On (𝔺2N,ω2) , the Lax matrix Vμ satisfies a Lax equation

dVμdτλ=[W(λ,μ),Vμ],λ,μ𝔺,λμ. (3.14)

Besides,

{Fμ,Fλ}=0,λ,μ𝔺,λμ, (3.15)
{Fj,Fk}=0,j,k=0,1,2,. (3.16)

Proof. Only for simplifying the description, we denote

ɛ1j=ψ1jψ2jσ1+ψ1j2σ2ψ2j2σ3,ɛ2j=ψ¯1jψ¯2jσ1+ψ¯2j2σ2ψ¯1j2σ3.

It follows from (3.11) and (3.12) that

dɛ1jdτλ=[W(λ,λj),ɛ1j],dɛ2jdτλ=[W(λ,λ¯j),ɛ2j]. (3.17)

Resorting to (3.3), (3.13) and (3.17), a direct calculation yields

dVμdτλ=j=1N1μλjdɛ1jdτλ+j=1N1μλ¯jdɛ2jdτλ=j=1N1μλj[W(λ,λj),ɛ1j]+j=1N1μλ¯j[W(λ,λ¯j),ɛ2j]=2λμj=1N((1μλj1λλj)[Vλ,ɛ1j]+(1μλ¯j1λλ¯j)[Vλ,ɛ2j])=2λμ[Vλ,j=1N(ɛ1jμλj+ɛ2jμλ¯jɛ1jλλjɛ2jλλ¯j)]=2λμ[Vλ,VμVλ]=[W(λ,μ),Vμ].

Furthermore, from (3.14) we arrive at

{Fμ,Fλ}=dFμdτλ=ddτλ(12trVμ2)=trVμtr[W(λ,μ),Vμ]=0. (3.18)

Substituting (3.5) into (3.18) leads to the identity (3.16), which completes the proof.

Apart from the involutivity of integrals of motion, the other essential element to the Liouville integrability of FDHSs is the functional independence, which means that solutions of the FDHSs can be obtained by solving a finite number of algebraic equations and computing a finite number of integrals. Below, we turn to the functional independence of Fk (0 ≤ k ≤ 2N −1).

Lemma 3.2.

The integrals of motion {F0, F1, ⋯, F2N−1} given by (3.7)(3.10) are functionally independent in a dense open subset of (𝔺2N,ω2) .

Proof. It is known from (3.5) that

Fk=j=1N(λjkEj+λ¯jkE¯j),0k2N1. (3.19)

Let P0=(ψ11,,ψ1N,ψ¯21,,ψ¯2N;ψ21,,ψ2N,ψ¯11,,ψ¯1N)T be a fixed point in 𝔺2N with ψ1j = 0, ψ2j ≠ 0, (1 ≤ jN). And then,

Ejψ1k|P0=2iδjkψ2j,E¯jψ¯1k|P0=2iδjkψ¯2j,1j,kN. (3.20)

By (3.20), we arrive at the Jacobi determinant of {Ej,E¯j} associated with {ψ1j,ψ¯1j} at P0

(E1,,EN,E¯1,,E¯N)(ψ11,,ψ1N,ψ¯11,,ψ¯1N)|P0=22Nj=1N|ψ2j|2, (3.21)
which signifies the linear independence of {dE1,,dEN,dE¯1,,dE¯N} over a dense open subset of 𝔺2N [3]. It is supposed that there are 2N constants γ0, γ1, ⋯, γ2N−1 such that
k=02N1γkdFk=0, (3.22)
which is in agreement with
k=02N1γkλjk=0,k=02N1γkλ¯jk=0,1jN, (3.23)
in view of (3.19) and the linear independence of {dEj,dE¯j} . It is noted that the determinant of coefficients of γk is the Vandermonde determinant. Namely, γ0 = ⋯ = γ2N−1 = 0, which means that {Fk}(0 ≤ k ≤ 2N − 1) are functionally independent in a dense open subset of (𝔺2N,ω2) .

On one hand, it is seen from (2.21), (2.23), (2.24), and (3.7)(3.10) that

H1=12F1+18F02,H2=F2+12F1F018F03,H3=F3+12F0F2+14F1218F02F1+164F04, (3.24)
on the other hand, by (3.16), (3.24) and the Leibniz rule of Poisson bracket we obtain
dFλdt={Fλ,H(2,3)}={Fλ,αH2+2βH3}=0, (3.25)
which indicates that {Fk} are also integrals of motion for the complex FDHSs (2.22). We attain the Liouville integrability to the complex FDHSs (2.20) and (2.22).

Proposition 3.1.

The complex FDHSs (H1,ω2,𝔺2N) and (H(2,3),ω2,𝔺2N) are completely integrable in the Liouville sense.

Based on Proposition 3.1, it is known that two complex FDHSs (2.20) and (2.22) reduced from the Hirota equation (1.2) are compatible over (𝔺2N,ω2) . This means that there exists a smooth function in x and t giving an involutive solution for complex FDHSs (H1,ω2,𝔺2N) and (H(2,3),ω2,𝔺2N) . To progress further, from the commutability of Hamiltonian flows, we are in a position to establish the relation between the Hirota equation and the complex FDHSs, and then to confirm the existence of a finite number of spectral bands for the eigenvalue problem (2.1).

Proposition 3.2.

Let (ψ1(x, t), ψ2(x, t))T be an involutive solution of integrable complex FDHSs (2.20) and (2.22). Then

v(x,t)=ψ¯1(x,t),ψ¯1(x,t)+ψ2(x,t),ψ2(x,t), (3.26)
is a finite parametric solution of the Hirota equation (1.2).

Proof. Resorting to the complex FDHSs (2.20) and (2.22), we compute

vx=2i(Λψ2,ψ2+Λ¯ψ¯1,ψ¯1)2(ψ¯1,ψ¯1+ψ2,ψ2)(ψ1,ψ2ψ¯1,ψ¯2), (3.27)
vxx=4i(Λψ1,ψ2Λ¯ψ¯1,ψ¯2)(ψ¯1,ψ¯1+ψ2,ψ2)4(Λ2ψ2,ψ2+Λ¯2ψ¯1,ψ¯1)+4(i(Λψ2,ψ2+Λ¯ψ¯1,ψ¯1)+(ψ¯1,ψ¯1+ψ2,ψ2)(ψ1,ψ2ψ¯1,ψ¯2))(ψ1,ψ2ψ¯1,ψ¯2), (3.28)
vxxx=8i(Λ3ψ2,ψ2+Λ¯3ψ¯1,ψ¯1)8(ψ2,ψ2+ψ¯1,ψ¯1)(ψ1,ψ2ψ¯1,ψ¯2)38i(Λψ2,ψ2+Λ¯ψ¯1,ψ¯1)(ψ1,ψ2ψ¯1,ψ¯2)2+8(ψ2,ψ2+ψ¯1,ψ¯1)(Λ2ψ1,ψ2Λ¯2ψ¯1,ψ¯2)+4i|ψ1,ψ1+ψ¯2,ψ¯2|2(Λψ2,ψ2+Λ¯ψ¯1,ψ¯1)+8(Λψ2,ψ2+Λ¯ψ¯1,ψ¯1)(Λψ1,ψ2Λ¯ψ¯1,ψ¯2)16i(ψ1,ψ2ψ¯1,ψ¯2)(ψ2,ψ2+ψ¯1,ψ¯1)(Λψ1,ψ2Λ¯ψ¯1,ψ¯2)4i(ψ2,ψ2+ψ¯1,ψ¯1)2(Λψ1,ψ1+Λ¯ψ¯2,ψ¯2)+8(ψ1,ψ2ψ¯1,ψ¯2)(Λ2ψ2,ψ2+Λ¯2ψ¯1,ψ¯1), (3.29)
vt=α[4i(Λ2ψ2,ψ2+Λ¯2ψ¯1,ψ¯1)+2i(ψ1,ψ1+ψ¯2,ψ¯2)(ψ¯1,ψ¯1+ψ2,ψ2)24(Λψ2,ψ2+Λ¯ψ¯1,ψ¯1)(ψ¯1,ψ¯2ψ¯1,ψ¯2)+4i(ψ¯1,ψ¯1+ψ2,ψ2)(ψ1,ψ2ψ¯1,ψ¯2)24(ψ¯1,ψ¯1+ψ2,ψ2)(Λψ1,ψ2Λ¯ψ¯1,ψ¯2)]+β[8(ψ¯1,ψ¯1+ψ2,ψ2)(Λ2ψ1,ψ2Λ¯2ψ¯1,ψ¯2)8(Λψ1,ψ2Λ¯ψ¯1,ψ¯2)(Λψ2,ψ2+Λ¯ψ¯1,ψ¯1)+16i(ψ¯1,ψ¯1+ψ2,ψ2)(ψ1,ψ2ψ¯1,ψ¯2)(Λψ1,ψ2Λ¯ψ¯1,ψ¯2)8(ψ1,ψ2ψ¯1,ψ¯2)(Λ2ψ2,ψ2+Λ¯2ψ¯1,ψ¯1)+8i(Λψ2,ψ2+Λ¯ψ¯1,ψ¯1)(ψ1,ψ2ψ¯1,ψ¯2)2+8(ψ¯1,ψ¯1+ψ2,ψ2)(ψ1,ψ2ψ¯1,ψ¯2)38i(Λ3ψ2,ψ2+Λ¯3ψ¯1,ψ¯1)+12(ψ1,ψ1+ψ¯2,ψ¯2)(ψ1,ψ2ψ¯1,ψ¯2)(ψ¯1,ψ¯1+ψ2,ψ2)2+8i|ψ1,ψ1+ψ¯2,ψ¯2|2(Λψ2,ψ2+Λ¯ψ¯1,ψ¯1)+4i(ψ¯1,ψ¯1+ψ2,ψ2)2(Λψ1,ψ1+Λ¯ψ¯2,ψ¯2)]. (3.30)

Substituting (3.27)(3.30) back into the Hirota equation (1.2), it is shown that the expression (3.26) exactly solves the Hirota equation (1.2).

Remark 3.1.

As a concrete application of Proposition 3.2, the derivation of explicit solutions to the Hirota equation is transformed to the problem of solving two complex FDHSs.

Proposition 3.3.

Let (ψ1(x), ψ2(x))T be a solution of the complex FDHSs (2.20). Then

v=ψ¯1(x),ψ¯1(x)+ψ2(x),ψ2(x), (3.31)
is a finite-gap solution to the complex Novikov (high-order stationary NLS) equation
X2N+a˜1X2N1+c˜2X2N2++c˜2NX0=0,N2, (3.32)
where
c˜j=a˜j+k=0j2a˜kc^jk,j=2,3,,2N,a˜0=1,a˜1=j=1N(λj+λ¯j),a˜2=1i<jN(λiλj+λ¯iλ¯j)+i,j=1Nλiλ¯j,a˜j=(1)j1i1<<ijN(λi1λij+λ¯i1λ¯ij)+1sN11k1<<ksN1l1<<ljsNλk1λksλ¯l1λ¯ljs,3jN,a˜j=(1)jjNsN1k1<<ksN1l1<<ljsNλk1λksλ¯l1λ¯ljs,N+1j2N,
and c^2,c^3,,c^2N are some constants of integration.

Proof. One one hand, take into account an auxiliary polynomial in λ

a(λ)=j=1N(λλj)(λλ¯j)=a˜0λ2N+a˜1λ2N1++a˜2N. (3.33)

Applying the operator J−1K on the symmetric constraint (2.16) k times, we derive

j=1N(λjkλj+λ¯jkλ¯j)=gk+c^2gk2++c^kg0,k3, (3.34)
in view of the Lenard eigenvalue equations (2.15). On the other hand, it follows from (2.16), (3.33) and (3.34) that
0=j=1N[a(λj)λj+a(λ¯j)λ¯j]=j=1N[(λj2Nλj+λ¯j2Nλ¯j)+a˜1(λj2N1λj+λ¯j2N1λ¯j)++a˜2N(λj+λ¯j)]=(g2N+c^2g2N2++c^2Ng0)+a˜1(g2N1+c^2g2N3++c^2N1g0)++a˜2Ng0=g2N+a˜1g2N1+c˜2g2N2++c˜2Ng0, (3.35)
which immediately becomes the complex Novikov equation (3.32) after being acted with the Lenard operator J. This completes the proof.

4. STRAIGHTENING OUT OF HIROTA FLOW

It is shown that the Hirota equation (1.2) has been reduced to two complex FDHSs with real-valued Hamiltonians on (𝔺2N,ω2) . And further, the Bargmann map (2.17) results in a finite-gap potential to the complex Novikov (high-order stationary NLS) equation (3.32). In this section, the complex FDHSs (2.20) and (2.22) serve as a basis to display the evolution picture of Hirota flow on the Jacobi variety of a Riemann surface.

For the sake of succinctness in writing, let us make the notation

VλVλ11σ1+Vλ12σ2+Vλ21σ3.

From Lemma 3.1, we know that the Lax matrix Vμ satisfies a Lax equation along with τλ-flow. In particular, after a direct but tedious calculation, the Lax matrix Vλ also satisfies two Lax equations associated with the variables of x and t, respectively.

Lemma 4.1.

xVλ=[U˜,Vλ],U˜=iλσ1+(ψ1,ψ1+ψ¯2,ψ¯2)σ2(ψ¯1,ψ¯1+ψ2,ψ2)σ3, (4.1)
tVλ=[V˜(2,3),Vλ],V˜(2,3)=V˜11(2,3)σ1+V˜12(2,3)σ2+V˜21(2,3)σ3,t=/t, (4.2)
where
V˜11(2,3)=α[2iλ2i|ψ1,ψ1+ψ¯2,ψ¯2|2]+β[4iλ32iλ|ψ1,ψ1+ψ¯2,ψ¯2|22(ψ1,ψ1+ψ¯2,ψ¯2)((ψ¯1,ψ¯1+ψ2,ψ2)(ψ1,ψ2ψ¯1,ψ¯2)+i(Λψ2,ψ2+Λ¯ψ¯1,ψ¯1))2(ψ¯1,ψ¯1+ψ2,ψ2)×(i(Λψ1,ψ1+Λ¯ψ¯2,ψ¯2)+(ψ1,ψ1+ψ¯2,ψ¯2)(ψ1,ψ2ψ¯1,ψ¯2))],V˜12(2,3)=α[2λ(ψ1,ψ1+ψ¯2,ψ¯2)+2(Λψ1,ψ1+Λ¯ψ¯2,ψ¯2)2i(ψ1,ψ1+ψ¯2,ψ¯2)(ψ1,ψ2ψ¯1,ψ¯2)]+β[4λ2(ψ1,ψ1+ψ¯2,ψ¯2)+4λ((Λψ1,ψ1+Λ¯ψ¯2,ψ¯2)i(ψ1,ψ1+ψ¯2,ψ¯2)(ψ1,ψ2ψ¯1,ψ¯2))2(ψ1,ψ1+ψ¯2,ψ¯2)2(ψ¯1,ψ¯1+ψ2,ψ2)+4(Λ2ψ1,ψ1+Λ¯2ψ¯2,ψ¯2)4i(ψ1,ψ1+ψ¯2,ψ¯2)(Λψ1,ψ2Λ¯ψ¯1,ψ¯2)4i(Λψ1,ψ1+Λ¯ψ¯2,ψ¯2)(ψ1,ψ2ψ¯1,ψ¯2)4(ψ1,ψ1+ψ¯2,ψ¯2)(ψ1,ψ2ψ¯1,ψ¯2)2],V˜21(2,3)=α[2λ(ψ¯1,ψ¯1+ψ2,ψ2)2(Λψ2,ψ2+Λ¯ψ¯1,ψ¯1)+2i(ψ¯1,ψ¯1+ψ2,ψ2)(ψ1,ψ2ψ¯1,ψ¯2)]β[4λ2(ψ¯1,ψ¯1+ψ2,ψ2)+4λ((Λψ2,ψ2+Λ¯ψ¯1,ψ¯1)i(ψ¯1,ψ¯1+ψ2,ψ2)(ψ1,ψ2ψ¯1,ψ¯2))2(ψ1,ψ1+ψ¯2,ψ¯2)(ψ¯1,ψ¯1+ψ2,ψ2)2+4(Λ2ψ2,ψ2+Λ¯2ψ¯1,ψ¯1)4i(ψ¯1,ψ¯1+ψ2,ψ2)(Λψ1,ψ2Λ¯ψ¯1,ψ¯2)4i(Λψ2,ψ2+Λ¯ψ¯1,ψ¯1)(ψ1,ψ2ψ¯1,ψ¯2)4(ψ¯1,ψ¯1+ψ2,ψ2)(ψ1,ψ2ψ¯1,ψ¯2)2]. (4.3)

It follows from (3.3) and (3.5) that Fλ and Vλ21 are the rational polynomial functions of λ with simple poles at {λj,λ¯j}(j=1,2,,N) . As a result, we define

Fλ=Vλ12Vλ21(Vλ11)2=b(λ)a(λ)=a(λ)b(λ)a2(λ)=R(λ)a2(λ), (4.4)
Vλ21=Qλ(ψ2,ψ2)Qλ¯(ψ¯1,ψ¯1)=(ψ¯1,ψ¯1+ψ2,ψ2)n(λ)a(λ), (4.5)
where
a(λ)=k=1N(λλk)(λλ¯k),b(λ)=k=12N(λλN+k),R(λ)=k=1N(λλk)(λλ¯k)k=12N(λλN+k),n(λ)=k=12N1(λνk), (4.6)
and v1, v2, …, v2N−1 are a set of elliptic variables for the complex FDHSs (2.20) and (2.22).

Lemma 4.2.

Λψ2,ψ2+Λ¯ψ¯1,ψ¯1ψ¯1,ψ¯1+ψ2,ψ2=k=1N(λk+λ¯k)k=12N1νk, (4.7)
Λ2ψ2,ψ2+Λ¯2ψ¯1,ψ¯1ψ¯1,ψ¯1+ψ2,ψ2=i<jνiνj+k=1N(λk+λ¯k)(k=1N(λk+λ¯k)k=12N1νk)i<j(λiλj+λ¯iλ¯j)i=1Nλij=1Nλ¯j, (4.8)
2i(ψ1,ψ2ψ¯1,ψ¯2)=k=1N(λk+λ¯k)k=12Nλk+N. (4.9)

Proof. Multiplied by –a(λ) on both sides of (4.5), the Right-hand Side (RHS) of (4.5) can be rewritten as

RHS=(ψ¯1,ψ¯1+ψ2,ψ2)(λ2N1λ2N2j=1Nνj+λ2N3i<jνiνj+j=12N1νj), (4.10)
simultaneously, the Left-hand Side (LHS) of (4.5) can be expanded as
LHS=l=1Nψ2l2(λ2N1λ2N2(i=1,ilNλi+j=1Nλ¯j)+λ2N3(i<j;i,jlλiλj+i<jλ¯iλ¯j+i=1,ilNλij=1Nλ¯j)+i=1,ilNλij=1Nλ¯j)+l=1Nψ¯1l2(λ2N1λ2N2(i=1Nλi+j=1,jlNλ¯j)+λ2N3(i<jλiλj+i<j;i,jlλ¯iλ¯j+i=1Nλij=1,jlNλ¯j)+i=1Nλij=1,jlNλ¯j), (4.11)

By comparing the coefficient of λ2N−2 and λ2N−3 in (4.10) and (4.11), we have

(ψ¯1,ψ¯1+ψ2,ψ2)j=1Nνj=l=1Nψ2l2(k=1N(λk+λ¯k)λl)+l=1Nψ¯1l2(k=1N(λk+λ¯k)λ¯l),(ψ¯1,ψ¯1+ψ2,ψ2)i<jνiνj=l=1Nψ2l2(λl2λlk=1N(λk+λ¯k)+i<j(λiλj+λ¯iλ¯j)+i=1Nλij=1Nλ¯j)+l=1Nψ¯1l2(λ¯l2λ¯lk=1N(λk+λ¯k)+i<j(λiλj+λ¯iλ¯j)+i=1Nλij=1Nλ¯j),
which leads to the formulas (4.7) and (4.8). It is seen from (3.4), (3.7) and (3.25) that the LHS of (4.9) is the constant of motion F0 both in x and t. Similar to the treatment as (4.10) and (4.11), the coefficient of λ2N−1 in the expansion of (4.4) reads
k=1N(λk+λ¯k)l=1N(El+E¯l)=k=12Nλk+N,
which gives the formula (4.9) since Ej are described by (3.6).

Replacing λ with vk in (4.4) gives rise to

Vλ11|λ=νk=R(νk)a(νk),1k2N1. (4.12)

Considering the (2, 1)-entry of Lax equations (4.1) and (4.2), we derive

xVλ21=2(ψ¯1,ψ¯1+ψ2,ψ2)Vλ112iλVλ21,tVλ21=2Vλ11V˜21(2,3)2Vλ21V˜11(2,3). (4.13)

By combining (4.5), (4.12), (4.13) and Lemma 4.2, we attain the Dubrovin type equations

dνkdx=2R(νk)j=1,jk2N1(νkνj),1k2N1, (4.14)
dνkdt=2R(νk)j=1,jk2N1(νkνj)[α(2νk2j=12N1νj+j=1N(λj+λ¯j)+j=12Nλj+2N)+2β(2νk22νkj=12N1νj+2i<jνiνj+(j=1N(λj+λ¯j)+j=12Nλj+N)×(νkj=12N1νj)+12j=1N(λj2+λ¯j2)+12j=12Nλj+N2+14(j=1N(λj+λ¯j)+j=12Nλj+N)2)],1k2N1, (4.15)
which control the dynamics of elliptic variables {vk}.

To solve the Dubrovin type equations (4.14) and (4.15), the subsequent attention in this section is instructed to the theory of algebraic curves. From the generating function of integrals of motion, we define a hyperelliptic curve of Riemann surface Γ: ξ 2 + R(λ) = 0, which allows with 2N − 1 linearly independent holomorphic differentials

ω˜l=λl1dλ2R(λ),1l2N1.

Thanks to deg R(λ) = 4N by Eq. (4.6), the genus of Γ is 2N − 1 that coincides with the number of elliptic variables {vk}. For any λ(λj,λ¯j(1jN);λN+k(1k2N))𝔺 , there exist two points P+(λ)=(λ,R(λ)) and P(λ)=(λ,R(λ)) on the upper and lower sheets of Γ. In particular, there are two infinite points ∞1 and ∞2 as λ = ∞, which are not the branch points and can be expressed as (0, −1) and (0, 1) in the local coordinate λ = z−1.

Introduce a set of canonical basis of cycles {aj,bj}j=12N1 on Γ, which are independent with the intersection numbers aiaj = bibj = 0, aibj = δij, (i, j = 1, 2, ⋯, 2N − 1). By the canonical basis of cycles, let us bring in the integral Aij=ajω˜i,(1i,j2N1), which yields a (2N − 1) by (2N − 1) nondegenerate matrix C = (Cij) = (Aij)−1 [21,30]. And then, the holomorphic differential ω˜l can be converted into a normalized one

ωj=l=12N1Cjlω˜l,1j2N1, (4.16)
with the property
aiωj=l=12N1Cjlaiω˜l=l=12N1CjlAli=δji={1,i=j,0,ij. (4.17)

Write ω = (ω1, ω2, ⋯ω2N−1)T for short, and define

δj=ajω,Bjbjω,1j2N1. (4.18)

It is found that δ = (δij)2N−1×2N−1 is a unit matrix, and B = (Bij)2N−1×2N−1 is a symmetric matrix (Bij = Bji) with positive-definite imaginary part [21,30]. Moreover, the 4N − 2 periodic vectors {δj, Bj} span a lattice   𝒯 in 𝔺2N1 that specifies the Jacobi variety J(Γ)=𝔺2N1/𝒯 of Riemann surface Γ.

After the above preparations, we suitably select out the Abel–Jacobi variable with a fixed point p0(i(i=1,2);λj,λ¯j(1jN);λN+k(1k2N)) on Γ

ρj(x,t)=k=12N1p0νk(x,t)ωj=k=12N1l=12N1Cjlp0νk(x,t)λl1dλ2R(λ). (4.19)

By using (4.14) and (4.15), a direct calculation results in

xρj(x,t)=k=12N1l=12N1Cjlνkl1j=1,jkN(νkνj), (4.20)
tρj(x,t)=k=12N1l=12N1Cjlνkl1j=1,jkN(νkνj)[α(2νk2j=12N1νj+j=1N(λj+λ¯j)+j=12Nλj+N)+2β(2νk22νkj=12N1νj+2i<jνiνj+(j=1N(λj+λ¯j)+j=12Nλj+N)(νkj=12N1νj)+12j=1N(λj2+λ¯j2)+12j=12Nλj+N2+14(j=1N(λj+λ¯j)+j=12Nλj+N)2)]. (4.21)

With the aid of the algebraic formulas [31]

Is=k=12N1νksj=1,jk2N1(νkνj)=δs,2N2,1s2N2,I2N1=I2N2j=12N1νj,I2N=I2N1j=12N1νjI2N2i<j;i,j=12N1νiνj, (4.22)
we arrive at
xρj(x,t)=Cj2N1Ωj(1),tρj(x,t)=α[2Cj2N2+Cj2N1(j=1N(λj+λ¯j)+j=12Nλj+N)]β[4Cj2N3+2Cj2N2(j=1N(λj+λ¯j)+j=12Nλj+N)+Cj2N1×(j=1N(λj2+λ¯j2)+j=12Nλj+N2+12(j=1N(λj+λ¯j)+j=12Nλj+N)2)]Ωj(2). (4.23)

By using (4.23), the H1-, H(2,3)- and the Hirota-flows are represented as

H1flow:ρj(x)=Ωj(1)x+ρ0j,H2flow:ρj(t)=Ωj(2)t+ρ0j,Hirotaflow:ρj(x,t)=Ωj(1)x+Ωj(2)t+ρ0j, (4.24)
where ρ0j=k=12N1p0νk(0,0)ωj is a constant of integration.

It has been shown that the evolution velocities Ωj(1) and Ωj(2) are the combination of constants of motion and constants of integration. Having a look at the shape of (4.24), the Abel–Jacobi variable ρj(x, t) can be understood as the angle variable, which exhibit the linearity of Hirota flow on the Jacobi variety J(Γ) of a Riemann surface.

5. QUASI-PERIODIC SOLUTIONS

Followed by the Bargmann map (2.17) and Lemma 4.2, we bridge the gap between the Hirota equation (1.2) and the complex FDHSs (2.20) and (2.22), and further connect the eigenfunctions with the symmetric functions of elliptic variables. It is noted from (4.24) that the Hirota equation has been integrated with the Abel–Jacobi solution over J(Γ), which stimulates us to discuss the Riemann–Jacobi inversion from ρj(x, t) to {vk}.

We turn to the Abel map from the divisor group to the Jacobi variety 𝒜:Div(Γ)J(𝒯)

𝒜(p)=p0pω,𝒜(k=12N1nkpk)=k=12N1nk𝒜(pk).

Let us choose a special divisor p=k=12N1p(νk) , where p(νk)=(νk,ξ(νk)). Denote ρ = (ρ1, ρ2, ⋯, ρ2N−1) for short. The Abel–Jacobi variable can be rewritten as

ρ=k=12N1p0p(νk)ω=𝒜(k=12N1p(νk))=k=12N1𝒜(p(νk)). (5.1)

By the symmetric matrix B, we introduce the Riemann theta function of Γ [21,30]

θ(ζ)=z𝕑2N1expπi(Bz,z+2ζ,z),ζ𝔺2N1.

According to the Riemann theorem [21], it is known from the Abel–Jacobi variable (5.1) that there exists a vector of Riemann constant M=(M1,M2,,M2N1)T𝔺2N1 such that f (λ) = θ (A(p(λ)) − ρM) has 2N – 1 simple zeros at v1, v2, ⋯, v2N−1. To make the function f (λ) single value, the Riemann surface Γ should be suitably cut along with the contours aj and bj to form a simply connected region with the boundary γ, which is consisted of 8N − 4 edges in the order a1+b1+a1b1a2+b2+a2b2a2N1+b2N1+a2N1b2N1, where the symbols +, − denote the orientation. And then, the positive power sums of {νj}j=12N1 can be figured out by the calculation of residues of f (λ) at ∞1 and ∞2, namely

j=12N1νjk=Ik(Γ)s=12Resλ=sλkdlnf(λ), (5.2)
where
Ik(Γ)=12πiγλkdlnf(λ)=j=12N1ajλkωj
is a constant independent of the Abel–Jacobi variable ρ [15].

Lemma 5.1.

Let Sk=j=1N(λjk+λ¯jk)+j=12Nλj+Nk . The coefficients in the expansion

λ2NR(λ)=k=0Λkλk,|λ|>max{|λ1|,,|λN|,|λN+1|,,|λ3N|}, (5.3)
are given by the recursive formulae
Λk=0(k1),Λ0=1,Λ1=12S1,Λ2=14(S2+S1Λ1)=14S2+18S12,Λk=12k(Sk+i+j=k,i,j1SiΛj),k3. (5.4)

Lemma 5.2.

Nears (s = 1, 2), under the local coordinate z = λ−1 the holomorphic differential ω˜l can be described by

ω˜l=i2k=0Λkz2N1l+kdz. (5.5)

We denote the jth component of f(λ) by ζj, j = /∂ζj, jk2=2/ζjζk , etc. With the Einstein summation convention, in the neighborhood of λ = ∞s(s = 1, 2) the Riemann theta function f(λ) has the asymptotic expansion (z = λ1)

f(λ)=θs()+(1)s2iCj2N1zjθs()+z22(14Cj2N1Ck2N1jk2θs()+(1)s2i(Cj2N1Λ1+Cj2N2)jθs())+z36((1)s+18iCj2N1Ck2N1Cl2N1jkl3θs()34Cj2N1(Ck2N1Λ1+Ck2N2)jk2θs()+(1)si(Λ2Cj2N1+Λ1Cj2N2+Cj2N3)jθs())+O(z4), (5.6)
which together with (4.23) gives rise to
dlnf(λ)dλ=(1)s+12ixlnθs()14(2(1)s+1iΩ˜(2)Ω(2)tlnθs()+x2lnθs())z+O(z2), (5.7)
where Ω˜ j(2)=Cj2N1Λ1+Cj2N2 , θs()=θs()(ρ+M+χs) and χs=sP0ω . Resorting to (5.2) and (5.7), we attain the trace formulae
k=12N1νk=I1(Γ)+12ixlnθ2()θ1(),k=12N1νk2=I2(Γ)+12iΩ˜(2)Ω(2)tlnθ2()θ1()+14x2lnθ1()θ2(). (5.8)

By using the Bargmann map (2.17) and the complex FDHS (2.20), a direct calculation yields

xlnv=2iΛ¯ψ¯1,ψ¯1+Λψ2,ψ2ψ¯1,ψ¯1+ψ2,ψ22(ψ1,ψ2ψ¯1,ψ¯2), (5.9)
which together with Lemma 4.2 and the trace formula (5.8) results in
xlnv=N1+xlnθ(Ω(1)x+Ω(2)t+α2)θ(Ω(1)x+Ω(2)t+α1), (5.10)
where
N1=2iI1(Γ)i(j=1N(λj+λ¯j)+j=12Nλj+2N),αs=ρ0+M+χs,s=1,2.

Taking one integration on (5.10) with respect to x, we obtain

v=v0eN1xθ(Ω(1)x+Ω(2)t+α2)θ(Ω(1)x+Ω(2)t+α1), (5.11)
where v0 is independent of x, but may depend on t. On the other hand, taking one partial derivative with respect to t on (5.11), we also have
tlnv=N2+tlnθ(Ω(1)x+Ω(2)t+α2)θ(Ω(1)x+Ω(2)t+α1),N2=tlnv0. (5.12)

Analogous to the treatment conducted as in Cao et al. [6] (see Theorem 11.1), it is found that N2 is also a constant of motion with regards to t. Finally, based on the presentations (5.10) and (5.12), we obtain the quasi-periodic solution for the Hirota equation

v(x,t)=v(0,0)eN1x+N2tθ(α1)θ(α2)θ(Ω(1)x+Ω(2)t+α2)θ(Ω(1)x+Ω(2)t+α1). (5.13)

Remark 5.1.

It looks like that only the quasi-periodic solution of odd genus (g = 2N − 1) has been attained in the above constructing scheme. In fact, the solution in the case of even genus (g = 2N − 2) can be obtained by the degeneration procedure of v2N−1 = 0, because the finite-genus solution can be embedded into an invariant torus with one more genus (for more details, see the subsection 2.4 in Chen and Pelinovsky [9]).

In conclusion, an explicit quasi-periodic solution has been constructed for the Hirota equation (1.2) with the aid of two complex FDHSs (2.20) and (2.22). In particular, as α = 1 and β = 0, the quasi-periodic solution (5.13) becomes the exact solution of the focusing NLS equation that coincides with the one in the book [4] [see Eq. (4.1.22)]; whereas α = 0 and β = 1, for real v(x, t) the quasi-periodic solution (5.13) delivers a new solution for the focusing mKdV equation.

CONFLICTS OF INTEREST

The authors declare they have no conflicts of interest.

ACKNOWLEDGMENT

This work was supported by the National Natural Science Foundation of China (No. 11971103).

REFERENCES

[3]VI Arnold, Mathematical Methods of Classical Mechanics, Springer-Verlag, Berlin, 1978.
[4]ED Belokolos, AI Bobenko, VZ Enol’skii, AR Its, and VB Matveev, Algebro-Geometric Approach to Nonlinear Integrable Equations, Springer-Verlag, Berlin, 1994.
[7]J Cen and A Fring, Asymptotic and scattering behaviour for degenerate multi-solitons in the Hirota equation, Physica D, Vol. 397, 2019, pp. 17-24.
[15]LA Dickey, Soliton Equations and Hamiltonian Systems, World Scientific, Singapore, 1991.
[19]F Gesztesy, Quasi-periodic, finite-gap solutions of the modified Korteweg-de Vries equation, S Albeverio, JE Fenstad, H Holden, and T Lindstròm (editors), Ideas and Methods in Mathematical Analysis, Stochastics, and Applications, Cambridge University Press, Cambridge, 1992, pp. 428-471.
[21]P Griffiths and J Harris, Principles of Algebraic Geometry, John Wiley & Sons, New York, 1994.
[30]D Mumford, Tata Lectures on Theta II, Birkhäuser, Boston, 1984.
[31]AC Newell, Solitons in Mathematics and Physics, SIAM, Philadelphia, PA, 1985.
Journal
Journal of Nonlinear Mathematical Physics
Volume-Issue
28 - 1
Pages
134 - 149
Publication Date
2020/12
ISSN (Online)
1776-0852
ISSN (Print)
1402-9251
DOI
https://doi.org/10.2991/jnmp.k.200922.010How to use a DOI?
Copyright
© 2020 The Authors. Published by Atlantis Press B.V.
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

Cite this article

TY  - JOUR
AU  - Jinbing Chen
AU  - Rong Tong
PY  - 2020
DA  - 2020/12
TI  - The Complex Hamiltonian Systems and Quasi-periodic Solutions in the Hirota Equation
JO  - Journal of Nonlinear Mathematical Physics
SP  - 134
EP  - 149
VL  - 28
IS  - 1
SN  - 1776-0852
UR  - https://doi.org/10.2991/jnmp.k.200922.010
DO  - https://doi.org/10.2991/jnmp.k.200922.010
ID  - Chen2020
ER  -