Akhmediev Breathers and Kuznetsov–Ma Solitons in the Cubic-Quintic Nonlinear Schrödinger Equation

Finite background solitons are a significant area of research in nonlinear dynamics, as they are commonly found in various complex physical systems. Understanding how finite background solitons are generated and determining the conditions required for their excitation is crucial for detecting and applying dynamic characteristics. We used the Darboux transformation method to obtain explicit analytical solutions for the Akhmediev breather, the Kuznetsov-Ma soliton, and the Peregrine soliton of the cubic-quintic nonlinear Schrödinger equation. This equation is typically used as a model to control the propagation of ultrashort pulses in highly nonlinear optical fibers. We also provide the conditions required for the existence of these different breather solutions and discuss their interesting dynamical properties, such as oscillation period, propagation direction, and peak amplitude. We systematically discuss the excitation conditions and phase diagrams of the breathers by analyzing modulation instability. These results and associated formulas can also be extended to vector or multi-component systems, the breathing dynamics of which remain to be explored.

as transient wave-packets that keep oscillatory on a nonzero background.While conventional solitons have been intensively studied over more than 50 years, only recently did the SFB draw great attention, because the latter can, to some extent, depict the mysterious and unpredictable rogue wave events occurring in the real world [12], [13], [14], [15], [16].
Peregrine soliton, named after Howell Peregrine who first constructed the rational soliton in 1983 [17], is an example of such deterministic SFB.It features a doubly-localized peak and two side troughs on a finite background [1].This bizarre spatiotemporal structure is particularly suitable for explaining the fleeting nature of realistic rogue waves and thus has generally been considered as the prototype of rogue waves [18].To date, the concept of Peregrine soliton has been extended to vector systems [19], [20], [21], [22], [23] and was experimentally confirmed in different settings including the water-wave tank [24], multi-component plasmas [25], optical fibers [12], [26], and irregular oceanic sea states [27], to name a few.Besides, there are also intensive researches on two other types of SFB, namely, the Akhmediev breathers [28] and the Kuznetsov-Ma (KM) solitons [29], [30].It is now widely accepted that the excitation of nonlinear waves (Peregrine soliton, Akhmediev breather, and KM soliton) on a plane-wave background relies on the modulation instability (MI) of the system [28], [31], [32], and, as a classical MI analytical tool, the linear stability analysis [33] can intuitively observe the MI's spectral range and perturbation gain.Recently, Li-Chen Zhao et al. established a quantitative correspondence between fundamental nonlinear wave excitations and MI gain distributions by systematically analyzing the distributions of MI in nonlinear Schrödinger (NLS) systems in correspondence with fundamental nonlinear waves on a planewave background [34].Since Akhmediev breathers and KM solitons are closely related to the formation of extreme wave events in the real world [11], [35], these breather states have attracted much attention in various fields such as optics and fluid dynamics [36], [37], [38], [39].
In the previous work of our group [40], the nth-order rogue wave solution, and particularly its explicit solution forms up to the third order, was presented for the first time by utilizing the non-recursive DT technique.It were found that super rogue waves with peak amplitudes up to 2n + 1 times the background level, and super rogue waves involve a frequency chirp that is also localized in both time and space.However, the question as to what form the allowed analytical Akhmediev breathers and KM solitons take in the cubic-quintic (CQ) NLS equation is still open.In this work, we are dedicated to addressing this issue, with an emphasis on the explicit analytic solutions of the breather, and discuss their intriguing dynamics as well as the parametric conditions for such dynamics.
In this paper, we derive a general breather solution of the CQ-NLS equation.The latter is a generalization of several previously known partial differential equations used in mathematical physics.The new solution covers all particular breather solutions of these equations.The solution is illustrated in Figs.1-5 for special values of parameters involved in the solution.In Section II, we report the theoretical model and basic framework.In Section III, we presented the finite background soliton solutions (KM solitons, Akhmediev breathers, and Peregrine solitons) and dynamic characteristics analysis for degenerate and non-degenerate cases, respectively, and discussed the generation mechanism and phase diagram of these nonlinear waves.This paper is concluded in Section IV.

II. THEORETICAL FRAMEWORK
The propagation of ultrashort pulses in highly nonlinear optical fibers can be modeled by the following dimensionless CQ-NLS equation where u(z, t) represents the normalized complex envelope of an optical pulse, z and t stand for propagation distance and retarded time, respectively, and the subscripts z and t denote partial derivatives.The coefficient of 1/2 represents the group-velocity dispersion (GVD) effect, while σ denotes the Kerr nonlinear coefficient that can typically be normalized to −1, 0, or 1.The coefficient γ represents the pulse self-steepening effect (we assume γ ≥ 0 without loss of generality) [41].To ensure integrability, we use two independent real parameters μ and γ that are related to the effects of self-steepening, nonlinear dispersion, and quintic nonlinearity.μ is associated with the nonlinearity dispersion that can cause a self-frequency shift if it is complex [41], while the quintic nonlinearity is commonly found in highly nonlinear materials such as chalcogenide glasses [42] and organic polymers [43].In the context of fiber optics, the term |u| 2 u in (1) is often referred to as self-phase modulation.In this case, the coefficient σ can be scaled out to the GVD term, which is termed anomalous dispersion if σ > 0 and normal dispersion if σ < 0 [9], [33].To evaluate the impact of nonlinearity factors on the breather dynamics, we have excluded higher-order dispersion terms from (1).These terms typically appear in the high NLS equation hierarchy [44].By doing so, we can analyze (1) using the Painlevé method, and make it completely integrable [45].This equation can be simplified into a series of complete integrable equations, such as the well-known standard NLS equation (μ = γ = 0), the Chen-Lee-Liu (CLL) type NLS equation (μ = γ = 0) [46], the Kaup-Newell (KN) type NLS equation (μ = 2γ) [47], the Gerdjikov-Ivanov (GI) equation (μ = 0) [48], and the Kundu-Eckhaus (KE) equation (γ = 0) [49].Researchers have extensively explored the rogue wave solutions of the latter equations [50], [51], [52] and the generalized derivative NLS equations [53], [54], [55] in recent years.This CQ-NLS equation can model the propagation of ultrashort pulses in single-mode long-range fibers [40] or in quadratic nonlinear media [56], taking into account group-velocity mismatch.
We can use the DT method to solve the breather solutions of (1).Usually, we transform (1) into a linear eigenvalue problem, with the Lax pair being where R = [r, s] T is the eigenvalue function (T represents a matrix transpose, r, and s are the functions of z, t, and complex spectral parameter λ).The matrices U and V are given by the following equation with σ 3 = diag(1, −1), and For any spectral parameter λ, the compatibility condition of this Lax pair is , which can easily be deduced to (1).
We take the following form as the initial plane-wave solution the amplitude a, wave number k, and frequency ω follow the dispersion relationship According to the literature [40], the exact fundamental firstorder solution can be obtained Therefore, it is only necessary to solve the expressions of r and s based on ( 2) and ( 3), and then bring them into (6) to solve the breather solutions.However, in direct solving, it is not Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
convenient to separate the solution into real and imaginary parts.So, we need to transform the matrices U and V into a constant form, to effectively simplify the breather solutions.
First of all, one matrix S is introduced to convert the matrices U and V into constant matrices U and V .Then the Lax pair is by combining ( 2) and ( 7), the matrices U and V can be obtained as with the selected matrix S being therefore, the expressions of U and V can be obtained at this moment, for any spectral parameter λ, this Lax pair also satisfies the compatibility condition Next, we can further diagonalize the constant matrices U and V .However, at this point, we need to consider whether the matrix U has equal roots, so that we can convert the matrix U into a diagonal matrix or a Jordan matrix.By Det[U − τ E] = 0 (E is the identity matrix of 2 × 2), the eigenvalue equation of the constant matrix U can be solved, and the specific expression is as follows According to the above equation, two roots of the eigenvalue equation can be obtained as . Therefore, we need to consider the cases of non-equal root and equal root to correspondingly transform the constant matrix U into a diagonal matrix and a Jordan matrix, respectively.

A. Non-Degenerate Case
We first discuss the case of non-equal roots, i.e., τ 1 = τ 2 .To convert constant matrices U and V into diagonal matrices U d and V d , we introduce one matrix D and apply D −1 simultaneously on both sides of (7), which yields the diagonal matrix U d and V d are here we set D −1 SR = R (where R = [r , s ] T , r , and s are also functions of z, t, and complex spectral parameter λ), then the Lax pair after diagonalization is the selected matrix D has the following form it should be noted that the matrix D is arbitrary, and thus the diagonal matrix U d and V d can be solved as according to ( 14) and ( 16), r and s can be solved as substituting ( 18) into (6) and simplifying yields the exact fundamental first-order solution on a plane-wave background the expressions for H 1 , H 2 , D 1 and D 2 are all real functions and other parameter expressions are as follows Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
It should be noted that during the above solving process, we set the complex spectral parameter λ = c 1 + iω 1 /2 (where c 1 is any real number and ω 1 is a non-zero real number), so when a, ω, c 1 , ω 1 , σ, γ and μ are confirmed, the exact solution of the CQ-NLS equation can be obtained by (19), which includes KM solitons, Akhmediev breathers, and Peregrine solitons.At the same time, it was found that there is a peak amplitude at the origin, expressed as 1 + 2c 1 /aγ.Undoubtedly, this solution includes the NLS equation (μ = γ → 0), the KN equation (μ = 2γ), the CLL equation (μ = γ), the GI equation (μ = 0), and the KE equation (γ → 0).However, it is difficult to directly obtain the corresponding patterns of KM solitons, Akhmediev breathers, and Peregrine solitons through assignment.Therefore, it is necessary to further analyze and obtain the explicit forms of KM soliton, Akhmediev breather, and Peregrine soliton solutions based on the above analytical solutions.Since Peregrine solitons and breathers are generated by the widely existing MI [11], [36], [57], we first analyzed the MI here.Small amplitude Fourier modes p 1 and q 1 are added to the plane wave solution (4), i.e., u Ω is a positive modulation frequency and β is a propagation parameter.Substitute of the above perturbated solution into (1) and linearization yields two coupled linear equations of p 1 and q 1 .Only when β satisfies the following dispersion relationship does the system have a non-trivial solution, This is the quadratic equation for β.In principle, as long as the β in ( 22) has an imaginary part, the plane wave solution (4) becomes unstable.Obviously, within the limit of Ω = 0, Peregrine soliton and breather solution can exist only when (22) satisfies the condition of a 2 γ(μ − γ) − η < 0.
Equation ( 19) is a combination of trigonometric and hyperbolic functions.The trigonometric function determines the periodicity, while the hyperbolic function determines the localization.Let ω 1 = 2 −a 2 μγ +c 2  1 +η, then we have According to the relationship obtained by the MI, when c 1 > aγ, then ℘ r = 0; when c 1 < aγ, it corresponds to ℘ i = 0, which further simplifies the exact solutions.
Therefore, when c 1 > aγ, ω 1 = 2 −a 2 μγ + c 2 1 + η, we have ℘ r = 0.The analytic form of KM solitons can still be expressed by ( 19) and ( 20), but the corresponding parameters (21) are simplified as Meanwhile, from Fig. 1, it is found that the peak amplitudes of the KM solitons are all larger than 3, which can be confirmed by the expression of the peak amplitude 1 + 2c 1 /aγ.Furthermore, when ω = −a 2 μ, ω 1 = 2 c 2 1 + σ, corresponding to V g = 0 in (23), it is easy to obtain KM solitons oscillating periodically in the direction of t = 0.
Correspondingly, when c 1 < aγ, ω 1 = 2 −a 2 μγ +c 2 1 +η, then ℘ i = 0, the analytic form of the Akhmediev breathers can also be expressed by ( 19) and ( 20), but the corresponding parameters ( 21) can be simplified to the following expressions The structures of the Akhmediev breather in the KN equation, the CLL equation, and the GI equation are presented in Fig. 2. According to (24) it can be found that the Akhmediev breathers oscillate in the t direction (it can also be seen in Fig. 2), and the local peak follows the direction of t − (a 2 μ + ω)z = 0, with the oscillation period being γπ/℘ r .The smaller the |c 1 |, the stronger the periodic oscillation.In addition, it can be noticed from Fig. 2 that the peak amplitudes of the Akhmediev breathers are all less than 3, which can also be confirmed from the expression for the peak amplitude  form is as follows where H 1 , H 2 , D 1 , and D 2 are all real functions, expressed by The relationship a 2 γ(μ − γ) − η < 0 obtained from MI indicates that there is no singular solution for D 1 > 0. As shown in Fig. 3, we present the Peregrine soliton structures for the KN equation, the CLL equation, and the GI equation.It is not difficult to see from Fig. 3 that this is a typical Peregrine soliton with local characteristics in the z and t directions, and its peak amplitude is equal to 3, which can be easily confirmed by the expression of peak amplitude 1 + 2c 1 /aγ.

B. Degenerate Case
Next, considering the degenerate case, i.e., τ 1 = τ 2 , the complex spectral parameter λ 2 = −a 2 μγ + η + 2ia γ η − a 2 γ(μ − γ) can be solved.The complex spectral parameter λ 2 can be parameterized through the Jukowsky transform [58], and its imaginary part can be transformed, where ξ = R exp(iα), with R and α being the radius and angle of the polar coordinate system in the sector defined by R ≥ 1, α ∈ (−π/2, π/2).It can be deduced that the real part m and the imaginary part of λ 2 are given by the following equations, respectively let λ = √ Λ ≡ m + i , i.e., Λ = m + i , then the exact fundamental first-order solution of the degenerate case can still be expressed by (19)   in (20) have changed to the other parameters are We found that the exact fundamental first-order solution of the degenerate case also has a peak amplitude at the origin with the expression a + 2 /γ, where = Im[( . The exact solution depends on the background parameters a, ω, σ, γ, and μ, as well as the radius R and the angle α in the polar coordinate system.Similar to the situation in the literature [59], the growth and decay of the periodic structure of plane-waves are described.Under specific spectral parameters, the exact solution can be degenerated into KM solitons (R = 1, α = 0), Akhmediev breathers (R = 1, α = 0), and Peregrine solitons (R = 1, α → 0 or R → 1, α = 0).As shown in Fig. 4, we also obtain the structures of KM soliton, Akhmediev breather, and Peregrine soliton solution in the KN equation, the CLL equation, and the GI equation, with the background parameters a = 1, γ = 1, σ = 1, and ω = −3/2.At this time, the KM soliton oscillates more violently with a larger R, while the Akhmediev breather vibrates more frequently with a larger α.The oscillation periods of KM soliton and Akhmediev breather are still 2π/|V p | and γπ/℘ r .It is easy to observe from Fig. 4 that the peak amplitudes of KM solitons are all greater than 3, the peak amplitudes of Akhmediev breather are all less than 3, and the peak amplitudes of Peregrine solitons are all equal to 3.This can also be confirmed by the expression a + 2 /γ for peak amplitudes.

C. The Generation Mechanism and Phase Diagram of Nonlinear Waves
From the quadratic equation of β in (22), it can be obtained that When β has an imaginary part, that is, the plane wave (4) becomes unstable in the region of −2a −a 2 γ(μ − γ) + η < Ω < 2a −a 2 γ(μ − γ) + η, from which the gain γ h = Ω|Im(β)| in any dispersion can be calculated.Fig. 5 shows the distribution of MI gain and nonlinear waves in the (a, Ω) plane.It can be observed from Fig. 5 that under the background amplitude of a = 0, any disturbance frequency is stable, and the dashed line at Ω = 0 is the resonance line.
From the above discussion, we know that the parameters related to the MI include background amplitude a, background frequency ω, disturbance frequency Ω, dispersion coefficient σ, and two independent real parameters μ and γ.Except for the disturbance frequency Ω, all other parameters are directly manifested from the solution.Thus we can establish a corresponding relationship between these parameters and the MI gain.According to the corresponding relationship between μ and γ, it can be seen from Fig. 5 that for the CLL equation, the distribution region of MI gain presents two symmetrically triangular regions, while for the KN equation and the GI equation, this irregularly symmetrical region appears.In addition, KM soliton and Peregrine soliton excitations are located on the resonance line of the MI gain distribution plane (excluding the point of a = 0), and the Akhmediev breather is located on both sides of the resonance line.This was also confirmed by the study of Li-Chen Zhao's team [34].

IV. CONCLUSION
Within the framework of the CQ-NLS equation, we study its fundamental first-order breather solutions and its dynamical properties.In general, the CQ-NLS equation is a suitable model for controlling the propagation of ultrashort pulses in highly nonlinear fibers.It should be noted that this equation contains the standard NLS equation (μ = γ = 0), the CLL equation (μ = γ = 0), the KN equation (μ = 2γ), the GI equation (μ = 0), and the KE equation (γ = 0).We obtain the exact fundamental first-order solution for KM soliton, Akhmediev breather, and Peregrine soliton by DT method, and demonstrate their interesting dynamical properties.It is found that KM solitons periodically propagate in the direction of t − (a 2 μ + ω)z = 0, with an oscillation period of 2π/|V p |. Since the propagation direction is related to μ, the KM soliton structures under different equations are different.The Akhmediev breathers always oscillate in the t direction and its local peak value follows the direction of t − (a 2 μ + ω)z = 0, and the oscillation period is γπ/℘ r .Peregrine soliton structures have local properties in the z and t directions.Secondly, we find that the peak amplitude of KM solitons is greater than 3, the peak amplitude of Akhmediev breathers is less than 3, and the peak amplitude of Peregrine solitons is equal to 3.Although Akhmediev breather subsolutions for Manakov systems (vector NLS equations) have been explored by the DT method [60], there are still many breather solutions in vector systems that remain to be studied, such as the vector KN equation and vector CLL equation for Peregrine soliton solutions.We expect these results and related formulas will stimulate interest in extending them to vector or multi-component systems.
) from the equation D −1 SR = R , it can deduce to R = S −1 DR .Substituting matrices D, S, and (17) into the initial Lax pair of (2) yields

Fig. 2 .
Fig. 2. Three-dimensional surface (top) and contour (bottom) plots of the Akhmediev breathers in (a) the KN equation, (b) the CLL equation, and (c) the GI equation.c 1 = 5/6, and the background parameters are the same as those in Fig. 1.
corresponding to the Peregrine soliton solutions, the specific Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

Fig. 3 .
Fig. 3. Three-dimensional surface (top) and contour (bottom) plots of the Peregrine solitons in (a) the KN equation, (b) the CLL equation, and (c) the GI equation.c 1 → 1, and the background parameters are the same as those in Fig. 1.

Fig. 4 .
Fig. 4. Three-dimensional surface (top) and contour (bottom) plots of the KM solitons, Akhmediev breathers, and Peregrine solitons in (a) the KN equation, (b) the CLL equation, and (c) the GI equation.The first line corresponds to the KN equation, the second line corresponds to the CLL equation, and the third line corresponds to the GI equation.The first column corresponds to the KM solitons (R = 2, α = 0), the second column corresponds to the Akhmediev breathers (R = 1, α = 3/4), and the third column corresponds to the Peregrine solitons (R = 1, α = 1 × 10 −5 ).The background parameters are the same as those in Fig. 1.