# Breather Positons and Rogue Waves for the Nonlocal Fokas-Lenells Equation

Authors: Chun Wang; Rong Fan; Zhao Zhang; Biao Li
Journal: Advances in Mathematical Physics
 (2021)

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## Abstract

In this paper, we investigate breather positons and higher-order rogue waves for the nonlocal Fokas-Lenells equation. In this nonlocal optical system, rogue waves can be generated when periods of breather positons go to infinity. In addition, we find two very interesting phenomena: one is that rogue waves sitting on a periodic line wave background are derived; the other is that a hybrid of rogue waves and a periodic kink wave is also constructed. We believe that these interesting findings exist in the optical system corresponding to the nonlocal Fokas-Lenells equation.


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## Body

## 1. Introduction

In the field of mathematics and physics, the research on the degenerate solutions of integrable equations has always been a hot topic. These degenerate solutions include lump solutions, smooth positon solutions, breather positon solutions, and rogue waves. In general, rogue waves localized in both time and space may produce three to five times more amplitude than background waves in a very short time [1]. Rogue waves have also been observed in space plasmas, as well as in optics, when propagating high-power optical radiation through photonic crystal fibers [1–4]. In recent years, many scholars have paid much attention to rogue wave [1] solutions and positon solutions of integrable nonlinear partial differential equations. In 2013, He et al. proposed the generation mechanism of higher-order rogue waves for the nonlinear Schrödinger equation (2). Similarly, their team applied this mechanism to other integrable equations, such as the modified KdV equation (5) and Fokas-Lenells (FL) equation (7), to obtain higher-order rogue waves. Since Matveev [5, 6] proposed singular positon solutions for the Korteweg-de Vries equation in 1992, many experts [7–11] have been working in this field. Among the numerous studies on positons, Wang et al. [10] elaborate the connection between breather positons and rogue waves for a local integrable system.In this research, we study the nonlocal FL equation [12]:
(1)qxt+q−2iqq−x,−tqx=0,to find some new results. Equation (1) describes the promulgation of ultrashort pulsation in visual fiber. It must be mentioned here that the famous FL equation was proposed by Fokas [13]. Soliton solutions and breather solutions to equation (1) have been given in Ref. [12]. And we have used this nonlocal optical system to obtain some relatively satisfactory results [14, 15]: two types of smooth positons with a periodic line wave background and smooth positons with a nonzero background. However, as far as we know, the breather solutions and rogue wave solutions of this important mathematical and physical model (1) have not been studied.Combining Refs. [4, 12], a new solution qn of the nonlocal FL equation (1) through n-fold Darboux transformation with a plane wave solution q0=aeiθ, θ=bx+2a2b+1t/b, is given by
(2)qn=q0−N2nD2n,with
(3)D2n=λ1nϕ1λ1n−1φ1⋯λ1−n−2ϕ1λ1−n−1φ1λ2nϕ2λ2n−1φ2⋯λ2−n−2ϕ2λ2−n−1φ2⋮⋮⋱⋮⋮λ2n−1nϕ2n−1λ2n−1n−1φ2n−1⋯λ2n−1−n−2ϕ2n−1λ2n−1−n−1φ2n−1λ2nnϕ2nλ2nn−1φ2n⋯λ2n−n−2ϕ2nλ2n−n−1φ2n,where
(4)ϕj=2abλjeiσΩ+−λj2+b+2σe−iσΩeiθ/2,φj=−λj2+b+2σeiσΩ+2abλje−iσΩe−iθ/2,Ω=−tbλj2+x,σ=12λj4+−4a2b2−2bλj2+b2.Here,a∈ℂ and b∈ℝ. And N2n is a determinant given by D2n through replacing its last column with vector λ1−nϕ1λ2−nϕ2⋯λ2n−1−nϕ2n−1λ2n−nϕ2nT.In the second section, we will show that the relation between breather positons and rogue waves mentioned in Ref. [10] still holds true for nonlocal integrable systems. Then, we apply different constraints and degenerate mechanisms to equation (2) to produce some novel exact solutions, like rogue waves sitting on a periodic line wave background, a nonlinear superposition between rogue waves and a periodic kink wave.


## 2. The Relation between Breather Positons and Rogue Waves

Although in Refs. [10, 16] the authors obtained breather positons in the local integrable system, these ideas are still valid in the nonlocal system studied in this paper. Simply put, the breather positon is a limit case of equation (2). Hence, we give the following proposition about breather positons without proof. At the same time, we may wish to set a=αi,b=1,α∈ℝ in order to facilitate discussion and simplify the calculation.Proposition 1.
Based on the degenerate Darboux transformation, thenth-order breather positons qn−bp are given by
(5)qn−bp=q0−N2n′D2n′,with
(6)N2n′=∂hi∂εhiε=0N2nijλj+ε22n×2n,D2n′=∂hi∂εhiε=0D2nijλj+ε22n×2n,hx=2x−12,where
(7)α∈−1,1,λ2j−1=−λ2j∗=λ1,j=1,⋯,n.
Here,x denotes the floor function of x, and D2n, N2n, and q0 are given by equation (2).From Figure1, we can have a general understanding of the spatiotemporal characteristics of breather positons. Since breather positons were proposed by Wang et al. [10], it has been a difficult problem to study the change trend of breather positons over time. It is always a difficult problem to find the trajectories of breather positons [10, 16, 17]. Not to mention, let us study the dynamic properties of breather positons like decomposing smooth positons [7–9].Figure 1
Breather positons described by Proposition1 with parameter selections a=i/2, b=1, and λ1=1+i (color online). (a) A first-order breather positon q1−bp. (b) A second-order breather positon q2−bp. (c) A third-order breather positon q3−bp.
(a)(b)(c)Whenλ=ba±a2b2+b, we find ϕj=0 and φj=0. This implies that eigenfunctions ϕj,φj degenerate at λ=ba±a2b2+b. In particular, when b=1,a=αi,α∈−1,1, we can get ba+a2b2+b=−ba−a2b2+b∗. Combining with the main idea of Ref. [10], we can say that the period of breather positons described by Proposition 1 goes to infinity when λ1→ba+a2b2+ba=αi,α∈−1,1. In this particular case just mentioned, breather positons are converted to rogue waves.Proposition 2.
Based on the degenerate Darboux transformation, thenth-order rogue waves qn−RW are given by
(8)qn−RW=q0−N2n′D2n′,with
(9)N2n′=∂hi∂εhiε=0N2nijλj+ε22n×2n,D2n′=∂hi∂εhiε=0D2nijλj+ε22n×2n,hx=2x+12.where
(10)α∈−1,1,λ2j−1=−λ2j∗=λ1=ba+a2b2+b,j=1,⋯,n.Remark 3.
There are many conclusions [2, 3, 7, 10, 11, 16] like Propositions 1 and 2, so this research will not prove them in detail. Essentially, all the propositions in this paper are different limit cases of equation (2). The most difficult step to derive these propositions is to obtain the constraints such as equation (10) and the function hx that controls the degree of degradation.Figures1 and 2 intuitively illustrate that the transformation relationship between breather positons and rogue waves mentioned in Ref. [10] is still valid in the nonlocal FL system. As λ1 approaches to ba+a2b2+b, the other peaks near the center region of breather positons gradually disappear until only the central profile survives. In other words, as the period of a breather positon goes to infinity, its central region will turn into a rogue wave.Figure 2
Rogue waves described by Proposition2 with parameters λ1=i/2+3/2, a=i/2, and b=1 (color online). (a) First-order rogue wave. (b) Second-order rogue wave. (c) Third-order rogue wave.
(a)(b)(c)According to Proposition2, the mathematical expression of a first-order rogue wave q1−RW can be derived:
(11)q1−RW=−i16α4tx+4iα2t+12iα2x+4α2t2−8α2tx+4α2x2−3αe−i2α2t−t−x16α4tx+4iα2t−4iα2x+4α2t2−8α2tx+4α2x2+1.q1−RW has three critical points. q1−RW reaches the maximum value 3α at point x=0,t=0; the minimum value 0 is obtained at points t=−33/4−3α2+4α,x=3/4−3α2+4α and t=33/4−3α2+4α,x=−3/4−3α2+4α. When x→∞,t→∞, it is easy to find that q1−RW=α.Whenn=2, a second-order rogue wave q2−RW can be derived from Proposition 2:
(12)q2−RW=αi−L1L2e−i2α2t−t−x,where
(13)L1=768α7x5+6144α9x4t−3072α7x4t−12288α9x3t2+12288α11x3t2+4608α7x3t2+6144α9x2t3−3072α7x2t3+768α7xt4+1536iα7x4+192iα5x4−768iα5tx3+6144iα9x3t+1152iα5x2t2+9216iα9x2t2−4608iα7t2x2−768iα5t3x+3072iα7xt3+192iα5t4−768α7x3−768α5x3−4608α7x2t+1152α5x2t+6912α7xt2−384α5t3−576iα5x2+288iα3x2−576iα3tx+3456iα5xt+576iα5t2+288iα3t2−432α3x−288α3t−36iα,L2=64α6x6+768α8x5t−384α6x5t+3072α10x4t2−3072α8x4t2+960α6x4t2−6144α10x3t3+4096α12x3t3+4608α8x3t3−1280α6x3t3+3072α10x2t4−3072α8x2t4+960α6x2t4+768α8xt5−384α6xt5+64α6t6−192iα6x5+960iα6x4t−1536iα8tx4−1920iα6t2x3−3072iα10t2x3+4608iα8x3t2+1920iα6x2t3+3072iα10x2t3−4608iα8t3x2−960iα6t4x+1536iα8xt4+192iα6t5+48α4x4−192α6x4−192α4x3t−384α6x3t+2304α8x3t+288α4x2t2+1152α6x2t2+2304α8x2t2−192α4xt3−384α6xt3+2304α8xt3+48α4t4−192α6t4−192iα6x3−96iα4x3+288iα4x2t+1728iα6x2t−288iα4t2x−1728iα6t2x+192iα6t3+96iα4t3+144α4x2+108α2x2−216α2xt+1296α4xt+144α4t2+108α2t2−108iα2x+108iα2t+9.The maximum amplitude ofq2−RW is 5α at 0,0. In addition to the maximum value, q2−RW has four other local maximum points, which can be easily found in Figure 2(b).The mathematical expressions for higher-order rogue waves are too verbose to be given in this paper. By settingt=0,x=0, the maximum amplitude of qn−RW is 2n+1α. On each side of the straight line t=0, qn−RW has nn+1/2−1 local maximum points. The following conclusion about higher-order rogue waves can also be found in Figure 2. Starting from t=∞, before the maximum amplitude in the center, there are a series of peaks in qn−RW and the height of these peaks gradually increases. Comparing the relevant conclusions in Ref. [3], the basic description of the rogue waves in the nonlocal FL system is roughly the same as that in the local system.To be honest, Propositions1 and 2 simply apply the mature method [3, 4, 10] to the new equation. The rogue waves shown in Figure 2 are technically known as fundamental patterns of rogue waves. However, the nonlocal equation studied in this paper must satisfy the condition ϕjx,t=φj−x,−t,j=1,⋯,2n [12]. Therefore, a small perturbation parameter cannot be introduced to generate triangular patterns and ring patterns as in Refs. [3, 4, 18].


## 3. Rogue Waves Sitting on Other Forms of Waves

A rogue wave with a plane wave background is commonplace [1–4]. In order to better explain some phenomena in mathematical and physical models, some scholars have explored the generation mechanism of hybrid solutions consisting of rogue waves and breather solutions [10, 18].In equation (2), there is no constraint relation between λ2j and λ2j−1. This is the biggest difference between the nonlocal FL systems and the local systems in Refs. [1–4, 18]. Hence, a hybrid of rogue waves and other forms of solutions can be derived. We have the following conclusion.Proposition 4.
Based on the degenerate Darboux transformation, a novel nonlinear superposition between thenth-order rogue waves and other forms of solutions qn−RW−hyb is given by
(14)qn−RW−hyb=q0−N2n′D2n′,with
(15)N2n′=∂hi∂εhiε=0N2nijλj+ε22n×2n,D2n′=∂hi∂εhiε=0D2nijλj+ε22n×2n,hx=0,x≤2,2x+12−2,x>2,where
(16)α∈−1,1,λ1=ξ1+η1i,λ2=ξ2+η2i,λ2j−1=−λ2j∗=λ3=ba+a2b2+b,j=2⋯n+1.According to relevant conclusions in Refs. [12, 15], we can get rogue waves sitting on periodic line waves when ξ1=0,ξ2=0. A nonlinear superposition between rogue waves and a periodic kink wave can be derived when ξ2=0,η1=−η2. We have to say that it is difficult to give constraints that guarantee the analyticity of hybrid solutions described by Proposition 4. On the basis of some conclusions [12, 15], the improvement of Proposition 4 is to get rogue waves sitting on periodic line waves or a periodic kink wave. If the precondition ξ1=0,ξ2=0 or ξ2=0,η1=−η2 is satisfied and the solution is an analytic function, then the solution shown in Figure 3 will be derived. As far as we know, the interaction between rogue waves and breathers is relatively common [18], but the results shown in Figure 3 have not been found.Figure 3
(a) A first-order rogue wave sitting on periodic line waves described by Proposition4 with parameters n=1, a=i/2, λ1=i, λ2=3i/2, and λ3=3/2+3/2. (b) A nonlinear superposition between a first-order rogue wave and a periodic kink wave described by Proposition 4 with parameter selections n=1, a=i/2, λ1=1−3i, λ2=3i, and λ3=3/2+3/2. Color online.
(a)(b)


## 4. Conclusion

In addition to studying the breather positons and rogue waves for the nonlocal Fokas-Lenells equation, we also find some new and interesting results shown in Figure3: rogue waves sitting on periodic line waves and a hybrid of rogue waves and a periodic kink wave. Figures 1 and 2 vividly illustrate that a rogue wave can be generated when the period of a breather positon goes to infinity. We have tried our best to analyze the dynamic properties of rogue waves from two aspects: specific mathematical expressions (equations (11) and (12)) and exquisite images (Figure 2).Although our research is concise and the graphics are beautiful, the dynamic properties of these breather positon solutions are worthy of further study. Can we find the trajectories of breather positons as in Ref. [17]? If this works, then we can make some important conclusions about the dynamics. Can the nth-order breather positons be approximately decomposed into n single breather waves with the phase shift when ∣t∣→∞? Can we find a constraint that guarantees that the solution represented by Proposition 4 is analytic? Further, how do we investigate the dynamic properties of singular solutions in Proposition 4? As far as we know, this problem has not been resolved. We look forward to many colleagues working together to solve them.



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