Whitham modulation theory and periodic solutions for the fifth-order nonlinear Schrödinger equation in the Heisenberg ferromagnetic spin chain

In this paper, we investigate the periodic solutions and Whitham modulation theory for the fifth-order nonlinear Schrödinger equation, which can describe the one-dimensional anisotropic Heisenberg ferromagnetic spin chain. First, we introduce the principle of the finite-gap integration method. Then, we discuss the single-phase periodic solutions and their degenerate forms in two limit cases. In addition, we analyze the influence of higher-order term parameters on the propagation of periodic solutions and solitons. Further, we derive the single-phase Whitham modulation equation for the fifth-order nonlinear Schrödinger equation. Moreover, we systematically derive the two-phase periodic solutions and the corresponding Whitham modulation equations.


Introduction
The nonlinear evolution equations have been used to describe various complex phenomena in many branches of modern science such as fluid mechanics, condensed matter, biology and optical commu-Y. Zhang [1][2][3]. One of the most important models in the nonlinear evolution equation is the nonlinear Schrödinger (NLS) equation, which describes a large number of phenomena and dynamic processes in computer science, life science, plasma dynamics and chemistry [4][5][6][7]. However, the NLS equation is an approximation of different physical systems and contains only the lowest order dispersion and nonlinear terms. If the characteristics of the solution in the research process exceed the simple approximation used to derive the NLS equation, self-steepening, self-frequency shift, pulse deformation and other phenomena need to be considered in order to obtain a better approximation to describe the corresponding physical properties [8,9]. High-order dispersion and nonlinear effects will play an important role and may provide some different characteristics for wave propagation behavior. The influence of third-order effects in the field of femtosecond pulses has been intensively studied [10], and the fourthorder effects also play an important role in anisotropic Heisenberg ferromagnetic spins [11]. Moreover, the quintic effects should be considered when the propagation of ultrashort pulses with attosecond duration is investigated [12]. In general, the effects of each of the quintic terms should be analyzed separately, since quintic terms are more complex than other terms. The integrable models give us the general forms of the quintic terms, but the new terms cannot be applied directly to the NLS equation in any arbitrary way but must be combined with certain parameter restrictions to strictly preserve the integrability of the equation [13]. Further, the quintic terms are independent of the cubic and quartic terms. In this work, we consider the fifth-order nonlinear Schrödinger equation [14] iu t + u x x + 2 |u| 2 u − iα(u x x x + 6 |u| 2 u x ) where the asterisk " * " signifies the complex conjugation, u(x, t) denotes the complex valued wave function with the independent variables x and t representing space and time, α, γ and δ are coefficients of the third, fourth and fifth order terms, respectively. In fact, Eq. (1.1) contains some important nonlinear differential equations. When α = γ = δ = 0, Eq. (1.1) will reduce to the well-known focused NLS equation [15]. When α = 0, γ = δ = 0, the simplest exactly solvable extension will appear, Eq. (1.1) will translate to the Hirota equation [16] that can be used to describe the subpicosecond and femtosecond pulse propagation. When γ = 0, α = δ = 0, Eq. (1.1) will become the Lakshmanan-Porsezian-Daniel (LPD) equation [17], which can be used to describe the nonlinear spin excitation with octupole-dipole interaction in the one-dimensional isotropic biquadratic Heisenberg ferromagnetic spin chain. Further, this paper focuses on the case of δ = 0, α = γ = 0, where Eq. (1.1) can be simplified into the fifth-order NLS equation [18] that can describe the Heisenberg ferromagnetic spin system. During the past serval decades, there have been many analyzes on the fifth-order NLS equation, in which soliton and periodic solutions have been extensively studied. Soliton solutions are caused by the cancellation of nonlinear and dispersion effects in media, usually propagates at a constant rate and maintains its shape, and can describe the solution of a class of weakly nonlinear dispersion partial differential equations widely existing in physical systems [19][20][21][22][23]. Equation (1.1) has been proved to be integrable when the coefficients satisfy certain constraints, the Lax pair and other required complete integrable properties have been recognized, and the exact expressions of soliton solutions have been obtained by the Darboux transformation [24]. In Ref. [25], combining the Darboux transformation with the nonlinearization of the spectral problem, the rogue wave solutions on the periodic wave background of Eq. (1.1) have been constructed by using two types of Jacobian elliptic functions. In addition, the exact analytical solutions for one-soliton, two-soliton, and three-soliton of Eq. (1.1) have been derived by the Hirota bilinear method, and the influence of higherorder term parameters on solitons and interactions has been studied [26]. In Ref. [27], the multi-soliton solutions of Eq. (1.1) have been derived via the Riemann-Hilbert method. Although the soliton and periodic solutions of integrable equations can be studied in many ways, one of the methods for obtaining periodic or quasi-periodic solutions from linear spectral problems related to integrable nonlinear dispersion equations is known as the finite-gap integration method [28]. The formulas and methods for the finite-gap integration have important application value when it involves various asymptotic problems in mathematical physics, including some applications of the Schrödinger operator of decreasing potential energy [29][30][31][32][33][34][35]. Finite-gap integration method relies on a set of parameters called the Riemann invariant, which can be obtained by reparameterizing the zero points λ i of the algebraic solution to the polynomial P (λ) on the hyperelliptic curve ω 2 = P (λ). This means that it is convenient to observe the evolution of periodic solutions through the evolution behavior of Riemann invariants. Kamchatnov extended this so that the method could be applied to a series of equations for the AKNS system [36][37][38][39][40][41][42]. In recent years, it is found that the periodic solutions derived by the finite-gap integration method can be further studied by the Whitham modulation theory [43][44][45][46][47]. The Whitham modulation theory is the basis for the development of dispersive hydrodynamics, and it is often used to analyze the slow evolution of nonlinear dispersive waves [48][49][50][51][52][53][54]. When using the periodic or quasi-periodic solutions obtained by the finite-gap integration method to study the Whitham modulation theory, if the spectral band endpoints of the quasi-periodic solutions vary slowly with x and t, they also can be called the Riemann invariants of the Whitham modulation equation, which allows us to obtain the Riemann invariant parameterization needed for periodic solutions of integrable equations relevant with linear spectral problems [55].
In this paper, we focus on the periodic solutions, solitons and the corresponding Whitham modulation equations for Eq. (1.1). The arrangement of this paper is designed as follows: In Sect. 2, we will introduce the basic principle of the finite-gap integration method and give the Lax pair of Eq. (1.1). In Sect. 3, we will study the single-phase periodic solutions and their soliton and harmonic limits in the m → 1 and m → 0 cases. In addition, we will study the effect of higher-order term parameters on the soliton and periodic solutions. Moreover, we will derive the single-phase Whitham modulation equations and their different limiting forms. We will verify the correctness of the results by numerical simulation. In Sect. 4, we will derive the twophase periodic solutions and Whitham equations, and the summary will be proposed in Sect. 5. Finally, we will give the formula for the elliptic functions in the appendix.

The finite-gap integration method
The Lax pair corresponding to Eq. (1.1) is given as follows [56]: where Ψ = (φ 1 , φ 2 ) T (T denotes the transpose of a vector) is the eigenfunction, λ is the complex spectral parameter, and the matrices U and V have the following forms: Ψ must meet the compatibility condition Ψ xt = Ψ t x , then we can obtain the zero curvature equation And then we find that Eq. (2.1) has two fundamental solutions (φ 1 , φ 2 ) T and (ψ 1 , ψ 2 ) T , which can be used to construct vectors with the spherical components which satisfy the following systems: Through simple calculation, we can obtain which means that the Wronskian ψ 1 ϕ 2 − ψ 2 ϕ 1 does not depend on x and t. Hence, which is the motion integral that only depends on the spectral parameter λ. In the following discussion of the multi-phase solution of Eq. (1.1), the number of phases is determined by the degree of the polynomial P (λ).

Single-phase solution and Whitham modulation equations
In the case of single-phase, we assume that the solution of Eq. (2.4) has the following form: According to Eq. (2.5) we can deduce where λ i (i=1, 2, 3, 4) are zero points of the polynomial P (λ), and the standard symmetric functions s i (i=1, 2, 3, 4) and λ i are related as follows:

Single-phase solution
According to Eqs. (3.1) and (2.4), comparing the coefficients of λ k on both sides of the equation yields the following equations: as well as their complex conjugate. According to Eq. (3.2), we can obtain When the condition of λ = μ is satisfied, by simple calculations, we find that where According to Eqs. (3.13) and (3.14), it can be seen that μ and |u| 2 depend only on the phase W = x − V t and can be determined by the following equations: According to Eq. (3.10), when μ and μ * are considered as two solutions of a quadratic equation, we can obtain the following discriminant By calculating that we get the relationship between the zero points of R (v) and P (λ), which means that v i (i=1, 2, 3) and λ i satisfy the following equations: Since the value of v = |u| 2 is a real number greater than zero, that is, v oscillates between two positive zero points of R (v) (see Fig. 1). Moreover, we assume that v consists of two complex conjugate pairs [50] Therefore, we obtain the trajectories of trace formula μ and μ * , as shown in Fig. 2 Hence, where Since v = v 3 will result in W = 0, which means ℘ (c) = e 3 and c = ω , where ω and ω are half-periods of ℘-function [44]. According to Eqs. (3.26) and (3.23), we can obtain .
Since σ -function is an odd function, we can simplify σ (2W+ω +κ) σ (2W +ω )σ (κ) according to the formula in the appendix as where Therefore, we can derive the periodic solution of Eq. (1.1) (see Fig. 3): where Further make γ → η, so that in the case of m → 1, we can get the soliton solution of Eq. (1.1) (see Fig. 3): (3.32) In this case, we have We assume that κ = ω + ω , ζ (κ) = η + η , so we can further obtain where Therefore the periodic solution of Eq. (1.1) in the form of cn-function can be obtained (see Fig. 4): where We can derive the soliton solution in the same form as Eq. (3.32) when α → o (see Fig. 4): We find that the periodic solutions will evolve from the harmonic limit to the soliton limit in the process from m → 0 to m → 1 if appropriate parameters are selected, as shown in Fig. 5. Furthermore, we find that the higher-order term parameter δ has certain influence on the velocity and propagation direction of solitons and periodic solutions. When δ = 0, Eq. (1.1) will be transformed into the NLS equation and the soliton will propagate along the negative direction of the xaxis. When δ = − α 4(5α 4 −10α 2 η 2 +η 4 ) , the velocity of the soliton will become 0. In addition, as δ is gradually smaller than − α 4(5α 4 −−10α 2 η 2 +η 4 ) , the velocity of the soliton gradually increases, and the soliton moves along the positive direction of the x-axis, as shown in Fig. 6.

The Whitham equations for Single-phase solution
The Whitham equation is based on the fast oscillations of the wave and the slow evolution of the parameters, and the slow evolution equations of the parameters can be averaged in the fast oscillations to obtain the system of first order partial differential equations. Since the resulting corresponding equations are quite complex, if one diagonalizes the slowly varying parameter, that is, the Riemann invariant, then the system can be simplified by an appropriate parameterization of the periodic solution to further modulate the periodic wave. Therefore, from Eq. (2.4) we can obtain the conservation law Moreover, We can take to normalize Eq. (3.1), and further derive the following formula: where By simple calculation, we can get (3.38) Using the formula = 1 L L 0 dW to average Eq. (3.37) and replace the integral variable W , we find that The coefficients of singular terms due to the differential of P (λ) with respect to x and t in Eq. (3.39) disappear, we have (3.40) Thus we derive the Whitham equations in Riemann diagonal form:  (3.43) where the wavelength When the zero point v i of R (v) is taken we can obtain and Further, we derive the Whitham velocities of Eq. (1.1) through calculations: , , , , (3.45) where E(m) is the complete elliptic integral of the second kind, and After observation, Eq. (3.44) can also be written in the following form Next, we discuss the degradation of Whitham velocities in different limit conditions: (3.48) (3) λ 3 → λ 2 (m → 1): In several cases of m → 0 and m → 1, we find that two Whitham equations transform into the same equation, whereas the remaining two equations degenerate into the dispersionless limit system as shown in the following formula: (3.50) The essential difference of Whitham modulation theory between Eq. (1.1) and the standard NLS equation lies in the effect of higher order parameters on Whitham velocities. When the defocusing fifth-order nonlinear Schrödinger equation is considered, the limit of Whitham velocities will correspond to the leading edge and trailing edge velocities of the dispersive shock wave. Since Eq. (1.1) is a modification of the focusing NLS equation, while the dispersion-free equation is elliptic type, it leads to the modulated unstable solution with complex Riemann invariants. The Whitham modulation equations no longer describe the dispersive shock wave but rather describe the nonlinear stage of the development of the modulation instability [44,55], but this is out of the range of this paper, which we will continue to study in the follow-up work.

Two-phase periodic solution
When discussing the two-phase solution for Eq. (1.1), we assume that f , g, h have the following form: Similar to the case of single-phase solution, substituting Eqs. (4.1) into (2.4) yields and Detailed expressions of (uμ 1 + uμ 2 ) t and (uμ 1 μ 2 ) t are not given, because of its complexity, but they can be obtained via comparing the coefficients of λ. Different from the single-phase solution, in this section, we assume that P (λ) has the following form: 2,3,4,5,6) are zero points of P (λ), we can derive the relationship between λ i and s i Further, we can also simplify Eqs. (4.2) and (4.3) to get  ) and μ 2 = μ 2 (x, t) satisfy the Dubrovin form [57,58] of the following equations: which can be solved with the Abel transform [46,[59][60][61]. When solving the two-phase solution, μ 1 (x, t) and μ 2 (x, t) can be represented by the Riemann θ function, which depend on the two phase variables where κ 1 , κ 2 , ω 1 and ω 2 are affected by integrating over a certain period on the Riemann surface of the hyperelliptic curve and θ 01 and θ 02 are the initial phases [49]. Next, we will focus on the construction of the Whitham equations for the two-phase periodic solution.

The Whitham equations for the two-phase solution
The conservation law of the two-phase solution equals to According to Eqs. (2.2) and (4.1), we have where where C 1 (a cycle from λ 5 to λ 4 ) and C 2 (a cycle from λ 3 to λ 2 ) are cycles in which solutions of twophase Dubrovin type equations are defined by Abelian transformations [49,62]. Further, we can take hyperelliptlc integrals U i j = C i μ j−1 √ P(μ) dμ, i, j = 1, 2 to simplify Eq. (4.9) and deduce the Whitham velocities of Eq. (1.1) in the two-phase case:

Conclusions
In this paper, the single-phase and two-phase periodic solutions of the fifth-order nonlinear Schrödinger equation have been comprehensively studied by the finite-gap integration method, and the corresponding Whitham modulation equations have been derived. In the limit cases of m → 0 and m → 1, we have found that the single-phase periodic solution degenerates into the small-amplitude harmonic solution and soliton solution, respectively. The numerical simulations are in good agreement with the analytical solutions. In addition, we have analyzed the effect of higherorder parameters on the periodic solutions and solitons.
In future work, we will focus on the Whitham modulation equations, which describes the nonlinear stage of the development of modulation instability. Further, we will study the periodic solution of the defocusing fifth-order Schrödinger equation and the structure of the dispersive shock wave and discuss the complete classification of the Riemann problem in future work.

Properties of the Weierstrass ℘-function:
The periods of the function ℘ (z) are 2ω and 2ω , and ℘ satisfies ℘ (z) 2 The Jacobian elliptic functions and σ -function can be related by the following relationships: . (A.11a)