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Integrable turbulence for a coupled nonlinear Schrödinger system
JID:PLA AID:126119 /SCO Doctopic: Mathematical physics
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Physics Letters A ••• (••••) ••••••
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Integrable turbulence for a coupled nonlinear Schrödinger system Xi-Yang Xie, Sheng-Kun Yang, Chun-Hui Ai, Ling-Cai Kong ∗ Department of Mathematics and Physics, North China Electric Power University, Baoding 071003, China
a r t i c l e
i n f o
Article history: Received 5 August 2019 Received in revised form 1 November 2019 Accepted 2 November 2019 Available online xxxx Communicated by B. Malomed Keywords: Integrable turbulence Coupled nonlinear Schrödinger system Breather soliton Rogue wave
a b s t r a c t In this manuscript, we aim to study turbulence for a coupled nonlinear Schrödinger system, and find the cause of the appearance of rogue waves in such system. The initial conditions are chosen as the constant backgrounds of unit amplitudes with random perturbations. We list some cases of the standard deviations of one component for the initial conditions, to investigate the influences of the initial conditions on the intensity profiles and peak amplitudes of the two chaotic fields. In addition, sets of complex eigenvalues are illustrated with three various standard deviations, from which we observe that Akhmediev breathers may split into solitons with the values of standard deviations increasing, which will lead to the probability of the appearance of rogue waves increases. © 2019 Elsevier B.V. All rights reserved.
1. Introduction As an irregular behavior of dynamical systems marked by chaotic changes of the flow parameters, integrable turbulence approaches the stationary state in an oscillatory way asymptotically [1–5]. The formation of the chaotic wave field is created by modulation instability or initial condition with high amplitude chaotic component, and the continuous wave (cw) component mixed with noise of different amplitudes in the initial condition can help to distinguish the two mechanisms [4,6,7]. Integrable turbulence can be analyzed mainly using numerical simulations [7,8]. The previous studies imply that cw component in the initial condition results in the excitation of breathers via modulation instability, and the excitation of solitons happens when level of initial noise increases [7]. It is understood that the probability of the appearance of rogue waves is determined by the initial condition [4,6]. For integrable systems, exact solutions (i.e., soliton and breather solutions) can be expressed in explicit forms, and serve as elementary nonlinear modes, so it is convenient to classify them [8,9]. Many efforts have been paid for researching the turbulence for integrable systems in oceanographic, atmospheric and optical communications [4,5,7,10–13], since it provides the way to investigate the influence of randomness on the propagation of waves [14,15]. In this manuscript, we pay our attention to study the turbulence for a coupled nonlinear Schrödinger system written in dimensionless form as [16–24]
*
Corresponding author. E-mail address: [email protected] (L.-C. Kong).
https://doi.org/10.1016/j.physleta.2019.126119 0375-9601/© 2019 Elsevier B.V. All rights reserved.
iq1,t + q1,xx + 2(|q1 |2 + |q2 |2 )q1 = 0 iq2,t + q2,xx + 2(|q1 |2 + |q2 |2 )q2 = 0
(1)
which appears in many physical situations such as nonlinear optics [16], oceanics [17], multi-component Bose-Einstein condensates [18], finance [19] and bio-physics [20], where q j ( j = 1, 2) describe orthogonally polarized complex waves and the subscripts x and t represent the retarded time and normalized distance, respectively [21]. Dynamics of localized nonlinear waves for System (1) have been investigated: Rogue wave, breather and brightdark-rogue solutions for System (1) have been derived via the Darboux transformation [21]; Bright N-soliton solutions for System (1) have been obtained by the Hirota method [22]; Inelastic collision properties of the bright two solitons for System (1) have been observed [23]; Energy exchanging collisions between the solitons for System (1) have been experimentally demonstrated [24]. However, to our knowledge, integrable turbulence for System (1) has not been reported in the previous literature. Integrable turbulence for the dimensionless nonlinear Schrödinger equation has been investigated [7]. Since more kinds of solutions existing, the analysis of the integrable turbulence for System (1) is essential and meaningful. Apart from the intensity profiles of the chaotic wave field, we further investigate the transformations between the solitons and breathers in this manuscript. 2. Integrable turbulence for System (1) According to the Refs. [7,8,25,26], we choose q1 (0, x) = 1 +
μ1 f (x) and q2 (0, x) = 1 + μ2 f (x), where f (x) is a normalized complex random function with the standard deviation is σ = 1. Then, μ1 and μ2 give the standard deviation for the functions q1
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Fig. 1. Initial field intensity of q1 (0, x) for three values of μ1 with fixed value of μ2 (i.e., μ2 = 0.5): (a) for three values of μ2 with fixed value of μ1 (i.e., μ1 = 0.5): (a) μ2 = 0.1; (b) μ2 = 0.5; (c) μ2 = 0.9.
and q2 , respectively. The numerical discretization of either the real or imaginary part of f (xk ) is taken in the form of Gaussian dis√ tribution, each with (2x/ πτ )exp[−2(xk /τ )2 ] [27], where τ is variable correlation length. When τ = 0, the random sequence is completely uncorrelated, while the sequence is correlated when τ = 0 [26]. Then, we use the two variable parameters μ1 and μ2 to illustrate the complex dynamics of integrable turbulence for System (1). With the split-step method [28], Figs. 1 illustrate the intensity profiles of q1 (0, x) and q2 (0, x) in three cases of the initial conditions with the simulations. With μ2 fixed, Fig. 1(a) shows relatively small random deviations from the continuous wave whose amplitude is 1, with μ1 = 0.1; Fig. 1(b) exhibits higher chaotic deviations around the continuous wave with μ1 = 0.5, while a large amount of noise appear in Fig. 1(c) with μ1 = 0.9. Similarly, in Figs. 1(d)–(f), it shows that with the value of μ2 increasing, higher chaotic behavior for q2 is observed when μ1 is fixed. Next, the peak amplitudes of the two fields for the above three cases are shown in Figs. 2. As the absolute maximum of q j ( j = 1, 2) over the entire x interval of simulation, the peak amplitudes are found to be the functions of t. In Fig. 2(a), there exists an exponential increase for the absolute maximum of q1 at the initial stage of evolution, which accord with the theory of modulation instability, and it becomes chaotically after evolving a certain propagation distance. Similar phenomenon appears in Fig. 2(b), but it converges to the chaotic stage faster than that in Fig. 2(a). With the higher standard deviation for q1 , it is shown in Fig. 2(c) that the evolution has become chaotic at the very beginning. In Figs. 2(d)–(f), when we amplify the standard deviation for q2 , similar phenomenon happens. Some types of waves exist in the chaotic wave field, such as the Akhmediev breathers, Kuznetsov-Ma solitons and solitons. Then we will analyze how the standard deviation affect the changes of the types of waves.
μ1 = 0.1; (b) μ1 = 0.5; (c) μ1 = 0.9; Initial field intensity of q2 (0, x)
Lax pair associated with System (1) is written as [21]
x = (i λU 1 + U 0 ),
(2)
2
t = [3i λ U 1 + 3λU 0 + i σ3 (U 0,x − where
⎛
0 U 0 = ⎝ −q∗1 −q∗2
q1 0 0
⎞
⎛
U 02 )],
(3)
⎞
q2 −2 0 0 1 0 ⎠, 0 ⎠ , U1 = ⎝ 0 0 0 1 0
(4)
σ3 = diag{1, −1, −1}, = (1 , 2 , 3 )T (T denotes the transpose of a vector) is the vector eigenfunction, 1 , 2 and 3 are all the scalar eigenfunctions of x and t, and λ is a spectral parameter. System (1) can be generated from the compatibility condition of the Lax Pair (2) and (3). According to the theory in Refs. [7] and [29], the spectrum of eigenvalues of Lax Pair (2) does not depend on t, then the dynamics of the chaotic fields at any t can be determined by the initial condition (i.e., t = 0). Substitute q1 (0, x) and q2 (0, x) into Lax Pair (2) for the same three values of μ1 as above, sets of eigenvalues λ and eigenfunctions are derived with the Fourier collocation method [28], and the eigenfunctions are either nonlocalized solutions corresponding to breathers or localized solutions corresponding to solitons [7]. Figs. 3 show sets of complex eigenvalues λ, which can be used to find how to enhance the probability of the appearance of rogue waves. It should be pointed out that only the upper half of the complex plane is shown in Figs. 3, since the eigenvalues appear in complex conjugate pairs. Fig. 3(a) reveals the set of eigenvalues with μ1 = 0.1 and μ2 = 0.1, the point λ = i in the spectrum corresponds to the Peregrine breather, while the points below λ = i on the imaginary axis correspond to the Akhmediev breather. Expressions of the Kuznetsov-Ma solution and its eigenvalue have been illustrated in Ref. [30], and it has proved that imaginary part of its
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Fig. 2. Evolutions of the maximum of the field amplitudes versus t, and parameters are the same as Figs. 1.
Fig. 3. Complex eigenvalues λ calculated for the initial conditions with (a)
eigenvalue greater than i, then the eigenvalues located above the point λ = i on the imaginary axis correspond to the excitation of Kuznetsov-Ma solitons. Compared with Fig. 3(a), the distribution of eigenvalues in Fig. 3(b) are not still on the imaginary axis strictly, which means that most of the excitations in the chaotic fields are Akhmediev breathers, while some solitons are transformed by them. Fig. 3(c) shows the set of eigenvalues with the larger standard deviation, from which one can find that little eigenvalues left exactly on the imaginary axis, which imply that almost all the excitations in the chaotic fields are the solitons. Then, we analyze the transformations of the waves in the chaotic fields in detail with μ2 = 0.5. Figs. 4 and 5 show the proportions of the Kuznetsov-Ma soliton, Akhmediev breather, Radiation wave and soliton in the chaotic fields, respectively. In Fig. 4(a), we find that a small number of the Kuznetsov-Ma solitons exist in the chaotic fields, and μ1 has little impact on them. From Figs. 4(b) and 5(a), the amount of the Akhmediev breathers and Radiation
μ1 = 0.1, μ2 = 0.1; (b) μ1 = 0.5, μ2 = 0.5; (c) μ1 = 0.9, μ2 = 0.9.
waves decreases distinctly with the value of μ1 increasing, and we observe that the amount of the solitons increases simultaneously in Fig. 5(b). 3. Discussions and conclusions In this manuscript, we have revealed the integrable turbulence in a coupled nonlinear Schrödinger system (i.e., System (1)). The initial conditions have been chosen as the random functions, and intensity profiles of the initial conditions with different standard deviations are illustrated in Figs. 1, from which we have found that chaotic deviations around the continuous wave become higher with the value of standard deviation lager. The peak amplitudes of the two fields for the above cases have been illustrated in Figs. 2, from which we have found that the evolution for q1 or q2 become chaotic faster with the value of their own standard deviation lager. Figs. 3 have shown sets of complex eigenvalues, with the
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Fig. 4. (a) The frequency distribution of λ on the imaginary axis, whose imaginary part larger than i; (b) The frequency distribution of λ on the imaginary axis, whose imaginary part less than i.
Fig. 5. (a) The frequency distribution of λ around the real axis; (b) The frequency distribution of λ on in the upper half of the complex plane except for the imaginary and real axis.
values of the standard deviations being small, Peregrine breathers and Akhmediev breathers have been observed in the chaotic fields. When we increase the values of the standard deviations, Akhmediev breathers have been found to split into solitons. The transformations of the waves in the chaotic fields have been illustrated in Figs. 4 and 5 in detail. Rogue waves are the result of the collision between either breathers or solitons with each other, which means that the probability of the appearance of rogue waves depends on the standard deviations of the initial conditions. Declaration of competing interest The authors declare no conflicts of interest. Acknowledgements This work has been supported by the National Natural Science Foundation of China under grant nos. 11905061 and by the Fundamental Research Funds for the Central Universities (No. 2017BD0094). References [1] G.K. Batchelor, The Theory of Homogeneous Turbulence, Cambridge University Press, Cambridge, 1953. [2] D.S. Agafontsev, V.E. Zakharo, Nonlinearity 28 (2015) 2791. [3] V.E. Zakharov, Stud. Appl. Math. 122 (2009) 219. [4] P. Walczak, S. Randoux, P. Suret, Phys. Rev. Lett. 114 (2015) 143903. [5] D.S. Agafontsev, V.E. Zakharov, Nonlinearity 29 (2016) 3551.
[6] J.M. Dudley, F. Dias, M. Erkintalo, G. Genty, Nat. Photonics 8 (2014) 755. [7] J.M. Soto-Crespo, N. Devine, N. Akhmediev, Phys. Rev. Lett. 116 (2016) 103901. [8] N. Akhmediev, A. Ankiewicz, J.M. Soto-Crespo, Phys. Rev. E 80 (2009) 026601. [9] X.Y. Xie, G.Q. Meng, Eur. Phys. J. Plus 134 (2019) 359; W.T. Yu, W.J. Liu, H. Triki, Q. Zhou, A. Biswas, Nonlinear Dyn. 97 (2019) 1253; L. Wang, X. Wu, H.Y. Zhang, Phys. Lett. A 382 (2018) 2650. [10] A.R. Osborne, Nonlinear Ocean Waves, Academic Press, New York, 2009. [11] J.C. Wyngaard, Turbulence in the Atmosphere, Academic Press, New York, 2010. [12] M. Vorontsov, J. Minet, in: Proceedings of the Directed Energy Systems Symposium, Directed Energy Professional Society (DEPS), California, 2013. [13] P. Suret, A. Picozzi, S. Randoux, Opt. Express 19 (2011) 17852. [14] S. Randoux, P. Walczak, M. Onorato, P. Suret, Phys. Rev. Lett. 113 (2014) 113902. [15] P. Suret, R. El Koussaifi, A. Tikan, C. Evain, S. Randoux, C. Szwaj, S. Bielawski, Nat. Commun. 7 (2016) 13136. [16] G.P. Agrawal, Nonlinear Fiber Optics, Academic, New York, 1995. [17] K. Dysthe, H.E. Krogstad, P. Müller, Annu. Rev. Fluid Mech. 40 (2008) 287. [18] E.P. Bashkin, A.V. Vagov, Phys. Rev. B 56 (1997) 6207. [19] Z. Yan, Financial rogue waves appearing in the coupled nonlinear volatility and option pricing model, arXiv:1101.3107v1, 2011. [20] A.C. Scott, Phys. Scr. 29 (1984) 279. [21] B.L. Guo, L.M. Ling, Chin. Phys. Lett. 28 (2011) 110202. [22] R. Radhakrishnan, M. Lakshmanan, J. Phys. A, Math. Gen. 28 (1995) 2683. [23] R. Radhakrishnan, M. Lakshmanan, J. Hietarinta, Phys. Rev. E 56 (1997) 2213. [24] C. Anastassiou, M. Segev, K. Steiglitz, J.A. Giordmaine, M. Mitchell, M.F. Shih, S. Lan, J. Martin, Phys. Rev. Lett. 83 (1999) 2332. [25] N. Akhmediev, A. Ankiewicz, J.M. Soto-Crespo, J.M. Dudley, Phys. Lett. A 375 (2011) 775. [26] N. Akhmediev, J.M. Soto-Crespo, A. Ankiewicz, Phys. Lett. A 373 (2009) 2137. [27] M. Rosenblatt, Gaussian and Non-Gaussian Linear Time Series and Random Fields, Springer-Verlag, New York, 2000. [28] J.K. Yang, Nonlinear Waves in Integrable and Nonintegrable Systems, SIAM, Philadelphia, 2010. [29] V.E. Zakharov, A.B. Shabat, J. Exp. Theor. Phys. 34 (1972) 62. [30] S. Randoux, P. Suret, G. El, Sci. Rep. 6 (2016) 29238.
[m5G; v1.261; Prn:15/11/2019; 12:37] P.1 (1-4)
Physics Letters A ••• (••••) ••••••
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Integrable turbulence for a coupled nonlinear Schrödinger system Xi-Yang Xie, Sheng-Kun Yang, Chun-Hui Ai, Ling-Cai Kong ∗ Department of Mathematics and Physics, North China Electric Power University, Baoding 071003, China
a r t i c l e
i n f o
Article history: Received 5 August 2019 Received in revised form 1 November 2019 Accepted 2 November 2019 Available online xxxx Communicated by B. Malomed Keywords: Integrable turbulence Coupled nonlinear Schrödinger system Breather soliton Rogue wave
a b s t r a c t In this manuscript, we aim to study turbulence for a coupled nonlinear Schrödinger system, and find the cause of the appearance of rogue waves in such system. The initial conditions are chosen as the constant backgrounds of unit amplitudes with random perturbations. We list some cases of the standard deviations of one component for the initial conditions, to investigate the influences of the initial conditions on the intensity profiles and peak amplitudes of the two chaotic fields. In addition, sets of complex eigenvalues are illustrated with three various standard deviations, from which we observe that Akhmediev breathers may split into solitons with the values of standard deviations increasing, which will lead to the probability of the appearance of rogue waves increases. © 2019 Elsevier B.V. All rights reserved.
1. Introduction As an irregular behavior of dynamical systems marked by chaotic changes of the flow parameters, integrable turbulence approaches the stationary state in an oscillatory way asymptotically [1–5]. The formation of the chaotic wave field is created by modulation instability or initial condition with high amplitude chaotic component, and the continuous wave (cw) component mixed with noise of different amplitudes in the initial condition can help to distinguish the two mechanisms [4,6,7]. Integrable turbulence can be analyzed mainly using numerical simulations [7,8]. The previous studies imply that cw component in the initial condition results in the excitation of breathers via modulation instability, and the excitation of solitons happens when level of initial noise increases [7]. It is understood that the probability of the appearance of rogue waves is determined by the initial condition [4,6]. For integrable systems, exact solutions (i.e., soliton and breather solutions) can be expressed in explicit forms, and serve as elementary nonlinear modes, so it is convenient to classify them [8,9]. Many efforts have been paid for researching the turbulence for integrable systems in oceanographic, atmospheric and optical communications [4,5,7,10–13], since it provides the way to investigate the influence of randomness on the propagation of waves [14,15]. In this manuscript, we pay our attention to study the turbulence for a coupled nonlinear Schrödinger system written in dimensionless form as [16–24]
*
Corresponding author. E-mail address: [email protected] (L.-C. Kong).
https://doi.org/10.1016/j.physleta.2019.126119 0375-9601/© 2019 Elsevier B.V. All rights reserved.
iq1,t + q1,xx + 2(|q1 |2 + |q2 |2 )q1 = 0 iq2,t + q2,xx + 2(|q1 |2 + |q2 |2 )q2 = 0
(1)
which appears in many physical situations such as nonlinear optics [16], oceanics [17], multi-component Bose-Einstein condensates [18], finance [19] and bio-physics [20], where q j ( j = 1, 2) describe orthogonally polarized complex waves and the subscripts x and t represent the retarded time and normalized distance, respectively [21]. Dynamics of localized nonlinear waves for System (1) have been investigated: Rogue wave, breather and brightdark-rogue solutions for System (1) have been derived via the Darboux transformation [21]; Bright N-soliton solutions for System (1) have been obtained by the Hirota method [22]; Inelastic collision properties of the bright two solitons for System (1) have been observed [23]; Energy exchanging collisions between the solitons for System (1) have been experimentally demonstrated [24]. However, to our knowledge, integrable turbulence for System (1) has not been reported in the previous literature. Integrable turbulence for the dimensionless nonlinear Schrödinger equation has been investigated [7]. Since more kinds of solutions existing, the analysis of the integrable turbulence for System (1) is essential and meaningful. Apart from the intensity profiles of the chaotic wave field, we further investigate the transformations between the solitons and breathers in this manuscript. 2. Integrable turbulence for System (1) According to the Refs. [7,8,25,26], we choose q1 (0, x) = 1 +
μ1 f (x) and q2 (0, x) = 1 + μ2 f (x), where f (x) is a normalized complex random function with the standard deviation is σ = 1. Then, μ1 and μ2 give the standard deviation for the functions q1
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Fig. 1. Initial field intensity of q1 (0, x) for three values of μ1 with fixed value of μ2 (i.e., μ2 = 0.5): (a) for three values of μ2 with fixed value of μ1 (i.e., μ1 = 0.5): (a) μ2 = 0.1; (b) μ2 = 0.5; (c) μ2 = 0.9.
and q2 , respectively. The numerical discretization of either the real or imaginary part of f (xk ) is taken in the form of Gaussian dis√ tribution, each with (2x/ πτ )exp[−2(xk /τ )2 ] [27], where τ is variable correlation length. When τ = 0, the random sequence is completely uncorrelated, while the sequence is correlated when τ = 0 [26]. Then, we use the two variable parameters μ1 and μ2 to illustrate the complex dynamics of integrable turbulence for System (1). With the split-step method [28], Figs. 1 illustrate the intensity profiles of q1 (0, x) and q2 (0, x) in three cases of the initial conditions with the simulations. With μ2 fixed, Fig. 1(a) shows relatively small random deviations from the continuous wave whose amplitude is 1, with μ1 = 0.1; Fig. 1(b) exhibits higher chaotic deviations around the continuous wave with μ1 = 0.5, while a large amount of noise appear in Fig. 1(c) with μ1 = 0.9. Similarly, in Figs. 1(d)–(f), it shows that with the value of μ2 increasing, higher chaotic behavior for q2 is observed when μ1 is fixed. Next, the peak amplitudes of the two fields for the above three cases are shown in Figs. 2. As the absolute maximum of q j ( j = 1, 2) over the entire x interval of simulation, the peak amplitudes are found to be the functions of t. In Fig. 2(a), there exists an exponential increase for the absolute maximum of q1 at the initial stage of evolution, which accord with the theory of modulation instability, and it becomes chaotically after evolving a certain propagation distance. Similar phenomenon appears in Fig. 2(b), but it converges to the chaotic stage faster than that in Fig. 2(a). With the higher standard deviation for q1 , it is shown in Fig. 2(c) that the evolution has become chaotic at the very beginning. In Figs. 2(d)–(f), when we amplify the standard deviation for q2 , similar phenomenon happens. Some types of waves exist in the chaotic wave field, such as the Akhmediev breathers, Kuznetsov-Ma solitons and solitons. Then we will analyze how the standard deviation affect the changes of the types of waves.
μ1 = 0.1; (b) μ1 = 0.5; (c) μ1 = 0.9; Initial field intensity of q2 (0, x)
Lax pair associated with System (1) is written as [21]
x = (i λU 1 + U 0 ),
(2)
2
t = [3i λ U 1 + 3λU 0 + i σ3 (U 0,x − where
⎛
0 U 0 = ⎝ −q∗1 −q∗2
q1 0 0
⎞
⎛
U 02 )],
(3)
⎞
q2 −2 0 0 1 0 ⎠, 0 ⎠ , U1 = ⎝ 0 0 0 1 0
(4)
σ3 = diag{1, −1, −1}, = (1 , 2 , 3 )T (T denotes the transpose of a vector) is the vector eigenfunction, 1 , 2 and 3 are all the scalar eigenfunctions of x and t, and λ is a spectral parameter. System (1) can be generated from the compatibility condition of the Lax Pair (2) and (3). According to the theory in Refs. [7] and [29], the spectrum of eigenvalues of Lax Pair (2) does not depend on t, then the dynamics of the chaotic fields at any t can be determined by the initial condition (i.e., t = 0). Substitute q1 (0, x) and q2 (0, x) into Lax Pair (2) for the same three values of μ1 as above, sets of eigenvalues λ and eigenfunctions are derived with the Fourier collocation method [28], and the eigenfunctions are either nonlocalized solutions corresponding to breathers or localized solutions corresponding to solitons [7]. Figs. 3 show sets of complex eigenvalues λ, which can be used to find how to enhance the probability of the appearance of rogue waves. It should be pointed out that only the upper half of the complex plane is shown in Figs. 3, since the eigenvalues appear in complex conjugate pairs. Fig. 3(a) reveals the set of eigenvalues with μ1 = 0.1 and μ2 = 0.1, the point λ = i in the spectrum corresponds to the Peregrine breather, while the points below λ = i on the imaginary axis correspond to the Akhmediev breather. Expressions of the Kuznetsov-Ma solution and its eigenvalue have been illustrated in Ref. [30], and it has proved that imaginary part of its
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Fig. 2. Evolutions of the maximum of the field amplitudes versus t, and parameters are the same as Figs. 1.
Fig. 3. Complex eigenvalues λ calculated for the initial conditions with (a)
eigenvalue greater than i, then the eigenvalues located above the point λ = i on the imaginary axis correspond to the excitation of Kuznetsov-Ma solitons. Compared with Fig. 3(a), the distribution of eigenvalues in Fig. 3(b) are not still on the imaginary axis strictly, which means that most of the excitations in the chaotic fields are Akhmediev breathers, while some solitons are transformed by them. Fig. 3(c) shows the set of eigenvalues with the larger standard deviation, from which one can find that little eigenvalues left exactly on the imaginary axis, which imply that almost all the excitations in the chaotic fields are the solitons. Then, we analyze the transformations of the waves in the chaotic fields in detail with μ2 = 0.5. Figs. 4 and 5 show the proportions of the Kuznetsov-Ma soliton, Akhmediev breather, Radiation wave and soliton in the chaotic fields, respectively. In Fig. 4(a), we find that a small number of the Kuznetsov-Ma solitons exist in the chaotic fields, and μ1 has little impact on them. From Figs. 4(b) and 5(a), the amount of the Akhmediev breathers and Radiation
μ1 = 0.1, μ2 = 0.1; (b) μ1 = 0.5, μ2 = 0.5; (c) μ1 = 0.9, μ2 = 0.9.
waves decreases distinctly with the value of μ1 increasing, and we observe that the amount of the solitons increases simultaneously in Fig. 5(b). 3. Discussions and conclusions In this manuscript, we have revealed the integrable turbulence in a coupled nonlinear Schrödinger system (i.e., System (1)). The initial conditions have been chosen as the random functions, and intensity profiles of the initial conditions with different standard deviations are illustrated in Figs. 1, from which we have found that chaotic deviations around the continuous wave become higher with the value of standard deviation lager. The peak amplitudes of the two fields for the above cases have been illustrated in Figs. 2, from which we have found that the evolution for q1 or q2 become chaotic faster with the value of their own standard deviation lager. Figs. 3 have shown sets of complex eigenvalues, with the
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Fig. 4. (a) The frequency distribution of λ on the imaginary axis, whose imaginary part larger than i; (b) The frequency distribution of λ on the imaginary axis, whose imaginary part less than i.
Fig. 5. (a) The frequency distribution of λ around the real axis; (b) The frequency distribution of λ on in the upper half of the complex plane except for the imaginary and real axis.
values of the standard deviations being small, Peregrine breathers and Akhmediev breathers have been observed in the chaotic fields. When we increase the values of the standard deviations, Akhmediev breathers have been found to split into solitons. The transformations of the waves in the chaotic fields have been illustrated in Figs. 4 and 5 in detail. Rogue waves are the result of the collision between either breathers or solitons with each other, which means that the probability of the appearance of rogue waves depends on the standard deviations of the initial conditions. Declaration of competing interest The authors declare no conflicts of interest. Acknowledgements This work has been supported by the National Natural Science Foundation of China under grant nos. 11905061 and by the Fundamental Research Funds for the Central Universities (No. 2017BD0094). References [1] G.K. Batchelor, The Theory of Homogeneous Turbulence, Cambridge University Press, Cambridge, 1953. [2] D.S. Agafontsev, V.E. Zakharo, Nonlinearity 28 (2015) 2791. [3] V.E. Zakharov, Stud. Appl. Math. 122 (2009) 219. [4] P. Walczak, S. Randoux, P. Suret, Phys. Rev. Lett. 114 (2015) 143903. [5] D.S. Agafontsev, V.E. Zakharov, Nonlinearity 29 (2016) 3551.
[6] J.M. Dudley, F. Dias, M. Erkintalo, G. Genty, Nat. Photonics 8 (2014) 755. [7] J.M. Soto-Crespo, N. Devine, N. Akhmediev, Phys. Rev. Lett. 116 (2016) 103901. [8] N. Akhmediev, A. Ankiewicz, J.M. Soto-Crespo, Phys. Rev. E 80 (2009) 026601. [9] X.Y. Xie, G.Q. Meng, Eur. Phys. J. Plus 134 (2019) 359; W.T. Yu, W.J. Liu, H. Triki, Q. Zhou, A. Biswas, Nonlinear Dyn. 97 (2019) 1253; L. Wang, X. Wu, H.Y. Zhang, Phys. Lett. A 382 (2018) 2650. [10] A.R. Osborne, Nonlinear Ocean Waves, Academic Press, New York, 2009. [11] J.C. Wyngaard, Turbulence in the Atmosphere, Academic Press, New York, 2010. [12] M. Vorontsov, J. Minet, in: Proceedings of the Directed Energy Systems Symposium, Directed Energy Professional Society (DEPS), California, 2013. [13] P. Suret, A. Picozzi, S. Randoux, Opt. Express 19 (2011) 17852. [14] S. Randoux, P. Walczak, M. Onorato, P. Suret, Phys. Rev. Lett. 113 (2014) 113902. [15] P. Suret, R. El Koussaifi, A. Tikan, C. Evain, S. Randoux, C. Szwaj, S. Bielawski, Nat. Commun. 7 (2016) 13136. [16] G.P. Agrawal, Nonlinear Fiber Optics, Academic, New York, 1995. [17] K. Dysthe, H.E. Krogstad, P. Müller, Annu. Rev. Fluid Mech. 40 (2008) 287. [18] E.P. Bashkin, A.V. Vagov, Phys. Rev. B 56 (1997) 6207. [19] Z. Yan, Financial rogue waves appearing in the coupled nonlinear volatility and option pricing model, arXiv:1101.3107v1, 2011. [20] A.C. Scott, Phys. Scr. 29 (1984) 279. [21] B.L. Guo, L.M. Ling, Chin. Phys. Lett. 28 (2011) 110202. [22] R. Radhakrishnan, M. Lakshmanan, J. Phys. A, Math. Gen. 28 (1995) 2683. [23] R. Radhakrishnan, M. Lakshmanan, J. Hietarinta, Phys. Rev. E 56 (1997) 2213. [24] C. Anastassiou, M. Segev, K. Steiglitz, J.A. Giordmaine, M. Mitchell, M.F. Shih, S. Lan, J. Martin, Phys. Rev. Lett. 83 (1999) 2332. [25] N. Akhmediev, A. Ankiewicz, J.M. Soto-Crespo, J.M. Dudley, Phys. Lett. A 375 (2011) 775. [26] N. Akhmediev, J.M. Soto-Crespo, A. Ankiewicz, Phys. Lett. A 373 (2009) 2137. [27] M. Rosenblatt, Gaussian and Non-Gaussian Linear Time Series and Random Fields, Springer-Verlag, New York, 2000. [28] J.K. Yang, Nonlinear Waves in Integrable and Nonintegrable Systems, SIAM, Philadelphia, 2010. [29] V.E. Zakharov, A.B. Shabat, J. Exp. Theor. Phys. 34 (1972) 62. [30] S. Randoux, P. Suret, G. El, Sci. Rep. 6 (2016) 29238.