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

^{*}, Rong Tong

^{*}Corresponding author. Email: cjb@seu.edu.cn

- DOI
- 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]

*v*is a scalar function depending on

*x*and

*t*

*i*is the imaginary unit, and

*α*,

*β*,

*δ*, and

*γ*are real positive constants satisfying

*αγ*=

*βδ*. Let

*δ*be 2

*α*and

*γ*be 2

*β*. The Hirota equation (1.1) can be rewritten as

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 *v _{xxx}* and a time-delay correction to the cubic term

*v*|

_{x}*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 [6,8–11,13,17,18,34]. 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 [9–11,17,34]. 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]

*v*, and

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

Let *a*_{0} = 2*i* and *b*_{0} = *c*_{0} = 0 be the initial values. Up to constants of integration, *a _{j}*,

*b*and

_{j}*c*can be uniquely determined by means of the recursive formula (2.3), for example

_{j}Based on the recurrence chain (2.3), we introduce the Lenard gradients {*g _{j}*} and the Lenard operator pair

*K*and

*J*:

*∂*=

_{x}*∂*/

*∂*under the condition

_{x}It is assumed that *φ* satisfies a spectral problem determined by the Lenard gradients {*g _{j}*}

The zero-curvature equation of spectral problems (2.1) and (2.8), i.e.

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

Let *λ*_{1,} *λ*_{2}, ⋯, λ* _{N}* be

*N*arbitrary distinct nonzero complex eigenvalues, namely,

*ψ*

_{1j},

*ψ*

_{2}

*)*

_{j}*be the vector eigenfunction pertinent to*

^{T}*λ*. Due to the symmetry of (2.1),

_{j}*λ*

_{1,}

*λ*

_{2}, ⋯, λ

*),*

_{N}*ψ*

_{1}= (

*ψ*

_{11},

*ψ*

_{12}, ⋯,

*ψ*

_{1}

*)*

_{N}*, and*

^{T}*ψ*

_{2}= (

*ψ*

_{21},

*ψ*

_{22}, ⋯,

*ψ*

_{2}

*)*

_{N}*. The diamond bracket 〈. , .〉 stands for the vector product:*

^{T}*ξ*= (

*ξ*

_{1},

*ξ*

_{2}, ⋯,

*ξ*)

_{N}*and*

^{T}*η*= (

*η*

_{1},

*η*

_{2}, ⋯,

*η*)

_{N}*. According to the nonlinearization of Lax pair [5], we consider*

^{T}*N*copies of spectral problem (2.1)

It follows from [37,39] that the functional gradients of *λ _{j}* and

*v*are

Recall the Bargmann (symmetric) constraint

*v*with the eigenfunctions (

*ψ*

_{1},

*ψ*

_{2})

On

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

*H*

_{(2,3)}=

*α H*+ 2

_{2}*β H*

_{3}together with

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

## 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

It follows from (2.15) that

Substituting *G*_{λ} back into the expression of *V*^{(}^{n}^{)} gives rise to a Lax matrix

*V*

_{λ}is a generating function of integrals of motion for the complex FDHSs (2.20) [36]. With |

*λ*| > max{|

*λ*

_{1}|, |

*λ*

_{2}|, ⋯, |

*λ*|}, we come to

_{N}Let *F _{λ}* be a real-valued Hamiltonian on

*τ*be the flow variable of

_{λ}*F*. From the Poisson bracket, a direct calculation results in two canonical Hamiltonian equations

_{λ}### Lemma 3.1.

*On
*

*Besides,*

*Proof.* Only for simplifying the description, we denote

It follows from (3.11) and (3.12) that

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

Furthermore, from (3.14) we arrive at

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 *F _{k}* (0 ≤

*k*≤ 2

*N*−1).

### Lemma 3.2.

*The integrals of motion* {*F*_{0}, *F*_{1}, ⋯, *F*_{2}_{N}_{−1}} *given by (3.7)–(3.10) are functionally independent in a dense open subset of
*.

*Proof.* It is known from (3.5) that

Let
*ψ*_{1}* _{j}* = 0,

*ψ*

_{2}

*≠ 0, (1 ≤*

_{j}*j*≤

*N*). And then,

By (3.20), we arrive at the Jacobi determinant of
*P*_{0}

*N*constants

*γ*

_{0}_{,}

*γ*

_{1,}

_{⋯,}

*γ*

_{2}

_{N}_{−1}such that

*γ*is the Vandermonde determinant. Namely,

_{k}*γ*

_{0}= ⋯ =

*γ*

_{2}

_{N}_{−1}= 0, which means that {

*F*}(0 ≤

_{k}*k*≤ 2

*N*− 1) are functionally independent in a dense open subset of

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

*F*} 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).

_{k}### Proposition 3.1.

*The complex FDHSs
*

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
*x* and *t* giving an involutive solution for complex FDHSs

### Proposition 3.2.

*Let* (*ψ _{1}*(

*x*,

*t*),

*ψ*

_{2}(

*x*,

*t*))

^{T}be an involutive solution of integrable complex FDHSs (2.20) and (2.22). Then*is a finite parametric solution of the Hirota equation (1.2).*

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

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* (*ψ _{1}*(

*x*),

*ψ*

_{2}(

*x*))

^{T}be a solution of the complex FDHSs (2.20). Then*is a finite-gap solution to the complex Novikov (high-order stationary NLS) equation*

*where*

*and*c ^ 2 , c ^ 3 , ⋯ , c ^ 2 N
are some constants of integration.

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

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

*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

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

From Lemma 3.1, we know that the Lax matrix *V _{μ}* satisfies a Lax equation along with

*τ*-flow. In particular, after a direct but tedious calculation, the Lax matrix

_{λ}*V*also satisfies two Lax equations associated with the variables of

_{λ}*x*and

*t*, respectively.

### Lemma 4.1.

*where*

It follows from (3.3) and (3.5) that *F _{λ}* and

*λ*with simple poles at

*v*

_{1},

*v*

_{2}, …,

*v*

_{2}

_{N}_{−1}are a set of elliptic variables for the complex FDHSs (2.20) and (2.22).

### Lemma 4.2.

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

By comparing the coefficient of *λ ^{2}^{N−}*

^{2}and

*λ*

^{2}

^{N−}^{3}in (4.10) and (4.11), we have

*F*

_{0}both in

*x*and

*t*. Similar to the treatment as (4.10) and (4.11), the coefficient of

*λ*

^{2}

^{N−}^{1}in the expansion of (4.4) reads

*E*are described by (3.6).

_{j}Replacing *λ* with *v _{k}* in (4.4) gives rise to

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

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

*v*}.

_{k}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 2*N* − 1 linearly independent holomorphic differentials

Thanks to deg *R*(*λ*) = 4*N* by Eq. (4.6), the genus of Γ is 2*N* − 1 that coincides with the number of elliptic variables {*v _{k}*}. For any

_{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
*a _{i}* ○

*a*=

_{j}*b*○

_{i}*b*= 0,

_{j}*a*○

_{i}*b*=

_{j}*δ*, (

_{ij}*i*,

*j*= 1, 2, ⋯, 2N − 1). By the canonical basis of cycles, let us bring in the integral

*N*− 1) by (2

*N*− 1) nondegenerate matrix

*C*= (

*C*) = (

_{ij}*A*)

_{ij}^{−1}[21,30]. And then, the holomorphic differential

Write *ω* = (*ω _{1}*,

*ω*

_{2}, ⋯

*ω*

_{2}

_{N}_{−1})

*for short, and define*

^{T}It is found that *δ* = (*δ _{ij}*)

_{2}

_{N}_{−1}

_{×2}

_{N}_{−1}is a unit matrix, and

*B*= (

*B*)

_{ij}_{2}

_{N}_{−1}

_{×2}

_{N}_{−1}is a symmetric matrix (

*B*) with positive-definite imaginary part [21,30]. Moreover, the 4

_{ij}= B_{ji}*N*− 2 periodic vectors {

*δ*,

_{j}*B*} span a lattice

_{j}_{Γ}.

After the above preparations, we suitably select out the Abel–Jacobi variable with a fixed point

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

With the aid of the algebraic formulas [31]

By using (4.23), the *H*_{1}-, *H*_{(}_{2,3)}- and the Hirota-flows are represented as

It has been shown that the evolution velocities
*ρ _{j}*(

*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 *ρ _{j}*(

*x*,

*t*) to {

*v*}.

_{k}We turn to the Abel map from the divisor group to the Jacobi variety

Let us choose a special divisor
*ρ* = (*ρ*_{1}, *ρ*_{2}, ⋯, *ρ*_{2}_{N}_{−1}) for short. The Abel–Jacobi variable can be rewritten as

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

According to the Riemann theorem [21], it is known from the Abel–Jacobi variable (5.1) that there exists a vector of Riemann constant
*f *(*λ*) = *θ *(*A*(*p*(*λ*)) − *ρ* − *M*) has 2*N* – 1 simple zeros at *v*_{1}, *v*_{2}, ⋯, *v*_{2}_{N}_{−1}. To make the function *f* (*λ*) single value, the Riemann surface Γ should be suitably cut along with the contours *a _{j}* and

*b*to form a simply connected region with the boundary

_{j}*γ*, which is consisted of 8

*N*− 4 edges in the order

*f*(

*λ*) at ∞

_{1}and ∞

_{2}, namely

*ρ*[15].

### Lemma 5.1.

*Let
*

*are given by the recursive formulae*

### Lemma 5.2.

*Near* ∞* _{s}* (

*s*= 1, 2)

*, under the local coordinate z*=

*λ*

^{−1}

*the holomorphic differential*ω ˜ l
can be described by

We denote the *j*^{th} component of *f*(*λ*) by *ζ _{j}*,

*∂*=

_{j}*∂*/

*∂ζ*,

_{j}*λ*= ∞

*(*

_{s}*s*= 1, 2) the Riemann theta function

*f*(

*λ*) has the asymptotic expansion (

*z*= λ

^{−}1)

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

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

*v*is independent of

_{0}*x*, but may depend on

*t*. On the other hand, taking one partial derivative with respect to

*t*on (5.11), we also have

Analogous to the treatment conducted as in Cao et al. [6] (see Theorem 11.1), it is found that *N*_{2} 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

### Remark 5.1.

It looks like that only the quasi-periodic solution of odd genus (*g* = 2*N* − 1) has been attained in the above constructing scheme. In fact, the solution in the case of even genus (*g* = 2*N* − 2) can be obtained by the degeneration procedure of *v*_{2}_{N}_{−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

## REFERENCES

### Cite this article

TY - JOUR AU - Jinbing Chen AU - Rong Tong PY - 2020 DA - 2020/12/10 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 - 10.2991/jnmp.k.200922.010 ID - Chen2020 ER -