Superregular breathers, characteristics of nonlinear stage of modulation instability induced by higher-order effects

Abstract

We study the higher-order generalized nonlinear Schrödinger (NLS) equation describing the propagation of ultrashort optical pulse in optical fibres. By using Darboux transformation, we derive the superregular breather solution that develops from a small localized perturbation. This type of solution can be used to characterize the nonlinear stage of the modulation instability (MI) of the condensate. In particular, we show some novel characteristics of the nonlinear stage of MI arising from higher-order effects: (i) coexistence of a quasi-Akhmediev breather and a multipeak soliton; (ii) two multipeak solitons propagation in opposite directions; (iii) a beating pattern followed by two multipeak solitons in the same direction. It is found that these patterns generated from a small localized perturbation do not have the analogues in the standard NLS equation. Our results enrich Zakharov’s theory of superregular breathers and could provide helpful insight on the nonlinear stage of MI in presence of the higher-order effects.

1. Introduction

Rogue wave, which originally described the mysterious giant ocean wave phenomenon that can lead to water walls taller than 20–30 m and that represents a catastrophe for ships and offshore oil platforms [1,2], has received a lot of attention in nonlinear optical system [3–5] and other related fields [6–12]. This wave shows the propensity to appear from nowhere and disappear without a trace [13]. As the lowest-order rational solution of the standard nonlinear Schrödinger (NLS) equation, the Peregrine soliton (PS) [14], which was found over 30 years ago, plays an important role in characterizing the rogue wave in different physical areas [15]. The PS solution is an outlandish doubly localized wave packet featuring a peaked hump three times the significant wave height and two side holes that appear as a result of energy conservation. It is the limiting case of two kinds of breather solutions, namely, the Kuznetsov–Ma breather (KMB) [16–18] and Akhmediev breather (AB) [19] solutions. The AB solution is developed from a weak periodic modulation, which is localized in the longitudinal dimension as it undergoes growth and decay. By contrast, the KMB undergoes periodic evolution with propagation [16–19]. The rogue wave is localized in both transverse and longitudinal dimensions.

It is well known that the PS, KMB and AB solutions are associated with modulation instability (MI), i.e. the instability of a constant background with respect to long wavelength perturbations [20]. This effect has been studied by theoretical and experimental observations in the 1960s [20–26]. MI is one of the most ubiquitous kinds of instabilities in nature. In the context of hydromechanics, it is known as the Benjamin–Feir instability [25]. In addition, MI was also discovered in other physical settings ranging from nonlinear optics [27], plasma physics [28–30], in electrical transmission lines [31] and in Bose–Einstein condensates [32]. It is agreed that a rogue wave may appear as a result of MI. However, not every type of MI necessarily results in rogue-wave excitation. In 2014, Baronio et al. [33–35] proposed the concept of baseband MI which is defined as the condition where a continuous-wave background is unstable with respect to perturbations having infinitesimally small frequencies. Conversely, they defined passband MI as the situation where the perturbation experiences gain in a spectral region not including ω=0 as a limiting case [33–35]. They further showed that MI is a necessary but not a sufficient condition for the existence of rogue waves, namely, rogue waves can exist if and only if the MI gain band also contains the zero-frequency perturbation as a limiting case (baseband MI) [33–35]. In the passband regime, MI only leads to the birth of nonlinear oscillations [33–35]. The corresponding numerical simulations have confirmed that rogue waves can indeed be excited from a noisy input continuous-wave background in the baseband-MI regime, whereas one can only observe the generation of nonlinear wave oscillations in the passband-MI regime [33,34].

The NLS equation has a simple solution, i.e. the monochromatic wave with frequency depending on amplitude (it is called the condensate). The condensate is unstable with respect to MI. The linear stability analysis can be used to identify the instability criterion and estimate the initial growth rate of the sidebands. Nevertheless, the linearization ceases to be valid as soon as the perturbations have grown to the point that they are of comparable size compared to the background. Thus, this simplistic analysis provides only snapshots of the initial steps of MI and the question of the nonlinear stage of MI has remained open for almost 50 years. The AB, KMB and rogue-wave solutions could be seen as the signs of the nonlinear stage of elementary MI evolved from periodic perturbations. Specifically, the AB describes an elementary MI with only one growth-return cycle evolved from a weak periodic perturbation, and the Peregine soliton is the single-cycle limiting case of the AB. While the KMB shows the elementary MI dynamics appearing in the longitudinal growth and decay of the individual modulation cycles of a strongly periodic perturbation. From the physical and practical significance perspective, these solutions have their limitation. To address this issue, the theory of the superregular breather solutions was proposed by Zakharov & Gelash [36,37]. In contrast to the previous solutions, the superregular breather solution describes the MI scenario developed from a localized weak perturbation, which subsequently exhibits a long-time complex nonlinear evolution (long-time pulsating behaviour). Thus, it provides a global understanding of MI development of localized small perturbations [36,37]. Recent experiments [38] have demonstrated the existence of the superregular breathers in real physical systems such as water waves and nonlinear fibre optics. These results are in excellent agreement with analytic predictions.

Recent studies showed that rogue waves or breathers of the NLS equations with higher-order effects can display diverse structures beyond the reach of the standard NLS system. Higher-order effects not only produce the compression effects of the breather and rogue waves [39], but also affects the spatial distributions of the rogue waves [40]. Moreover, Akhmediev and co-workers [41,42] have shown that a breather solution of the third- and fifth-order equations can be converted into a non-pulsating soliton solution that does not have an analogue in the standard NLS equation. Wang et al. [43,44] have found that the breather solutions of the fourth-order NLS and variable-coefficient Hirota equations can be converted into four types of nonlinear waves on constant backgrounds, including the multi-peak solitons, antidark soliton, periodic wave and W-shaped soliton. Liu et al. [45,46] have discovered that state transitions between the Peregrine rogue wave and W-shaped travelling wave in the Hirota as well as the coupled Hirota equations appear as a result of higher-order effects. For the Sasa–Satsuma equation with the third-order dispersion, self-steepening and delayed nonlinear response, the high-order effects could make the rogue waves twisted [47,48]. Zhao et al. [49] have further reported a striking dynamical process where two W-shaped solitons are generated from a weak modulation signal on the continuous wave backgrounds for the Sasa–Satsuma equation. The striking process involves both MI and modulation stability (MS) regimes, in contrast to the rogue waves and W-shaped soliton reported before which involve MI and MS [49], respectively. Indeed, with certain higher-order perturbation terms such as the third-order dispersion and delayed nonlinear response term, the MI growth rate shows a non-uniform distribution characteristic in the low perturbation frequency region, in particular, it opens up a stability region as the background frequency changes [45,46].

In this paper, we study the higher-order generalized NLS (HGNLS) equation [50],

$$\mathrm{i}{q}_t+q_{xx}+2q|q{|}^2+\gamma (q_{xxxx}+6q_x^2{q}^{\ast }+4q|q_x{|}^2+8|q{|}^2{q}_{xx}+2q^2{q}_{xx}^{\ast }+6|q{|}^{4}q)=0,$$ 1.1

which is also called the Lakshmanan–Porsezian–Daniel equation. Hereby, q(x,t) denotes the complex envelope and γ stands for the strength of higher-order linear and nonlinear effects. The HGNLS equation emerges very often in various physical problems such as in the molecular systems, nonlinear optics and fluid dynamics. In a long-distance and high-speed optical fibre transmission system, equation (1.1) models the propagation of ultrashort optical pulses with the fourth-order dispersion, cubic-quintic nonlinearity, self-steepening and self-frequency shift [51,52]. In order to extend the Davydov model for the energy transfer of local vibrational modes in α-helical proteins [53,54], the effects of higher-order molecular excitations [55] should be considered. The result of such a generalization leads to the study of equation (1.1). In addition, equation (1.1) can also describe the nonlinear spin excitations in one-dimensional isotropic biquadratic Heisenberg ferromagnetic spin with the octupole–dipole interaction [50,56–59]. Porsezian et al. [58] and Zhang et al. [60] have investigated the Painleve property, derived the Lax pair, discussed the infinite conservation laws and obtained some soliton solutions for equation (1.1). The rogue-wave solutions of equation (1.1) have been derived via the modified Darboux transformation (mDT) in [39]. Guo & Hao [61] have presented breathers and multi-soliton solutions of equation (1.1). Breather-to-soliton transitions, nonlinear wave interactions and MI of equation (1.1) have been studied in [43]. Compared with the standard NLS equation, the previous studies have shown that equation (1.1) has certain additional properties. The higher-order effects could make the rogue waves compressed [39] and convert the breathers into other types of nonlinear waves [43]. Moreover, a linear stability analysis is performed to identify that the MI growth rate shows a non-uniform distribution characteristic in the low perturbation frequency region due to the higher-order effects [43]. Further, an interesting question is, what will happen in the nonlinear stage of MI with the higher-order terms for equation (1.1)?

In this paper, our first aim is to show that superregular breather solutions exist in the NLS equation in the presence of higher-order effects. This type of solution starts with infinitesimally small localized perturbation and develops into a nonlinear superposition of pairs of quasi-ABs. Then, we focus on an interesting mechanism where a small localized perturbation can develop into three different types of states in the nonlinear stage of MI: (i) coexistence of a quasi-AB and a multipeak soliton; (ii) two stable multipeak solitons in opposite directions; and (iii) a beating structure followed by two stable multipeak solitons in the same direction. These patterns do not have the analogues in the standard NLS equation. Our results reveal the characteristics of the nonlinear stage of MI induced by higher-order effects.

2. Superregular breather solutions

In this section, based on the theory proposed by Zakharov & Gelash [36,37], we will construct the superregular breather solutions of equation (1.1) on a plane wave solution with the form q₀=c e^i(ax+bt), where c, a and b represent the amplitude, wave number and frequency, respectively. We first perform the Jukowsky transform to the eigenvalue λ. By using the following transformation:

$$\mathrm{\lambda }=-\mathrm{i}\frac{c}{2}(\xi +\frac{1}{\xi })-\frac{a}{2},\xi =R\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}\alpha },$$ 2.1

the plane of λ is mapped onto the outer-ring of the circle of unit radius. Here, the parameters R and α stand for the radius and angle, and define the location of the eigenvalue λ. The graphical explanatory of such transformation is depicted in figure 1. Consequently, the general two-breather solution can be described by four polar parameters, R_1,2 and α_1,2, instead of λ_Re1,2 and λ_Im1,2. The amplitude of the perturbation is controlled by the difference of R_1,2 from unity. When R₁ and R₂ are close to 1, the quasi-annihilation occurs, i.e. two quasi-ABs almost annihilate at the moment of collision with the formation of a small perturbation of the plane wave. It should be pointed out that the solutions in this paper are basically the same as those of Wang et al. [43] in essence except for the expression ways.

Figure 1.

(a,b) Uniformization of the plane of spectral parameter with the help of Jukowsky transform. (Online version in colour.)

By using the Darboux transformation of equation (1.1) [39,61] and the above transformation (2.1), we present the first-order breather solution with four real parameters R, α, μ and θ as follows:

$$q_B^{[1]}=c(1-2(R+\frac{1}{R})\cos \alpha \frac{{\psi }_1{\psi }_2^{\ast }}{|{\psi }_1{|}^2+|{\psi }_2{|}^2}){\mathrm{e}}^{\mathrm{i}\rho },$$ 2.2

with

$$\begin{array}{rl}\rho & =ax+bt,\\ b& =(a^4-12a^2{c}^2+6c^4)\gamma +2c^2-a^2,\\ {\psi }_1& =\frac{{\mathrm{e}}^{A-\mathrm{i}\alpha }}{R}+{\mathrm{e}}^{-A},{\psi }_2=\frac{{\mathrm{e}}^{-A-\mathrm{i}\alpha }}{R}+{\mathrm{e}}^A,A=hx+\omega t+\frac{1}{2}(\mu -\mathrm{i}\theta ),\\ h& =h_R+\mathrm{i}{h}_I=\frac{c}{2}(R-\frac{1}{R})\cos \alpha +\mathrm{i}\frac{c}{2}(R+\frac{1}{R})\sin \alpha ,\\ \omega & =h({\omega }_R+\mathrm{i}{\omega }_I)=h(2{\mathrm{\lambda }}_1-a+\gamma [a(a^2-6c^2)-8{\mathrm{\lambda }}_1^3+4a{\mathrm{\lambda }}_1^2+{\mathrm{\lambda }}_1(4c^2-2a^2)],\\ {\mathrm{\lambda }}_1& =-\frac{a}{2}+\frac{c}{2}(R-\frac{1}{R})\sin \alpha -\mathrm{i}\frac{c}{2}(R+\frac{1}{R})\cos \alpha .\end{array}$$

Hereby, the parameters θ, μ, respectively, define the location and phase of the breather, and play an important role in the general N-breather solution.

The parameters R and α are the polar coordinates of the point in the area R≥1 and −π/2<α<π/2 and determine the breather types. In the Kuznetsov case, R>1 and α=0, while R=1 and α≠0 correspond to the Akhmediev case. The AB becomes the PS in the limit $\alpha \to 0$. The carrying wave moves with the phase velocity V_P=(h_Rω_I/h_I)+ω_R while the envelope moves with the group velocity V_G=ω_R−h_Iω_I/h_R. The oscillation period is π/(h_Rω_I+h_Iω_R). To obtain the quasi-AB solution, we assume the parameter R=1+ε, where ε is a small parameter. Figure 2 demonstrates the intensity distribution (I=|q|²) of a quasi-AB with the parameters R=1.13 and α=0.6. In general, the N-breather solution includes the similar four parameters, R_i, α_i, μ_i, θ_i, so that the total solution is described by 4N parameters. Here, we present the one-pair breather solution analytically. For simplicity, we set R₁=R₂=1+ε, α₁=−α₂=α (figure 3). In this symmetrical case, the expression of the second-order breather solution is given by

$$q_B^{[2]}=c(1-\mathrm{i}(R^2-\frac{1}{R^2})\sin 2\alpha \frac{N_R+\mathrm{i}{N}_I}{\mathrm{\Delta }}){\mathrm{e}}^{\mathrm{i}\rho },$$ 2.3

with

$$\begin{array}{rl}{N}_R& =(R+\frac{1}{R})\cos \alpha [(|{\psi }_2{|}^2-|{\psi }_1{|}^2){\psi }_3{\psi }_4^{\ast }+(|{\psi }_3{|}^2-|{\psi }_4{|}^2){\psi }_1{\psi }_2^{\ast }],\\ N_I& =-(R-\frac{1}{R})\sin \alpha [(|{\psi }_2{|}^2+|{\psi }_1{|}^2){\psi }_3{\psi }_4^{\ast }+(|{\psi }_3{|}^2+|{\psi }_4{|}^2){\psi }_1{\psi }_2^{\ast }],\\ \mathrm{\Delta }& ={\sin }^2\alpha {(R-\frac{1}{R})}^2(|{\psi }_1{|}^2|{\psi }_3{|}^2+|{\psi }_2{|}^2|{\psi }_4{|}^2)+4{\cos }^2\alpha (|{\psi }_1{|}^2|{\psi }_4{|}^2+|{\psi }_2{|}^2|{\psi }_3{|}^2)\\ & -{\cos }^2\alpha {(R+\frac{1}{R})}^2({\psi }_1{\psi }_4{\psi }_2^{\ast }{\psi }_3^{\ast }+{\psi }_2{\psi }_3{\psi }_1^{\ast }{\psi }_4^{\ast })+{(R-\frac{1}{R})}^2(|{\psi }_1{|}^2|{\psi }_4{|}^2+|{\psi }_2{|}^2|{\psi }_3{|}^2),\\ {\psi }_1& =\frac{{\mathrm{e}}^{A_1-\mathrm{i}\alpha }}{R}+{\mathrm{e}}^{-A_1},{\psi }_2=\frac{{\mathrm{e}}^{-A_1-\mathrm{i}\alpha }}{R}+{\mathrm{e}}^{A_1},A_1=hx+\omega t+\frac{1}{2}({\mu }_1-\mathrm{i}{\theta }_1),\\ {\psi }_3& =\frac{{\mathrm{e}}^{A_2+\mathrm{i}\alpha }}{R}+{\mathrm{e}}^{-A_2},{\psi }_4=\frac{{\mathrm{e}}^{-A_2+\mathrm{i}\alpha }}{R}+{\mathrm{e}}^{A_2},A_2=h^{\ast }x+{\omega }^{†}t+\frac{1}{2}({\mu }_2-\mathrm{i}{\theta }_2),\\ {\omega }^{†}& =h^{\ast }({\omega }_R^{†}+\mathrm{i}{\omega }_I^{†})=h^{\ast }(2{\mathrm{\lambda }}_2-a+\gamma [a(a^2-6c^2)-8{\mathrm{\lambda }}_2^3+4a{\mathrm{\lambda }}_2^2+{\mathrm{\lambda }}_2(4c^2-2a^2)]).\end{array}$$

Figure 2.

Intensity distribution (I=|q|²) of a first-order quasi-AB with the parameters R=1.13, α=0.6, c=1 and γ=0.1. (Online version in colour.)

Figure 3.

Uniformization of symmetrical eigenvalue parameters of second-order breather with the help of Jukovsky transform with the same small parameter ε and symmetrical angle α. (Online version in colour.)

In expression (2.3), the small parameter ε is response for the amplitude of the perturbation at the moment of breather quasi-annihilation. The N pairs of quasi-ABs almost annihilate to a small localized perturbation as a result of their collision. This process is called as ‘quasi-annihilation’. The parameters μ_1,2 and θ_1,2 characterize the phase shifts in space and time and between the two breathers. As pointed by Zakharov & Gelash [36,37], such phase shifts have an effect on the shape and the amplitude of the perturbation. We can observe the breather quasi-annihilation if θ₁+θ₂ approximates to π. The difference between θ₁+θ₂ and π determines the complexity of the wave profile at the area of interaction. To obtain the superregular breather solution, we need to make the perturbation small enough.

Figure 4a–c, respectively, describes three types of interactions between two breathers (R=1.13, α=0.6, μ_1,2=0 for all cases), i.e. the ghost collision (θ_1,2=0), synchronized collision (θ_1,2=π) and quasi-annihilation (θ_1,2=π/2). All these interactions have been also presented in the standard NLS equation [36,37]. Especially, the second one has been investigated in optics experiments [62], by controlling the phase and velocity differences between the breathers. Our focus is the third one, namely, the quasi-annihilation of two superregular breathers. The reverse process of this case describes the nonlinear stage of MI in which a small localized perturbation develops into a pair of quasi-ABs propagating with a very fast group velocity in opposite directions, as shown in figure 5a. Very recently, Kibler et al. reported the first successful experimental evidence of such superregular breather waves in an optical fibre and in a water-wave tank [38]. Therefore, it is expected that the superregular breather solutions represent a significant step towards a global understanding of the nonlinear stage of MI in diverse different physical fields.

Figure 4.

Intensity distribution (I=|q|²) of three modes of second-order breather solution (R=1.13, α=0.6, μ_1,2=0, c=1, a=0.5 and γ=0.1) with the following phase shifts: (a) ghost case: θ_1,2=0, (b) synchronized case: θ_1,2=π, (c) superregular breather case: θ_1,2=π/2. Note that the intensities at central position (q^[2](0,0)|²) are (a) 5.41, (b) 18.71, (c) 1.16, respectively. (Online version in colour.)

Figure 5.

(a) The development of the quasi-annihilation of breathers with the same parameters as in figure 4c. Solid line is small perturbation at the moment t=0. Dashed line is the breathers solution at the moment t=4. (b) is the enlarged small perturbations presented in (a). (Online version in colour.)

Let us represent further the solution (2.3) as a condensate solution plus small deviation: $q_B^{[2]}=(c+\delta q)\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}\rho }$. The deviation δq is calculated in variable λ by the expression

$$\delta q=-c\mathrm{i}(R^2-\frac{1}{R^2})\sin 2\alpha \frac{N_{\mathrm{\lambda }}}{{\mathrm{\Delta }}_{\mathrm{\lambda }}}.$$ 2.4

Here N_λ is a variation of the numerator while Δ_λ is a modified version of the denominator calculated with higher accuracy. By assuming |ε|≪1, we further consider the approximate modified function of the deviation δq. Firstly, we pay close attention to spatial term and neglect γ because we find that the higher-order effect has no effect on the perturbation as the time changes. Note that the eigenvalue is simplified to

$${\mathrm{\lambda }}_{1,2}\approx -\frac{a}{2}\pm c\epsilon \sin \alpha -\mathrm{i}c\cos \alpha .$$ 2.5

By analogy

$$\delta q\approx -2\mathrm{i}c\epsilon \sin 2\alpha \frac{N_{\mathrm{\lambda }}}{{\mathrm{\Delta }}_{\mathrm{\lambda }}},$$ 2.6

with

$$\begin{array}{rl}{N}_{\mathrm{\lambda }}& =2\cos \alpha [(|{\psi }_2{|}^2-|{\psi }_1{|}^2){\psi }_3{\psi }_4^{\ast }+(|{\psi }_3{|}^2-|{\psi }_4{|}^2){\psi }_1{\psi }_2^{\ast }]\\ & -\mathrm{i}\epsilon \sin \alpha [(|{\psi }_2{|}^2+|{\psi }_1{|}^2){\psi }_3{\psi }_4^{\ast }+(|{\psi }_3{|}^2+|{\psi }_4{|}^2){\psi }_1{\psi }_2^{\ast }],\\ {\mathrm{\Delta }}_{\mathrm{\lambda }}& =-4{\cos }^2\alpha ({\psi }_1{\psi }_4{\psi }_2^{\ast }{\psi }_3^{\ast }+{\psi }_2{\psi }_3{\psi }_1^{\ast }{\psi }_4^{\ast }-|{\psi }_1{|}^2|{\psi }_4{|}^2-|{\psi }_2{|}^2|{\psi }_3{|}^2).\end{array}$$

In what follows in this paragraph, we put μ_1,2=0, which corresponds to solitons colliding at the point (0,0). Then the exponential powers A₁ and A₂ in the potential function ψ_1,2,3,4 of N_λ and Δ_λ at t=0 have the following form:

$$A_1\approx (c\epsilon \cos \alpha +\mathrm{i}c\sin \alpha )x-\frac{\mathrm{i}}{2}{\theta }_1\mathrm{and}{A}_2\approx (c\epsilon \cos \alpha -\mathrm{i}c\sin \alpha )x-\frac{\mathrm{i}}{2}{\theta }_2.$$ 2.7

When θ₁+θ₂=π, the simplest case of the perturbation δq in terms of trigonometric and hyperbolic functions is given by

$$\delta q\approx 4\mathrm{i}c\epsilon \frac{\cos \alpha \cos (2cx\sin \alpha -({\theta }_1-{\theta }_2)/2)}{\cosh (2c\epsilon x\cos \alpha )}.$$ 2.8

The small parameter ε has two functions to the perturbation: decreasing ε results in the increasing of the temporal width of the initial perturbation while the decreasing of its amplitude. The perturbation is localized in space and contains many oscillations, which is plotted in figure 5b.

In spite of the higher-order effects, we can still derive the superregular breather solutions which have the similar dynamic features as those in the standard NLS equation. In particular, we find that the higher-order effects have no effect on the form of the perturbation. However, these effects can lead to the state transition between the soliton and breather, which will give rise to some new features of nonlinear stages of MI.

3. Characteristics of superregular breathers

In this section, we shall study the characteristics of superregular breathers induced by higher-order effects, which could describe the nonlinear stage of MI in the presence of higher-order effects. As shown in [41–46], the ABs, KMBs or rogue waves can be converted into the stable solitons on constant backgrounds in the higher-order NLS equations, which does not have an analogue in the standard NLS equation. The linear stability analysis indicates that such conversions are strictly associated with the MI analysis that involves an MI region and a stability region, i.e. the higher-order perturbation term opens up a stability region as the background frequency changes. Similarly, adding the higher-order effect also leads to the state transition between the soliton and superregular breather. Based on Zakharov’s theory [38], the superregular breather describes the nonlinear stage of the MI of the condensate. Thus, we shall demonstrate some new features of the nonlinear stage of MI in the presence of the higher-order effects.

In [43], we presented the state transition equation for equation (1.1), namely,

$${\mathrm{\lambda }}_{\mathrm{Im}}^2\gamma =\frac{1}{4}(a^2-2c^2-4a{\mathrm{\lambda }}_{\mathrm{Re}}+12{\mathrm{\lambda }}_{\mathrm{Re}}^2)\gamma -\frac{1}{4},$$ 3.1

where λ_Re and λ_Im denote, respectively, the real and imaginary parts of eigenvalue λ, a and c are the wavenumber and amplitude of the plane wave, and γ describes the higher-order effects. Equation (3.1) is a necessary condition to enable the state transition between breather and soliton. Especially, we should point out that the value of the higher-order γ must not be zero (if γ=0, equation (3.1) will lead to a conflicting result). This shows that such transition cannot exist in the standard NLS equation because it does not include the higher-order effect. In equation (3.1), we note that the parameters a, c and γ are fixed in the plane-wave solution while λ_Re and λ_Im are free parameters to be adjusted. In order to give rise to different types of transformed solitons, one can choose the suitable values of λ_Re and λ_Im. We have exhibited five kinds of transformed waves that correspond to different λ_Re and λ_Im in [43], including the multipeak soliton, W-shaped soliton and periodic wave.

After the Jukowsky transformation, the parameter R and α control the main breather properties. Thus, instead of λ_Re and λ_Im, we have to use R and α to describe equation (3.1) that is the necessary condition for the state transition between the superregular breather and soliton. The one-pair superregular solution can be generated from the general two-breather solution (2.3). We note that the parameters (R_j,α_j) in the solution (2.3) are independent of each other for j=1,2. This gives us the flexibility to implement the conversion. We first consider the parameter condition of one-pair superregular breathers as follows:

$$R_1=R_2=R=1+\epsilon \mathrm{and}{\alpha }_1=-{\alpha }_2=\alpha .$$ 3.2

In this case, the two sets of parameters (R_j,α_j) are related to each other for j=1,2. Furthermore, the state transition equation in polar coordinates is expressed by

$$\gamma =\frac{R^2}{c^2(1+R^4)+6(a^2-c^2)R^2-2c^2(1-R^2+R^4)\cos 2\alpha -8caR(R^2-1)\sin \alpha }.$$ 3.3

The above equation is another description of equation (3.1). It is easy to check that equation (3.3) is equivalent to

$$V_{\mathrm{P}1}=V_{\mathrm{G}1},$$ 3.4

which means the conversion occurs when the phase velocity is equal to the group velocity. With equations (3.3) and (3.4) satisfied, we find that only one breather can be transformed into a multipeak soliton while the other remains the character of oscillation. Meanwhile, we also keep the small localized perturbation at the moment of collision.

Figure 6a describes the scenario where an initially small localized perturbation of the condensate gives rise to a multi-peak soliton and a breather propagating in opposite directions. Different from the one-pair breather solution in figure 5, the combination of a breather and soliton characterizes the nonlinear stage of MI in the NLS equation with the higher-order effects, as depicted in figure 6b. The oscillatory behaviour of one of the breathers is completely suppressed. Instead, it is observed that a stable soliton propagates with constant amplitude. On the other hand, it also describes the quasi-annihilation of a breather and a soliton. At first a small localized perturbation grows exponentially, which is described by the well-known equations for the linear stage of MI. Then the linearization ceases to be valid as soon as the perturbations have grown to the point that they are of comparable size compared to the background. However, the subsequent dynamics is different from the case in NLS equation where the higher-order terms are absent. Instead of a pair of breathers, we observe a mixed state of a breather and a soliton in the nonlinear stage of MI for equation (1.1). The higher-order effects really add novel characteristic to the nonlinear stage of MI, which cannot be found in the standard NLS equation. Typical development of a small localized perturbation of the condensate is shown in figure 6c.

Figure 6.

The quasi-annihilation of a multipeak soliton and a breather with R₁= R₂=1.13, α₁=−α₂=0.6, μ_1,2=0, θ_1,2=π/2, c=1, a=0.5 and γ=−0.266. (b) The cross-sectional view of (a) at t=0 and t=30. (c) The enlarged small perturbation. (Online version in colour.)

Next, we assume that two sets of eigenvalue parameters (R_j,α_j) have different values, namely,

$$R_1=1+{\epsilon }_1,R_2=1+{\epsilon }_2,{\epsilon }_1\ne {\epsilon }_2,{\alpha }_1\ne -{\alpha }_2.$$ 3.5

The corresponding positions of these two points are plotted in figure 7. In this case, two eigenvalue parameters of breathers are independent of each other. Thus, we can realize the state transition between solitons and breathers completely if the parameters R_1,2, α_1,2 and γ meet the state transition equations as follows:

$$\begin{array}{rl}\gamma & =\frac{R_j^2}{c^2(1+R_j^4)+6(a^2-c^2)R_j^2-2c^2(1-R_j^2+R_j^4)\cos 2\alpha -8ca{R}_j(R_j^2-1)\sin \alpha },j=1,2.\end{array}$$ 3.6

For the fixed values of a and γ, one can choose two different sets of values of (R_j,α_j) to satisfy the above equations. Then both breathers can be converted into the stable multipeak solitons. This is different from the previous case for the associated eigenvalues (see equation (3.2)).

Figure 7.

The eigenvalue parameters is asymmetrical when ε₁≠ε₂ and α₁≠−α₂. (Online version in colour.)

As shown in figure 8, it is found that a small localized perturbation first grows and then evolves into two stable multipeak solitons in different directions. Under condition (3.6), the oscillating behaviour of the breathers is suppressed completely. Therefore, the nonlinear stage of MI is described by the two-soliton state. The reverse process can be viewed as the quasi-annihilation of two multipeak solitons on constant backgrounds. From figures 4, 6 and 8, it is concluded that the nonlinear stage of MI can involve different types of nonlinear wave structures in the framework of higher-order NLS equations, including the two breathers, the coexistence of the breather and soliton, and two stable solitons. Our result is beyond the MI of the standard NLS equation [36] and enriches the connotation of the nonlinear stage of MI.

Figure 8.

The quasi-annihilation of two multipeak solitons propagating in opposite directions with R₁=1.05, R₂=1.1, α₁=0.5, α₂=−0.501, μ_1,2=0, θ_1,2=π/2, c=1, a=0.01 and γ=−0.197. (b) is the cross-sectional view of (a) at t=0 and t=2200. (c) is the enlarged small perturbation. (Online version in colour.)

To end this section, we exhibit a type of partially coherent interaction that involves different kinds of nonlinear wave patterns successively. In [41], one can observe the overlap of two breathers with the same group velocity, i.e. the beating pattern. This means that we can study the interactions among different types of nonlinear waves by controlling the wave velocities. Hereby, we use the plane-wave wave number a to control the velocities of two transformed solitons. Instead of the same group velocity that results in the full-coherent interaction, we only consider two transformed solitons having the similar velocities. Meanwhile, we remain the small perturbation at the moment of collision. Then, we will observe some novel characteristics of the nonlinear stage of MI in equation (1.1).

In order to convert the superregular breathers into two solitons with the similar velocities, we set ε₁=0.08, ε₂=0.03, α₁=0.6, α₂=−0.472 and a=0.5 in figure 9a. Interestingly, it is found that a small localized perturbation experiences two major phases of developments in the nonlinear stage of MI. In the first stage, the small localized perturbation evolves into a type of breath-type structure (beating pattern) which is clearly presented in figure 9b. This structure is different from some known breathers such as the AB, KMB and superregular breather since it has oscillating tails and each unit has different shapes. In addition, the beating pattern is short-lived. As shown in figure 9c, the breath-type structure begins to split and then divides into two multipeak solitons at the branch points (BPs). In these points, the central peaks of two solitons begin to form. It is difficult to find the BPs analytically. Thus, we give coordinates for these points numerically, i.e. (t,x)=(±111.98,∓169.64). The formation of these two solitons is caused by the higher-order effects and they propagate with constant velocities and amplitudes (figure 9d). To exhibit the global scenario of the perturbation evolvement, we plot figure 10a from where a small localized perturbation develops and goes through two stages. Namely, the nonlinear stage of MI in this case includes two entirely different patterns in succession, i.e. a short-time unstable beating stage and a long-time stable multi-peak soliton stage. This new feature is also caused by the higher-order effects. We show the initial perturbation in a larger scale in figure 10b. It should be pointed out that we do not produce two stable solitons in opposite directions due to the larger localized perturbation at the moment of the solitons’ collision.

Figure 9.

(a) The quasi-annihilation of two multipeak solitons (they have similar velocities) propagating in the same directions with R₁=1.08, R₂=1.03, α₁=0.6, α₂=−0.472, μ_1,2=0, θ_1,2=π/2, c=1, a=0.5 and γ=−0.280. (b) The beating pattern. (c) The BPs. (d) Two stable multipeak solitons. (Online version in colour.)

Figure 10.

(a) The global scenario of development of a small localized perturbation. (b) The enlarged small perturbation. A small localized perturbation experiences two major phases of developments, i.e. the beating pattern and two multipeak soliton state. (Online version in colour.)

4. Conclusion

We have derived the superregular breather solutions of the HGNLS equation via the Darboux transformation. These solutions propagating in opposite directions start with infinitesimally small localized perturbations and describe the nonlinear stage of the MI of the condensate. Owing to the existence of the higher-order effects, we have successfully converted the superregular breathers into the stable solitons. We have found that the small localized perturbation can also develop into the other three types of patterns: (i) the coexistence of a quasi-AB and a multipeak soliton; (ii) two stable multipeak solitons in opposite directions and (iii) a beating structure followed by two stable multipeak solitons in the same direction. These structures have no analogues in the standard NLS equation. Our results have shown that the higher-order effects can provide some novel features to the nonlinear stage of MI.

Data accessibility

All the mathematical results are in analytic form and are reproducible.

Authors' contributions

L.W. and C.L. put forward the idea of this paper. L.W. contributed all mathematical calculation and physical analysis. L.W. wrote the paper. C.L. supplemented some physical analysis, modified the figures and polished the language. J.-H.Z. generated all the figures and was responsible for all simulations.

Competing interests

We declare we have no competing interests.

Funding

This work has been supported by the National Natural Science Foundation of China under grant no. 11305060, and by the Fundamental Research Funds of the Central Universities (grant no. 2015ZD16).