- Email: [email protected]
Novel conditions for soliton breathers of the complex modified Korteweg–de Vries equation with variable coefficients
Optik - International Journal for Light and Electron Optics 172 (2018) 1117–1122
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.com/locate/ijleo
Original research article
Novel conditions for soliton breathers of the complex modified Korteweg–de Vries equation with variable coefficients
T
⁎
V.N. Serkina, , T.L. Belyaevab a b
Benemerita Universidad Autonoma de Puebla, Av. 4 Sur 104, C. P. 72001, Puebla, Mexico Universidad Autonoma del Estado de Mexico, Av. Instituto Literarion 100, C. P. 50000, Toluca, Mexico
A R T IC LE I N F O
ABS TRA CT
Keywords: Complex modified Korteweg–de Vries Breather solutions General conditions for soliton breather solutions
We reveal novel unexpected conditions for soliton breathers of the generalized complex modified Korteweg–de Vries equation with variable coefficients and the loss (or gain) term (vc cmKdV). Novel relations between spectral parameters of two solitons giving rise to the breather substantially extend the well-known constraints for canonical soliton breathers. This finding allows us to systematically construct the variety of soliton breather solutions on a zero background of the considered model. These generalized breathers move with varying amplitudes and velocities adapted to variations of the dispersion, nonlinearity, and gain or losses. Among other things, we establish that both the standing generalized breathers and envelope breathers exist as well.
1. Introduction Nonlinear wave phenomena occur in a great number of physical situations in different branches of science and technology, which are often related to similar and even common mathematical formalism [1–5]. An additional point to emphasize is that the impressive progress has been achieved recently in the successful search for the exactly integrable mathematical models [6–12]. The Ablowitz–Kaup–Newell–Segur (AKNS) method [13] allows one to construct the hierarchy of the integrable nonlinear evolution equations. The complex modified Korteweg–de Vries (cmKdV) equation has been arose in plasma physics, and today, this model has a wide range of applications, from plasmas, elastic solids, molecular chains to the ultrashort pulses in nonlinear optics, etc. [2–4,14]. Both the cmKdV and its real version, the mKdV equation, belong to the AKNS hierarchy and represent the completely integrable nonlinear evolution equations [15–20]. The variable-coefficient version of the cmKdV was found to be also completely integrable equation with certain constraints to the coefficients [21–30]. Complex mKdV equation admits various soliton and multisoliton solutions, compactons, and rational (rogue wave) solutions [31–38]. In this paper, we consider a special type of the two-soliton solutions of the cmKdV equation with variable coefficients, which represents the periodically varying in time bound state of two bright solitons and is known as breather by virtue of its “breathering” localized periodic structure. Notice that a breather is a stable solitary wave (as an ordinary soliton), but with internal oscillation modes. We reveal novel unexpected conditions for these soliton breathers which substantially extend the well-known constraints for canonical soliton breathers. We systematically construct the variety of soliton breather solutions on a zero background of the considered model. These generalized breathers move with varying amplitudes and velocities adapted to variations of the dispersion, nonlinearity, and gain or losses. Among other things, we establish that both the standing generalized breathers and envelope breathers exist as well.
⁎
Corresponding author. E-mail address: [email protected] (V.N. Serkin).
https://doi.org/10.1016/j.ijleo.2018.07.139 Received 31 July 2018; Accepted 31 July 2018 0030-4026/ © 2018 Elsevier GmbH. All rights reserved.
Optik - International Journal for Light and Electron Optics 172 (2018) 1117–1122
V.N. Serkin, T.L. Belyaeva
2. Soliton breather solutions of the complex modified Korteweg–de Vries equations with variable coefficients 2.1. Integrability conditions Consider the complex mKdV equation with variable coefficients
qt + D3 (t ) qxxx + 6R3 (t )|q|2 qx =
1 Γ(t ) q, 2
(1)
where the gain/loss coefficient
Γ(t ) =
W (R3, D3) R3 D3
(2)
depends on the Wronskian W(R3, D3) = D3tR3 − R3tD3 of arbitrary real-valued time-varying functions D3(t) and R3(t), which represent the dispersion and nonlinearity coefficients of the third order. Complete integrability of Eq. (1) follows from the linear eigenvalue problem
Ψx = ˆ Ψ(x , t ),
Ψt = ˆ Ψ(x , t )
(3)
for the 2-component complex scattering function Ψ(x, t) = {ψ1, ψ2} with the complex-valued (2 × 2) matrices T
R3 ⎛− i Λ q⎞ D3 ˆ ⎟ ⎜ = ⎜− R3 q* i Λ ⎟ D3 ⎠ ⎝
(4)
ˆ = (CA −B A),
(5)
and
A = R3 (qq*x − q*qx ) + 2iR3|q|2 Λ − 4iD3 Λ3, B=
R3 D3
[−D3 qxx − 2R3 |q|2 q + 2i ΛD3 qx + 4Λ2D3 q],
C=
R3 D3
* + 2R3 |q|2 q* + 2i ΛD3 qx* − 4Λ2D3 q*], [D3 qxx
(6)
satisfying the compatibility condition
ˆ ˆ ˆˆ
t − x + [ , ] = 0.
(7)
Constant spectral parameter Λ = κ + iη defines the initial velocity κ and amplitude η of the soliton. The soliton solutions of order n of Eq. (1) can be calculated by applying the auto-Bäcklund transformation and the recurrent relation [39–42]
4 qn (x , t ) = −qn − 1 (x , t ) −
∼ D3 η Γn − 1 (x , R3 n
t) , ∼ 1 + |Γn − 1 (x , t )|2
(8) ∼ which connects the (n − 1) and n soliton solutions by means of the so-called pseudo-potential Γn − 1 (x , t ) = ψ1 (x , t )/ ψ2 (x , t ) for the (n − 1) soliton scattering functions Ψ(x, t) = (ψ1ψ2)T. A recurrent process begins at zero-valued potential q(x, t) = 0, for which Eq. (1) is evidently fulfilled. The one- and two-soliton solutions of Eq. (1) are given by
q1 (x , t ) = 2η1
D3 (t ) sech[ξ1 (x , t )]exp{ −iχ1 (x , t )} R3 (t )
(9)
and
q2 (x , t ) = 4
D3 (t ) N (x , t ) , R3 (t ) D (x , t )
(10)
where
N (x , t )= η1 cosh ξ2 exp(−iχ1 )[(κ2 − κ1)2 + 2iη2 (κ2 − κ1)tanh ξ2 + (η12 − η22)] + η2 cosh ξ1 exp( −iχ2 )[(κ2 − κ1)2 − 2iη1 (κ2 − κ1)tanh ξ1 − (η12 − η22)] and
D (x , t ) = cosh(ξ1 + ξ2)[(κ2 − κ1)2 + (η2 − η1 )2] + cosh(ξ1 − ξ2)[(κ2 − κ1)2 + (η2 + η1)2] − 4η1 η2 cos(χ2 − χ1 ). The arguments of hyperbolic functions and the phases of solitons are defined by the following equations: 1118
(11)
Optik - International Journal for Light and Electron Optics 172 (2018) 1117–1122
V.N. Serkin, T.L. Belyaeva
Fig. 1. The breather solutions of the cmKdV (1) with D3(t) = D30 exp(C0t), constant nonlinearity R3(t) = R30 = 1, and gain Γ = C0 = 0.06. (a) Moving with acceleration breather formed by two solitons with initial velocities κ1 = 0.25 and κ2 =−0.25, and amplitude η1 = η2 = 0.5. (b) The standing breather formed by two solitons with the same amplitudes η1 = η2 = 0.5, and velocities κ1 = 0.5/ 3 and κ2 = −0.5/ 3 . The space-temporal dynamics (top panels) and the counter plot (bottom panels).
2 2 ξ1,2 (x , t ) = 2(x − x 01,2) η1,2 + 8η1,2 (3κ1,2 − η1,2 )
∫0
t
2 2 χ1,2 (x , t ) = 2(x − x 01,2) κ1,2 + 8κ1,2 (κ1,2 − 3η1,2 )
∫0
t
D3 (τ ) dτ ,
(12)
D3 (τ ) dτ + χ01,2 .
(13)
2.2. Soliton breathers of the cmKdV equation Let us consider special types of the two-soliton solutions corresponding to the bound states of solitons: spatially localized, timeperiodic solutions that are referred to as breathers. The breather solution of the real canonical mKdV equation was found for the first time by Wadati [15] and corresponds to the bound two-soliton solution with complex conjugated eigenvalues Λ2 = −Λ1*, where Λ = κ + iη, the real part of the eigenvalue Λ denotes the velocity κ and the imaginary part of Λ corresponds to the amplitude η of the soliton. The Wadati breather solution of Eq. (1) at the conditions: κ1 =− κ2 = κ and η2 = η1 = η becomes
qbr (x , t ) = 4
D3 (t ) η {κ cosh ξ cos χ − η sinh ξ sin χ } , η2 R3 (t ) κ cosh2 ξ + sin2 χ
(
κ2
)
(14)
where ξ(x, t) and χ(x, t) are defined by Eqs. (12) and (13) with x 01 = x 02 . The soliton solution that does not change its initial position is known as a “standing soliton”. The standing breather of the cmKdV Eq. (1) is formed by two solitons with arbitrary amplitudes ηi and the velocities satisfying the relation: κ1,2 = ± η1,2 / 3 . In Fig. 1, we show the breather solutions of the cmKdV Eq. (1) with exponentially varying dispersion D3(t) = D30 exp(C0t), constant nonlinearity R3(t) = R30 = 1, and gain Γ = C0. The moving (Fig. 1a) Wadati breather Eq. (14) is formed by two solitons with equal initial amplitudes (in Fig. 1(a), we choose, for example: η1 = η2 = 0.5) and arbitrary equal initial velocities of opposite signs (in Fig. 1(a), we choose κ1 = 0.25, κ2 =−0.25). The moving Wadati breather has the velocity V(t) = 4(3κ2 − η2)D30[exp(C0t) − 1]/C0 and the acceleration Vt(t) = 4(3κ2 − η2)D30 exp(C0t). The standing Wadati breather Eq. (14) (see Fig. 1(b)) is formed by two solitons with equal amplitudes (in Fig. 1(b) η1 = η2 = η = 0.5) and the opposite velocities κ2 = −κ1 = −η/ 3 . 2.3. General conditions for the soliton breather solutions of the complex modified Korteweg–de Vries equation with variable coefficients The bound states of two solitons of Eq. (1) (and, in general, multisoliton bound states on the zero background) can be constructed under more general conditions. We reveal the general relation between spectral parameters of two solitons giving rise to the breather. It corresponds to the following requirements to the soliton initial parameters: 1119
Optik - International Journal for Light and Electron Optics 172 (2018) 1117–1122
V.N. Serkin, T.L. Belyaeva
Fig. 2. The breather solutions of the cmKdV (1) with D3(t) = D30 exp(C0t), constant nonlinearity R3(t) = R30 = 1, and gain Γ = C0 = 0.1. (a) The standing breather formed by two solitons with different amplitudes η1 = 0.5 and η2 = 0.3, and velocities κ1 = 0.5/ 3 and κ2 = 0.3/ 3 . (b) The moving with acceleration breather formed by two solitons with parameters κ1 = 0.29, κ2 = 0.22, P = 1.25, and η2 = 0.436, calculated in accordance with Eq. (15). The space-temporal dynamics (top panels) and the counter plot (bottom panels).
η1 = Pη2 , η22 (P 2 − 1) = 3(κ12 − κ 22),
(15)
where P is an arbitrary constant. Constraints (15) include the Wadati condition at P = 1 when η1 = η2 and κ1 =− κ2 (in the case of κ1 = κ2, the solution (10,11) vanishes). In general, if P2 ≷ 1 then κ12≷κ 22 . It is interesting to note that, in particular, one of the two velocities, κ1 or κ2, admits zero value. Fig. 2 shows typical examples of how the breather solutions of the cmKdV Eq. (1) are formed in accordance with Eq. (15). In particular, we consider the exponentially varying dispersion D3(t) = D30 exp(C0t), constant nonlinearity R3(t) = R30 = 1, and the constant gain Γ = C0. The standing breather shown in Fig. 2(a) is formed by two solitons with different amplitudes η1 = 0.5 and η2 = 0.3, and the velocities κ1 = 0.5/ 3 and κ2 = 0.3/ 3 . The moving with acceleration breather shown in Fig. 2(b) is formed by two solitons with different velocities κ1 = 0.29 and κ2 = 0.22, and the amplitudes calculated in accordance with the conditions (15), η1 = 0.545 and η2 = 0.436, corresponding to the parameter P = 1.25. 2.4. Envelope breathers One of the most interesting solutions of the cmKdV equation (having applications in nonlinear optics of extremely short pulses) is the envelope breather filled with high-frequency oscillations [3]. These solutions look like electromagnetic few-cycle pulses or video pulses. They are formed by two bound solitons with parameters satisfying the condition ϵ = η/κ ≪ 1. Then, Eq. (14) at ϵ ≪ 1 approximately becomes
qen (x , t ) ≈ 4η
η D3 (t ) sechξ ⎛cos χ − tanh ξ sin χ ⎞. κ R3 (t ) ⎝ ⎠
(16)
In general, the envelope breather can be constructed in accordance with the constraints (15) for different initial amplitudes and velocities of both solitons. Fig. 3 shows examples of the envelope breathers of the cmKdV Eq. (1) with the exponentially varying dispersion D3(t) = D30 exp(C0t), constant nonlinearity R3(t) = R30 = 1, and the constant gain Γ = C0. The breather solution Eq. (16) moving with acceleration (Fig. 3(a)) is formed by two solitons with equal amplitudes η1 = η2 = 0.1 and equal velocities of the opposite signs κ1 =− κ2 = 0.5. Fig. 3(b) shows accelerated breather formed by two solitons with different velocities κ1 =−0.51 and κ2 = 0.50, and the amplitudes satisfying to Eq. (15): η1 = 0.185 and η2 = 0.006 corresponding to the parameter P = 3. 3. Conclusion We have revealed novel unexpected conditions for soliton breathers of the generalized complex modified Korteweg–de Vries 1120
Optik - International Journal for Light and Electron Optics 172 (2018) 1117–1122
V.N. Serkin, T.L. Belyaeva
Fig. 3. The envelope breather solutions of the cmKdV (1) with D3(t) = D30 exp(C0t), constant nonlinearity R3(t) = R30 = 1, and gain Γ = C0 = 0.06. (a) The breather formed by two soliton with equal amplitudes η1 = η2 = 0.1 and the velocities κ1 = 0.5, κ2 =−0.5. (b) The breather formed by two soliton with different parameters κ1 =−0.51, κ2 = 0.502, P = 3, η1 = 0.185, and η2 = 0.006, in accordance with Eq. (15). The space-temporal dynamics (top panels) and the counter plot (bottom panels).
equation with variable coefficients and the loss (or gain) term. Novel relations between spectral parameters of two solitons giving rise to the breather substantially extend the well-known constraints for canonical soliton breathers. We have present the systematic construction of the variety of soliton breather solutions on a zero background. These generalized breathers move with varying amplitudes and velocities adapted to variations of the dispersion, nonlinearity, and gain or losses. Among other things, we establish that both the standing generalized breathers and envelope breathers exist as well. Generalization of the main results for the AKNS hierarchy including the quantum-mechanical like wave properties of solitons [43–45] will be published elsewhere. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]
M. Ablowitz, H. Segur, Solitons and the Inverse Scattering Transform, SIAM, 1981. P.L. Christiansen, M.P. Sorensen, A.C. Scott, Nonlinear Science at the Dawn of the 21st Century, Springer, Berlin, 2000. A.I. Maimistov, A.M. Basharov, Nonlinear Optical Waves, Kluwer Academic Publishers, Dordrecht, 1999. L.A. Ostrovsky, A. Potapov, Modulated Waves: Theory and Applications, John Hopkins University Press, 2002. A. Biswas, S. Konar, Introduction to non-Kerr law optical solitons, Chapman & Hall/CRC Applied Mathematics & Nonlinear Science, (2006). W. Djoudi, A. Zerarka, Exact structures for the KdV-mKdV equation with variable coefficients via the functional variable method, Optik 127 (2016) 9621–9626. H. Bulut, T.A. Sulaiman, H.M. Baskonus, A.A. Sandulyak, New solitary and optical wave structures to the (1+1)-dimensional combined KdV-mKdV equation, Optik 135 (2017) 327–336. S. Zhang, J. Li, Soliton solutions and dynamical evolutions of a generalized AKNS system in the framework of inverse scattering transform, Optik 137 (2017) 228–237. C.-K. Kuo, A novel method for finding new multi-soliton wave solutions of the completely integrable equations, Optik 139 (2017) 283–290. V.N. Serkin, A. Hasegawa, T.L. Belyaeva, Nonautonomous solitons in external potentials, Phys. Rev. Lett. 98 (2007) 074102. V.N. Serkin, T.L. Belyaeva, Exactly integrable nonisospectral models for femtosecond colored solitons and their reversible transformations, Optik 158 (2018) 1289–1294. V.N. Serkin, T.L. Belyaeva, Nontrivial Galilean-like invariance of the generalized higher-order nonlinear Schrödinger equation model with gravitation-like potential and the soliton analogies of the cosmic dark energy and antigravitation effects, Optik 160 (2018) 389–395. M.J. Ablowitz, D.J. Kaup, A.C. Newell, H. Segur, Nonlinear evolution equations of physical significance, Phys. Rev. Lett. 31 (1973) 125–127. O.B. Gorbacheva, L.A. Ostrovsky, Nonlinear vector waves in a mechanical model of a molecular chain, Physics D 8 (1983) 223–228. M. Wadati, The modified Korteweg–de Vries equation, J. Phys. Soc. Jpn. 32 (1972) 1681. R. Hirota, Exact solution of the modified Korteweg–de Vries equation for multiple collisions of solitons, J. Phys. Soc. Jpn. 33 (1972) 1456–1458. M. Wadati, The exact solution of the modified Korteweg–de Vries equation, J. Phys. Soc. Jpn. 34 (1973) 1289–1296. T. Chou, Symmetries and a hierarchy of the general modified KdV equation, J. Phys. A: Math. Gen. 20 (1987) 367–374. J. Yang, Stable embedded solitons, Phys. Rev. Lett. 91 (2003) 1–4. A.-M. Wazwaz, The tanh and the sine-cosine methods for the complex modified KdV and generalized KdV equations, Comput. Math. Appl. 49 (2005) 1101–1112. W.L. Chan, K.S. Li, Non-propagating solitons of the non-isospectral and variable coefficient modified KdV equation, Phys. A: Math. Gen. 27 (1994) 883–902. H. Triki, T.R. Taha, A.-M. Wazwaz, Solitary wave solutions for a generalized KdV-mKdV equation with variable coefficients, Math. Comput. Simul. 80 (2010) 1867–1873. Z.-Y. Sun, Y.-T. Gao, Y. Liu, X. Yu, Soliton management for a variable-coefficient modified Korteweg–de Vries equation, Phys. Rev. E 84 (2011) 026606. Y. Liu, Y.T. Gao, Z.Y. Sun, X. Yu, Multi-soliton solutions of the forced variable-coefficient extended Korteweg–de Vries equation arisen in fluid dynamics of internal solitary waves, Nonlinear Dyn. 66 (2011) 575–587.
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V.N. Serkin, T.L. Belyaeva
[25] G.-Q. Meng, Y.-T. Gao, X. Yu, Y.-J. Shen, Y. Qin, Painlevé analysis, Lax pair, Backlund transformation and multi-soliton solutions for a generalized variablecoefficient KdV-mKdV equation in fluids and plasmas, Phys. Scr. 85 (2012) 055010 (12 pp.). [26] P. Wang, B. Tian, W.-J. Liu, Y. Jiang, Y.-Sh. Xue, Interactions of breathers and solitons of a generalized variable-coefficient Korteweg–de Vries-modified Korteweg–de Vries equation with symbolic computation, Eur. Phys. J. D 66 (2012) 233. [27] L. Wang, Y.-T. Gao, F.-H. Qi, N-fold Darboux transformation and double-Wronskian-typed solitonic structures for a variable-coefficient modified Korteweg–de Vries equation, Annals Phys. 327 (2012) 1974–1988. [28] H. Kumar, F. Chand, Soliton solutions of complex modified KdV equation with time-dependent coefficients, Indian J. Phys. 89 (2013) 909–912. [29] A.-M. Wazwaz, Partial Differential Equations and Solitary Waves Theory, Springer-Verlag, Berlin Heidelberg, 2009. [30] V.N. Serkin, A. Hasegawa, T.L. Belyaeva, Hidden symmetry reductions and the Ablowitz-Kaup-Newell-Segur hierarchies for nonautonomous solitons, in: K. Porsezian, R. Ganapathy (Eds.), Odyssey of Light in Nonlinear Optical Fibers: Theory and Applications, CRC Press, Taylor & Francis Group, Boca Raton, FL, USA, 2015, pp. 145–187. [31] A. Biswas, Solitary wave solution for KdV equation with power law nonlinearity and time-dependent coefficients, Nonlinear Dyn. 58 (2009) 345. [32] F. Cooper, H. Shepard, P. Sodano, Solitary waves in a class of generalized Korteweg–de Vries equation, Phys. Rev. E 48 (1989) 4027–4032. [33] J. He, L. Wang, L. Li, K. Porsezian, R. Erdélyi, Few-cycle optical rogue waves: complex modified Korteweg–de Vries equation, Phys. Rev. E 89 (2014) 062917. [34] A. Chowdury, A. Ankiewicz, N. Akhmediev, Periodic and rational solutions of modified Korteweg–de Vries equation, Eur. Phys. J. D 70 (2016) 104. [35] Y.-F. Liu, R. Guo, H. Li, Breathers and localized solutions of complex modified Korteweg–de Vries equation, Mod. Phys. Lett. B 29 (23) (2014) 1550129. [36] S. Das, D. Ghosh, A.K.N.S. Formalism, Exact Solutions of KdV and Modified KdV Equations with Variable-Coefficients, Int. J. Adv. Res. Math. 6 (2016) 32–41. [37] C. Liu, Y. Ren, Z.-Y. Yang, W.-L. Yang, Superregular breathers in a complex modified Korteweg–de Vries system, Chaos 27 (2017) 083120. [38] N. Cheemaa, S.A. Mehmood, S.T. Raza Rizvi, M. Younis, Single and combined optical solitons with third order dispersion in Kerr media, Optik 127 (2016) 8203–8208. [39] H.H. Chen, General derivation of Bäcklund transformations from inverse scattering problems, Phys. Rev. Lett. 33 (1974) 925–928. [40] K. Konno, M. Wadati, Simple derivation of Backlund transformation from Riccati form of inverse method, Prog. Theor. Phys. 53 (1975) 1652–1656. [41] A.A. Rangwala, J.A. Rao, Complete soliton solutions of the ZS/AKNS equations of the inverse scattering method, Phys. Lett. A 112 (1985) 188–192. [42] K. Porsezian, A. Hasegawa, V. Serkin, T. Belyaeva, R. Ganapathy, Dispersion and nonlinear management for femtosecond optical solitons, Phys. Lett. A 361 (2007) 504–508. [43] V.N. Serkin, A. Hasegawa, T.L. Belyaeva, Soliton self-induced sub-barrier transparency and the controllable ‘shooting out’ effect, J. Mod. Opt. 60 (2013) 444–451. [44] T.L. Belyaeva, V.N. Serkin, Wave-particle duality of solitons and solitonic analog of the Ramsauer-Townsend effect, Eur. Phys. J. D 66 (2012) 153. [45] V.N. Serkin, A. Hasegawa, T.L. Belyaeva, Geiger-Nuttall law for Schrödinger solitons, J. Mod. Opt. 60 (2013) 116–127.
1122
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.com/locate/ijleo
Original research article
Novel conditions for soliton breathers of the complex modified Korteweg–de Vries equation with variable coefficients
T
⁎
V.N. Serkina, , T.L. Belyaevab a b
Benemerita Universidad Autonoma de Puebla, Av. 4 Sur 104, C. P. 72001, Puebla, Mexico Universidad Autonoma del Estado de Mexico, Av. Instituto Literarion 100, C. P. 50000, Toluca, Mexico
A R T IC LE I N F O
ABS TRA CT
Keywords: Complex modified Korteweg–de Vries Breather solutions General conditions for soliton breather solutions
We reveal novel unexpected conditions for soliton breathers of the generalized complex modified Korteweg–de Vries equation with variable coefficients and the loss (or gain) term (vc cmKdV). Novel relations between spectral parameters of two solitons giving rise to the breather substantially extend the well-known constraints for canonical soliton breathers. This finding allows us to systematically construct the variety of soliton breather solutions on a zero background of the considered model. These generalized breathers move with varying amplitudes and velocities adapted to variations of the dispersion, nonlinearity, and gain or losses. Among other things, we establish that both the standing generalized breathers and envelope breathers exist as well.
1. Introduction Nonlinear wave phenomena occur in a great number of physical situations in different branches of science and technology, which are often related to similar and even common mathematical formalism [1–5]. An additional point to emphasize is that the impressive progress has been achieved recently in the successful search for the exactly integrable mathematical models [6–12]. The Ablowitz–Kaup–Newell–Segur (AKNS) method [13] allows one to construct the hierarchy of the integrable nonlinear evolution equations. The complex modified Korteweg–de Vries (cmKdV) equation has been arose in plasma physics, and today, this model has a wide range of applications, from plasmas, elastic solids, molecular chains to the ultrashort pulses in nonlinear optics, etc. [2–4,14]. Both the cmKdV and its real version, the mKdV equation, belong to the AKNS hierarchy and represent the completely integrable nonlinear evolution equations [15–20]. The variable-coefficient version of the cmKdV was found to be also completely integrable equation with certain constraints to the coefficients [21–30]. Complex mKdV equation admits various soliton and multisoliton solutions, compactons, and rational (rogue wave) solutions [31–38]. In this paper, we consider a special type of the two-soliton solutions of the cmKdV equation with variable coefficients, which represents the periodically varying in time bound state of two bright solitons and is known as breather by virtue of its “breathering” localized periodic structure. Notice that a breather is a stable solitary wave (as an ordinary soliton), but with internal oscillation modes. We reveal novel unexpected conditions for these soliton breathers which substantially extend the well-known constraints for canonical soliton breathers. We systematically construct the variety of soliton breather solutions on a zero background of the considered model. These generalized breathers move with varying amplitudes and velocities adapted to variations of the dispersion, nonlinearity, and gain or losses. Among other things, we establish that both the standing generalized breathers and envelope breathers exist as well.
⁎
Corresponding author. E-mail address: [email protected] (V.N. Serkin).
https://doi.org/10.1016/j.ijleo.2018.07.139 Received 31 July 2018; Accepted 31 July 2018 0030-4026/ © 2018 Elsevier GmbH. All rights reserved.
Optik - International Journal for Light and Electron Optics 172 (2018) 1117–1122
V.N. Serkin, T.L. Belyaeva
2. Soliton breather solutions of the complex modified Korteweg–de Vries equations with variable coefficients 2.1. Integrability conditions Consider the complex mKdV equation with variable coefficients
qt + D3 (t ) qxxx + 6R3 (t )|q|2 qx =
1 Γ(t ) q, 2
(1)
where the gain/loss coefficient
Γ(t ) =
W (R3, D3) R3 D3
(2)
depends on the Wronskian W(R3, D3) = D3tR3 − R3tD3 of arbitrary real-valued time-varying functions D3(t) and R3(t), which represent the dispersion and nonlinearity coefficients of the third order. Complete integrability of Eq. (1) follows from the linear eigenvalue problem
Ψx = ˆ Ψ(x , t ),
Ψt = ˆ Ψ(x , t )
(3)
for the 2-component complex scattering function Ψ(x, t) = {ψ1, ψ2} with the complex-valued (2 × 2) matrices T
R3 ⎛− i Λ q⎞ D3 ˆ ⎟ ⎜ = ⎜− R3 q* i Λ ⎟ D3 ⎠ ⎝
(4)
ˆ = (CA −B A),
(5)
and
A = R3 (qq*x − q*qx ) + 2iR3|q|2 Λ − 4iD3 Λ3, B=
R3 D3
[−D3 qxx − 2R3 |q|2 q + 2i ΛD3 qx + 4Λ2D3 q],
C=
R3 D3
* + 2R3 |q|2 q* + 2i ΛD3 qx* − 4Λ2D3 q*], [D3 qxx
(6)
satisfying the compatibility condition
ˆ ˆ ˆˆ
t − x + [ , ] = 0.
(7)
Constant spectral parameter Λ = κ + iη defines the initial velocity κ and amplitude η of the soliton. The soliton solutions of order n of Eq. (1) can be calculated by applying the auto-Bäcklund transformation and the recurrent relation [39–42]
4 qn (x , t ) = −qn − 1 (x , t ) −
∼ D3 η Γn − 1 (x , R3 n
t) , ∼ 1 + |Γn − 1 (x , t )|2
(8) ∼ which connects the (n − 1) and n soliton solutions by means of the so-called pseudo-potential Γn − 1 (x , t ) = ψ1 (x , t )/ ψ2 (x , t ) for the (n − 1) soliton scattering functions Ψ(x, t) = (ψ1ψ2)T. A recurrent process begins at zero-valued potential q(x, t) = 0, for which Eq. (1) is evidently fulfilled. The one- and two-soliton solutions of Eq. (1) are given by
q1 (x , t ) = 2η1
D3 (t ) sech[ξ1 (x , t )]exp{ −iχ1 (x , t )} R3 (t )
(9)
and
q2 (x , t ) = 4
D3 (t ) N (x , t ) , R3 (t ) D (x , t )
(10)
where
N (x , t )= η1 cosh ξ2 exp(−iχ1 )[(κ2 − κ1)2 + 2iη2 (κ2 − κ1)tanh ξ2 + (η12 − η22)] + η2 cosh ξ1 exp( −iχ2 )[(κ2 − κ1)2 − 2iη1 (κ2 − κ1)tanh ξ1 − (η12 − η22)] and
D (x , t ) = cosh(ξ1 + ξ2)[(κ2 − κ1)2 + (η2 − η1 )2] + cosh(ξ1 − ξ2)[(κ2 − κ1)2 + (η2 + η1)2] − 4η1 η2 cos(χ2 − χ1 ). The arguments of hyperbolic functions and the phases of solitons are defined by the following equations: 1118
(11)
Optik - International Journal for Light and Electron Optics 172 (2018) 1117–1122
V.N. Serkin, T.L. Belyaeva
Fig. 1. The breather solutions of the cmKdV (1) with D3(t) = D30 exp(C0t), constant nonlinearity R3(t) = R30 = 1, and gain Γ = C0 = 0.06. (a) Moving with acceleration breather formed by two solitons with initial velocities κ1 = 0.25 and κ2 =−0.25, and amplitude η1 = η2 = 0.5. (b) The standing breather formed by two solitons with the same amplitudes η1 = η2 = 0.5, and velocities κ1 = 0.5/ 3 and κ2 = −0.5/ 3 . The space-temporal dynamics (top panels) and the counter plot (bottom panels).
2 2 ξ1,2 (x , t ) = 2(x − x 01,2) η1,2 + 8η1,2 (3κ1,2 − η1,2 )
∫0
t
2 2 χ1,2 (x , t ) = 2(x − x 01,2) κ1,2 + 8κ1,2 (κ1,2 − 3η1,2 )
∫0
t
D3 (τ ) dτ ,
(12)
D3 (τ ) dτ + χ01,2 .
(13)
2.2. Soliton breathers of the cmKdV equation Let us consider special types of the two-soliton solutions corresponding to the bound states of solitons: spatially localized, timeperiodic solutions that are referred to as breathers. The breather solution of the real canonical mKdV equation was found for the first time by Wadati [15] and corresponds to the bound two-soliton solution with complex conjugated eigenvalues Λ2 = −Λ1*, where Λ = κ + iη, the real part of the eigenvalue Λ denotes the velocity κ and the imaginary part of Λ corresponds to the amplitude η of the soliton. The Wadati breather solution of Eq. (1) at the conditions: κ1 =− κ2 = κ and η2 = η1 = η becomes
qbr (x , t ) = 4
D3 (t ) η {κ cosh ξ cos χ − η sinh ξ sin χ } , η2 R3 (t ) κ cosh2 ξ + sin2 χ
(
κ2
)
(14)
where ξ(x, t) and χ(x, t) are defined by Eqs. (12) and (13) with x 01 = x 02 . The soliton solution that does not change its initial position is known as a “standing soliton”. The standing breather of the cmKdV Eq. (1) is formed by two solitons with arbitrary amplitudes ηi and the velocities satisfying the relation: κ1,2 = ± η1,2 / 3 . In Fig. 1, we show the breather solutions of the cmKdV Eq. (1) with exponentially varying dispersion D3(t) = D30 exp(C0t), constant nonlinearity R3(t) = R30 = 1, and gain Γ = C0. The moving (Fig. 1a) Wadati breather Eq. (14) is formed by two solitons with equal initial amplitudes (in Fig. 1(a), we choose, for example: η1 = η2 = 0.5) and arbitrary equal initial velocities of opposite signs (in Fig. 1(a), we choose κ1 = 0.25, κ2 =−0.25). The moving Wadati breather has the velocity V(t) = 4(3κ2 − η2)D30[exp(C0t) − 1]/C0 and the acceleration Vt(t) = 4(3κ2 − η2)D30 exp(C0t). The standing Wadati breather Eq. (14) (see Fig. 1(b)) is formed by two solitons with equal amplitudes (in Fig. 1(b) η1 = η2 = η = 0.5) and the opposite velocities κ2 = −κ1 = −η/ 3 . 2.3. General conditions for the soliton breather solutions of the complex modified Korteweg–de Vries equation with variable coefficients The bound states of two solitons of Eq. (1) (and, in general, multisoliton bound states on the zero background) can be constructed under more general conditions. We reveal the general relation between spectral parameters of two solitons giving rise to the breather. It corresponds to the following requirements to the soliton initial parameters: 1119
Optik - International Journal for Light and Electron Optics 172 (2018) 1117–1122
V.N. Serkin, T.L. Belyaeva
Fig. 2. The breather solutions of the cmKdV (1) with D3(t) = D30 exp(C0t), constant nonlinearity R3(t) = R30 = 1, and gain Γ = C0 = 0.1. (a) The standing breather formed by two solitons with different amplitudes η1 = 0.5 and η2 = 0.3, and velocities κ1 = 0.5/ 3 and κ2 = 0.3/ 3 . (b) The moving with acceleration breather formed by two solitons with parameters κ1 = 0.29, κ2 = 0.22, P = 1.25, and η2 = 0.436, calculated in accordance with Eq. (15). The space-temporal dynamics (top panels) and the counter plot (bottom panels).
η1 = Pη2 , η22 (P 2 − 1) = 3(κ12 − κ 22),
(15)
where P is an arbitrary constant. Constraints (15) include the Wadati condition at P = 1 when η1 = η2 and κ1 =− κ2 (in the case of κ1 = κ2, the solution (10,11) vanishes). In general, if P2 ≷ 1 then κ12≷κ 22 . It is interesting to note that, in particular, one of the two velocities, κ1 or κ2, admits zero value. Fig. 2 shows typical examples of how the breather solutions of the cmKdV Eq. (1) are formed in accordance with Eq. (15). In particular, we consider the exponentially varying dispersion D3(t) = D30 exp(C0t), constant nonlinearity R3(t) = R30 = 1, and the constant gain Γ = C0. The standing breather shown in Fig. 2(a) is formed by two solitons with different amplitudes η1 = 0.5 and η2 = 0.3, and the velocities κ1 = 0.5/ 3 and κ2 = 0.3/ 3 . The moving with acceleration breather shown in Fig. 2(b) is formed by two solitons with different velocities κ1 = 0.29 and κ2 = 0.22, and the amplitudes calculated in accordance with the conditions (15), η1 = 0.545 and η2 = 0.436, corresponding to the parameter P = 1.25. 2.4. Envelope breathers One of the most interesting solutions of the cmKdV equation (having applications in nonlinear optics of extremely short pulses) is the envelope breather filled with high-frequency oscillations [3]. These solutions look like electromagnetic few-cycle pulses or video pulses. They are formed by two bound solitons with parameters satisfying the condition ϵ = η/κ ≪ 1. Then, Eq. (14) at ϵ ≪ 1 approximately becomes
qen (x , t ) ≈ 4η
η D3 (t ) sechξ ⎛cos χ − tanh ξ sin χ ⎞. κ R3 (t ) ⎝ ⎠
(16)
In general, the envelope breather can be constructed in accordance with the constraints (15) for different initial amplitudes and velocities of both solitons. Fig. 3 shows examples of the envelope breathers of the cmKdV Eq. (1) with the exponentially varying dispersion D3(t) = D30 exp(C0t), constant nonlinearity R3(t) = R30 = 1, and the constant gain Γ = C0. The breather solution Eq. (16) moving with acceleration (Fig. 3(a)) is formed by two solitons with equal amplitudes η1 = η2 = 0.1 and equal velocities of the opposite signs κ1 =− κ2 = 0.5. Fig. 3(b) shows accelerated breather formed by two solitons with different velocities κ1 =−0.51 and κ2 = 0.50, and the amplitudes satisfying to Eq. (15): η1 = 0.185 and η2 = 0.006 corresponding to the parameter P = 3. 3. Conclusion We have revealed novel unexpected conditions for soliton breathers of the generalized complex modified Korteweg–de Vries 1120
Optik - International Journal for Light and Electron Optics 172 (2018) 1117–1122
V.N. Serkin, T.L. Belyaeva
Fig. 3. The envelope breather solutions of the cmKdV (1) with D3(t) = D30 exp(C0t), constant nonlinearity R3(t) = R30 = 1, and gain Γ = C0 = 0.06. (a) The breather formed by two soliton with equal amplitudes η1 = η2 = 0.1 and the velocities κ1 = 0.5, κ2 =−0.5. (b) The breather formed by two soliton with different parameters κ1 =−0.51, κ2 = 0.502, P = 3, η1 = 0.185, and η2 = 0.006, in accordance with Eq. (15). The space-temporal dynamics (top panels) and the counter plot (bottom panels).
equation with variable coefficients and the loss (or gain) term. Novel relations between spectral parameters of two solitons giving rise to the breather substantially extend the well-known constraints for canonical soliton breathers. We have present the systematic construction of the variety of soliton breather solutions on a zero background. These generalized breathers move with varying amplitudes and velocities adapted to variations of the dispersion, nonlinearity, and gain or losses. Among other things, we establish that both the standing generalized breathers and envelope breathers exist as well. Generalization of the main results for the AKNS hierarchy including the quantum-mechanical like wave properties of solitons [43–45] will be published elsewhere. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]
M. Ablowitz, H. Segur, Solitons and the Inverse Scattering Transform, SIAM, 1981. P.L. Christiansen, M.P. Sorensen, A.C. Scott, Nonlinear Science at the Dawn of the 21st Century, Springer, Berlin, 2000. A.I. Maimistov, A.M. Basharov, Nonlinear Optical Waves, Kluwer Academic Publishers, Dordrecht, 1999. L.A. Ostrovsky, A. Potapov, Modulated Waves: Theory and Applications, John Hopkins University Press, 2002. A. Biswas, S. Konar, Introduction to non-Kerr law optical solitons, Chapman & Hall/CRC Applied Mathematics & Nonlinear Science, (2006). W. Djoudi, A. Zerarka, Exact structures for the KdV-mKdV equation with variable coefficients via the functional variable method, Optik 127 (2016) 9621–9626. H. Bulut, T.A. Sulaiman, H.M. Baskonus, A.A. Sandulyak, New solitary and optical wave structures to the (1+1)-dimensional combined KdV-mKdV equation, Optik 135 (2017) 327–336. S. Zhang, J. Li, Soliton solutions and dynamical evolutions of a generalized AKNS system in the framework of inverse scattering transform, Optik 137 (2017) 228–237. C.-K. Kuo, A novel method for finding new multi-soliton wave solutions of the completely integrable equations, Optik 139 (2017) 283–290. V.N. Serkin, A. Hasegawa, T.L. Belyaeva, Nonautonomous solitons in external potentials, Phys. Rev. Lett. 98 (2007) 074102. V.N. Serkin, T.L. Belyaeva, Exactly integrable nonisospectral models for femtosecond colored solitons and their reversible transformations, Optik 158 (2018) 1289–1294. V.N. Serkin, T.L. Belyaeva, Nontrivial Galilean-like invariance of the generalized higher-order nonlinear Schrödinger equation model with gravitation-like potential and the soliton analogies of the cosmic dark energy and antigravitation effects, Optik 160 (2018) 389–395. M.J. Ablowitz, D.J. Kaup, A.C. Newell, H. Segur, Nonlinear evolution equations of physical significance, Phys. Rev. Lett. 31 (1973) 125–127. O.B. Gorbacheva, L.A. Ostrovsky, Nonlinear vector waves in a mechanical model of a molecular chain, Physics D 8 (1983) 223–228. M. Wadati, The modified Korteweg–de Vries equation, J. Phys. Soc. Jpn. 32 (1972) 1681. R. Hirota, Exact solution of the modified Korteweg–de Vries equation for multiple collisions of solitons, J. Phys. Soc. Jpn. 33 (1972) 1456–1458. M. Wadati, The exact solution of the modified Korteweg–de Vries equation, J. Phys. Soc. Jpn. 34 (1973) 1289–1296. T. Chou, Symmetries and a hierarchy of the general modified KdV equation, J. Phys. A: Math. Gen. 20 (1987) 367–374. J. Yang, Stable embedded solitons, Phys. Rev. Lett. 91 (2003) 1–4. A.-M. Wazwaz, The tanh and the sine-cosine methods for the complex modified KdV and generalized KdV equations, Comput. Math. Appl. 49 (2005) 1101–1112. W.L. Chan, K.S. Li, Non-propagating solitons of the non-isospectral and variable coefficient modified KdV equation, Phys. A: Math. Gen. 27 (1994) 883–902. H. Triki, T.R. Taha, A.-M. Wazwaz, Solitary wave solutions for a generalized KdV-mKdV equation with variable coefficients, Math. Comput. Simul. 80 (2010) 1867–1873. Z.-Y. Sun, Y.-T. Gao, Y. Liu, X. Yu, Soliton management for a variable-coefficient modified Korteweg–de Vries equation, Phys. Rev. E 84 (2011) 026606. Y. Liu, Y.T. Gao, Z.Y. Sun, X. Yu, Multi-soliton solutions of the forced variable-coefficient extended Korteweg–de Vries equation arisen in fluid dynamics of internal solitary waves, Nonlinear Dyn. 66 (2011) 575–587.
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Optik - International Journal for Light and Electron Optics 172 (2018) 1117–1122
V.N. Serkin, T.L. Belyaeva
[25] G.-Q. Meng, Y.-T. Gao, X. Yu, Y.-J. Shen, Y. Qin, Painlevé analysis, Lax pair, Backlund transformation and multi-soliton solutions for a generalized variablecoefficient KdV-mKdV equation in fluids and plasmas, Phys. Scr. 85 (2012) 055010 (12 pp.). [26] P. Wang, B. Tian, W.-J. Liu, Y. Jiang, Y.-Sh. Xue, Interactions of breathers and solitons of a generalized variable-coefficient Korteweg–de Vries-modified Korteweg–de Vries equation with symbolic computation, Eur. Phys. J. D 66 (2012) 233. [27] L. Wang, Y.-T. Gao, F.-H. Qi, N-fold Darboux transformation and double-Wronskian-typed solitonic structures for a variable-coefficient modified Korteweg–de Vries equation, Annals Phys. 327 (2012) 1974–1988. [28] H. Kumar, F. Chand, Soliton solutions of complex modified KdV equation with time-dependent coefficients, Indian J. Phys. 89 (2013) 909–912. [29] A.-M. Wazwaz, Partial Differential Equations and Solitary Waves Theory, Springer-Verlag, Berlin Heidelberg, 2009. [30] V.N. Serkin, A. Hasegawa, T.L. Belyaeva, Hidden symmetry reductions and the Ablowitz-Kaup-Newell-Segur hierarchies for nonautonomous solitons, in: K. Porsezian, R. Ganapathy (Eds.), Odyssey of Light in Nonlinear Optical Fibers: Theory and Applications, CRC Press, Taylor & Francis Group, Boca Raton, FL, USA, 2015, pp. 145–187. [31] A. Biswas, Solitary wave solution for KdV equation with power law nonlinearity and time-dependent coefficients, Nonlinear Dyn. 58 (2009) 345. [32] F. Cooper, H. Shepard, P. Sodano, Solitary waves in a class of generalized Korteweg–de Vries equation, Phys. Rev. E 48 (1989) 4027–4032. [33] J. He, L. Wang, L. Li, K. Porsezian, R. Erdélyi, Few-cycle optical rogue waves: complex modified Korteweg–de Vries equation, Phys. Rev. E 89 (2014) 062917. [34] A. Chowdury, A. Ankiewicz, N. Akhmediev, Periodic and rational solutions of modified Korteweg–de Vries equation, Eur. Phys. J. D 70 (2016) 104. [35] Y.-F. Liu, R. Guo, H. Li, Breathers and localized solutions of complex modified Korteweg–de Vries equation, Mod. Phys. Lett. B 29 (23) (2014) 1550129. [36] S. Das, D. Ghosh, A.K.N.S. Formalism, Exact Solutions of KdV and Modified KdV Equations with Variable-Coefficients, Int. J. Adv. Res. Math. 6 (2016) 32–41. [37] C. Liu, Y. Ren, Z.-Y. Yang, W.-L. Yang, Superregular breathers in a complex modified Korteweg–de Vries system, Chaos 27 (2017) 083120. [38] N. Cheemaa, S.A. Mehmood, S.T. Raza Rizvi, M. Younis, Single and combined optical solitons with third order dispersion in Kerr media, Optik 127 (2016) 8203–8208. [39] H.H. Chen, General derivation of Bäcklund transformations from inverse scattering problems, Phys. Rev. Lett. 33 (1974) 925–928. [40] K. Konno, M. Wadati, Simple derivation of Backlund transformation from Riccati form of inverse method, Prog. Theor. Phys. 53 (1975) 1652–1656. [41] A.A. Rangwala, J.A. Rao, Complete soliton solutions of the ZS/AKNS equations of the inverse scattering method, Phys. Lett. A 112 (1985) 188–192. [42] K. Porsezian, A. Hasegawa, V. Serkin, T. Belyaeva, R. Ganapathy, Dispersion and nonlinear management for femtosecond optical solitons, Phys. Lett. A 361 (2007) 504–508. [43] V.N. Serkin, A. Hasegawa, T.L. Belyaeva, Soliton self-induced sub-barrier transparency and the controllable ‘shooting out’ effect, J. Mod. Opt. 60 (2013) 444–451. [44] T.L. Belyaeva, V.N. Serkin, Wave-particle duality of solitons and solitonic analog of the Ramsauer-Townsend effect, Eur. Phys. J. D 66 (2012) 153. [45] V.N. Serkin, A. Hasegawa, T.L. Belyaeva, Geiger-Nuttall law for Schrödinger solitons, J. Mod. Opt. 60 (2013) 116–127.
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