Mechanisms of stationary converted waves and their complexes in the multi-component AB system. (English) Zbl 1508.35098
Summary: Under investigation in this article is a multi-component AB system which models the self-induced transparency phenomenon. By using the modified Darboux transformation, we present the breather solutions of such system. We study the subtle mechanism that converts the breathing state into the solitary and periodic ones, through which we obtain various stationary nonlinear excitations such as the multi-peak solitons, (quasi) periodic waves, (quasi) anti-dark solitons, W-shaped solitons and M-shaped solitons which exhibit stationary feature. According to the analysis of the group velocity difference, we give the corresponding conversion rule and present the explicit correspondence of phase diagram of wave numbers for various converted waves, by which we show the gradient relation among these converted waves. Further, by separating the converted waves into the solitary wave as well as the periodic wave, we classify different kinds of nonlinear waves and indicate the difference of the superposition mechanism among them. We show that the breather and various converted waves are formed by different superposition modes between the solitary wave components with different localities and periodic wave components with different frequencies. By virtue of the second-order solutions, we consider all possible superposition situations of two nonlinear waves and present the corresponding nonlinear wave complexes. In particular, for the hybrid structure made of a breather and a nonlinear wave with variable velocity, we then discover that the nonlinear wave does not change its state under the conversion condition, leading to that an additional breathing structure or a dark structure is contained in the converted waves. Finally, we unveil the underlying relationship between the conversion and modulation instability.
MSC:
| 35Q35 | PDEs in connection with fluid mechanics |
| 35Q51 | Soliton equations |
| 35C08 | Soliton solutions |
| 76B25 | Solitary waves for incompressible inviscid fluids |
| 76U65 | Rossby waves |
| 35B20 | Perturbations in context of PDEs |
| 35B10 | Periodic solutions to PDEs |
| 86A05 | Hydrology, hydrography, oceanography |
Keywords:
multi-component AB system; state conversion; nonlinear superposition mechanism; nonlinear wave complex; modulation instability; mixed solutionReferences:
| [1] | Li, Y.; Mu, M., Baroclinic instability in the generalized Phillips’ model Part I: Two-layer model, Adv. Atmos. Sci., 13, 33 (1996) |
| [2] | Li, Y., Baroclinic instability in the generalized Phillips’ model Part II: Three-layer model, Adv. Atmos. Sci., 17, 413 (2000) |
| [3] | Gibbon, J. D.; James, I. N.; Moroz, I. M., An example of soliton behaviour in a rotating baroclinic fluid, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 367, 219 (1979) · Zbl 0423.76016 |
| [4] | Hide, R.; Mason, P. J., Sloping convection in a rotating fluid, Adv. Phys., 24, 47 (1974) |
| [5] | Hide, R.; Mason, P. J.; Plumb, R. A., Thermal convection in a rotating fluid subject to a horizontal temperature gradient: Spatial and temporal characteristics of fully developed baroclinic waves, J. Atmos. Sci., 34, 930 (1977) |
| [6] | Pedlosky, J., Geophysical Fluid Dynamics (1987), Springer: Springer Berlin · Zbl 0713.76005 |
| [7] | Pedlosky, J., Finite-amplitude baroclinic waves, J. Atmos. Sci., 27, 15 (1970) |
| [8] | Phillips, N. A., A simple three-dimensional model for the study of large-scale extratropical flow patterns, J. Met, 8, 381 (1951) |
| [9] | Hart, J. E., A laboratory study of baroclinic instability, Fluid Dyn., 3, 181 (1972) |
| [10] | Eady, E. T., Long waves and cyclone waves, Tellus, 1, 33 (1949) |
| [11] | Barcilon, V., Role of the Ekman layers in the stability of the symmetric regime obtained in a rotating annulus, J. Atmos. Sci., 21, 291 (1964) |
| [12] | R. Hide, Some laboratory experiments on free thermal convection in a rotating fluid subject to a horizontal temperature gradient and their relation to the theory of the global atmospheric circulation, The global circulation of the atmosphere. |
| [13] | Hide, R., An experimental study of thermal convection in a rotating liquid, Philos. Trans. R. Soc. Lond. A Math. Phys. Eng. Sci., 250, 441 (1958) |
| [14] | Dodd, R. K.; Eilkck, J. C.; Gibbon, J. D.; Moms, H. C., Solitons and Nonlinear Wave Equations (1982), Academic Press: Academic Press New York · Zbl 0496.35001 |
| [15] | Tan, B.; Boyd, J. P., Envelope solitary waves and periodic waves in the AB equations, Stud. Appl. Math., 109, 67 (2002) · Zbl 1114.76315 |
| [16] | Moroz, I. M.; Brindley, J., Evolution of baroclinic wave packets in a flow with continuous shear and stratification, Proc. R. Soc. London, 377, 379 (1981) · Zbl 0481.76119 |
| [17] | Mooney, C. J.; Swaters, G. E., Finite amplitude baroclinic instability of a mesoscale gravity current in a channel, Geophys. Astrophys. Fluid Dyn., 82, 173 (1996) · Zbl 1181.76072 |
| [18] | Wu, C. F.; Grimshaw, R. H.J.; Chow, K. W.; Chan, H. N., A coupled AB system: Rogue waves and modulation instabilities, Chaos, 25, Article 103113 pp. (2015) · Zbl 1374.35391 |
| [19] | Kamchatnov, A. M.; Pavlov, M. V., Periodic solutions and whitham equations for the AB system, J. Phys. A: Math. Gen., 28, 3279 (1995) · Zbl 0830.35128 |
| [20] | Guo, R.; Hao, H. Q.; Zhang, L. L., Dynamic behaviors of the breather solutions for the AB system in fluid mechanics, Nonlinear Dynam., 74, 701 (2013) |
| [21] | Wang, X.; Li, Y. Q.; Huang, F.; Chen, Y., Rogue wave solutions of AB system, Commun. Nonlinear Sci. Numer. Simul., 20, 434 (2015) · Zbl 1306.37085 |
| [22] | Wang, L.; Wang, Z. Z.; Jiang, D. Y.; Qi, F. H.; Guo, R., Semirational solutions and baseband modulational instability of the AB system in fluid mechanics, Eur. Phys. J. Plus, 130, 199 (2015) |
| [23] | Matsuno, Y., The Bilinear Transformation Method (1984), Academic Press · Zbl 0552.35001 |
| [24] | Hirota, R., The Direct Method in Soliton Theory (2004), Cambridge University Press · Zbl 1099.35111 |
| [25] | Xie, X. Y.; Meng, G. Q., Multi-dark soliton solutions for a coupled AB system in the geophysical flows, Appl. Math. Lett., 92, 201 (2019) · Zbl 1412.35058 |
| [26] | Su, J. J.; Gao, Y. T.; Ding, C. C., Darboux transformations and rouge wave solutions of a generalized AB system for the geophysical flows, Appl. Math. Lett., 88, 201 (2019) · Zbl 1448.76087 |
| [27] | Xu, Z. W.; Yu, G. F.; Zhu, Z. N., Bright-dark soliton solutions of the multi-component AB system, Wave Motion, 83, 134 (2018) · Zbl 1469.35201 |
| [28] | Geng, X. G.; Shen, J.; Xue, B., Dynamical behaviour of rogue wave solutions in a multi-component AB system, Wave Motion, 89, 1 (2019) · Zbl 1524.35505 |
| [29] | Kotlyarov, V. P., On equations gauge equivalent to the sine-Gordon and Pohlmeyer-Lund-Regge equations, J. Phys. Soc. Japan, 63, 3535 (1994) · Zbl 0972.35522 |
| [30] | Konno, K.; Oono, H., New coupled integrable dispersionless equations, J. Phys. Soc. Japan, 63, 377 (1994) |
| [31] | Lou, S. Y.; Yu, G. F., A generalization of the coupled integrable dispersionless equations, Math. Methods Appl. Sci., 39, 4025 (2016) · Zbl 1346.35182 |
| [32] | Agrawal, G. P., Nonlinear Fiber Optics (1993), Academic Press: Academic Press New York |
| [33] | Baronio, F.; Chen, S. H.; Grelu, P.; Wabnitz, S.; Conforti, M., Baseband modulation instability as the origin of rogue waves, Phys. Rev. A, 91, Article 033804 pp. (2015) |
| [34] | Chen, S. H., Dark and composite rogue waves in the coupled Hirota equations, Phys. Lett. A, 378, 2851 (2014) · Zbl 1298.35027 |
| [35] | Baronio, F.; Degasperis, A.; Conforti, M.; Wabnitz, S., Solutions of the vector nonlinear Schrödinger equations: Evidence for deterministic rogue waves, Phys. Rev. Lett., 109, Article 044102 pp. (2012) |
| [36] | Degasperis, A.; Lombardo, S., Rational solitons of wave resonant-interaction models, Phys. Rev. E, 88, Article 052914 pp. (2013) |
| [37] | Baronio, F.; Conforti, M.; Degasperis, A.; Lombardo, S.; Onorato, M.; Wabnitz, S., Vector rogue waves and baseband modulation instability in the defocusing regime, Phys. Rev. Lett., 113, Article 034101 pp. (2014) |
| [38] | Chen, S. H.; Soto-Crespo, J. M.; Grelu, P., Dark three-sister rogue waves in normally dispersive optical fibers with random birefringence, Opt. Express, 22, Article 27632 pp. (2014) |
| [39] | Chen, S. H.; Song, L. Y., Rogue waves in coupled Hirota systems, Phys. Rev. E, 87, Article 032910 pp. (2013) |
| [40] | Chen, S. H.; Grelu, P.; Soto-Crespo, J. M., Dark- and bright-rogue-wave solutions for media with long-wave-short-wave resonance, Phys. Rev. E, 89, Article 011201 pp. (2014) |
| [41] | Chen, S. H.; Soto-Crespo, J. M.; Grelu, P., Coexisting rogue waves within the (2+1)-component long-wave-short-wave resonance, Phys. Rev. E, 90, Article 033203 pp. (2014) |
| [42] | Chen, S. H., Darboux transformation and dark rogue wave states arising from two-wave resonance interaction, Phys. Lett. A, 378, 1095 (2014) · Zbl 1331.35070 |
| [43] | Radhakrishnan, R.; Lakshmanan, M.; Hietarinta, J., Inelastic collision and switching of coupled bright solitons in optical fibers, Phys. Rev. E, 56, 2213 (1997) |
| [44] | Sheppard, A. P.; Kivshar, Y. S., Polarized dark solitons in isotropic Kerr media, Phys. Rev. E, 55, 4773 (1997) |
| [45] | Soljac̆ić, M.; Steiglitz, K.; Sears, S. M.; Segev, M.; Jakubowski, M. H.; Squier, R., Collisions of two solitons in an arbitrary number of coupled nonlinear Schrödinger equations, Phys. Rev. Lett., 90, Article 254102 pp. (2003) |
| [46] | Jiang, Y.; Tian, B.; Liu, W. J.; Sun, K.; Li, M.; Wang, P., Soliton interactions and complexes for coupled nonlinear Schrödinger equations, Phys. Rev. E, 85, Article 036605 pp. (2012) |
| [47] | Si, L. G.; Yang, W. X.; Lu, X. Y.; Li, J. H.; Yang, X. X., Slow vector optical solitons in a cold four-level inverted-Y atomic system, Eur. Phys. J. D, 55, 161 (2009) |
| [48] | Kanna, T.; Lakshmanan, M.; Dinda, P. T.; Akhmediev, N., Soliton collisions with shape change by intensity redistribution in mixed coupled nonlinear Schrödinger equations, Phys. Rev. E, 73, Article 026604 pp. (2006) |
| [49] | Wang, L.; Wu, X.; Zhang, H. Y., Superregular breathers and state transitions in a resonant erbium-doped fiber system with higher-order effects, Phys. Lett. A, 382, 2650 (2018) |
| [50] | Chen, S. H.; Pan, C. C.; Grelu, P.; Baronio, F.; Akhmediev, N., Fundamental peregrine solitons of ultrastrong amplitude enhancement through self-steepening in vector nonlinear systems, Phys. Rev. Lett., 124, Article 113901 pp. (2020) |
| [51] | Ginzburg, N. S.; Rozental, R. M.; Sergeev, A. S.; Fedotov, A. E.; Zotova, I. V.; Tarakanov, V. P., Generation of rogue waves in gyrotrons operating in the regime of developed turbulence, Phys. Rev. Lett., 119, Article 034801 pp. (2017) |
| [52] | Peregrine, D. H., Water waves nonlinear Schrödinger equations and their solutions, J. Aust. Math. Soc. Ser. B, 25, 16 (1983) · Zbl 0526.76018 |
| [53] | Kibler, B.; Fatome, J.; Finot, C.; Millot, G.; Dias, F.; Genty, G.; Akhmediev, N.; Dudley, J. M., The peregrine soliton in nonlinear fiber optics, Nat. Phys., 6, 790 (2010) |
| [54] | Wang, X.; Wei, J.; Wang, L.; Zhang, J. L., Baseband modulation instability, rogue waves and state transitions in a deformed Fokas-Lenells equation, Nonlinear Dynam., 97, 343 (2019) · Zbl 1430.37077 |
| [55] | Kuznetsov, E. A., Solitons in a parametrically unstable plasma, Sov. Phys. Dokl., 22, 507 (1977) |
| [56] | Ma, Y. C., The perturbed plane-wave solutions of the cubic Schrödinger equation, Stud. Appl. Math., 60, 43 (1979) · Zbl 0412.35028 |
| [57] | Kibler, B.; Fatome, J.; Finot, C.; Millot, G.; Genty, G.; Wetzel, B.; Akhmediev, N.; Dias, F.; Dudley, J. M., Observation of Kuznetsov-Ma soliton dynamics in optical fibre, Sci. Rep., 2, 463 (2012) |
| [58] | Akhmediev, N. N.; Korneev, V. I., Modulation instability and periodic solutions of the nonlinear Schrödinger equation, Theoret. Math. Phys., 69, 1089 (1986) · Zbl 0625.35015 |
| [59] | Dudley, J. M.; Genty, G.; Dias, F.; Kibler, B.; Akhmediev, N., Modulation instability Akhmediev Breathers and continuous wave supercontinuum generation, Opt. Express, 17, 21497 (2009) |
| [60] | Zakharov, V. E.; Gelash, A. A., Nonlinear stage of modulation instability, Phys. Rev. Lett., 111, Article 054101 pp. (2013) |
| [61] | Kibler, B.; Chabchoub, A.; Gelash, A.; Akhmediev, N.; Zakharov, V. E., Superregular breathers in optics and hydrodynamics: Omnipresent modulation instability beyond simple periodicity, Phys. Rev. X, 5, Article 041026 pp. (2015) |
| [62] | Wang, L.; Liu, C.; Wu, X.; Wang, X.; Sun, W. R., Dynamics of superregular breathers in the quintic nonlinear Schrödinger equation, Nonlinear Dynam., 94, 977 (2018) |
| [63] | Liu, C.; Yang, Z. Y.; Zhao, L. C.; Yang, W. L., State transition induced by higher-order effects and background frequency, Phys. Rev. E, 91, Article 022904 pp. (2015) |
| [64] | Chowdury, A.; Ankiewicz, A.; Akhmediev, N., Moving breathers and breather-to-soliton conversions for the Hirota equation, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 471, Article 20150130 pp. (2015) · Zbl 1371.35270 |
| [65] | Chowdury, A.; Kedziora, D. J.; Ankiewicz, A.; Akhmediev, N., Breather-to-soliton conversions described by the quintic equation of the nonlinear Schrödinger hierarchy, Phys. Rev. E, 91, Article 032928 pp. (2015) |
| [66] | Wang, L.; Zhang, J. H.; Wang, Z. Q.; Liu, C.; Li, M.; Qi, F. H.; Guo, R., Breather-to-soliton transitions, nonlinear wave interactions, and modulational instability in a higher-order generalized nonlinear Schrödinger equation, Phys. Rev. E, 93, Article 012214 pp. (2016) |
| [67] | Wang, L.; Zhang, J. H.; Liu, C.; Li, M.; Qi, F. H., Breather transition dynamics peregrine combs walls and modulation instability in a variable-coefficient nonlinear Schrödinger equation with higher-order effects, Phys. Rev. E, 93, Article 062217 pp. (2016) |
| [68] | Wang, L.; Liu, C.; Zhang, J. H., Superregular breathers characteristics of nonlinear stage of modulation instability induced by higher-order effects, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 473, Article 20160681 pp. (2017) · Zbl 1404.35398 |
| [69] | Zhao, L. C.; Li, S. C.; Ling, L. M., W-shaped solitons generated from a weak modulation in the Sasa-Satsuma equation, Phys. Rev. E, 93, Article 032215 pp. (2016) |
| [70] | Gao, P.; Duan, L.; Zhao, L. C.; Yang, Z. Y.; Yang, W. L., Dynamics of perturbations at the critical points between modulation instability and stability regimes, Chaos, 29, Article 083112 pp. (2019) · Zbl 1420.37106 |
| [71] | Liu, C.; Yang, Z. Y.; Zhao, L. C.; Duan, L.; Yang, G. Y.; Yang, W. L., Symmetric and asymmetric optical multipeak solitons on a continuous wave background in the femtosecond regime, Phys. Rev. E, 94, Article 042221 pp. (2016) |
| [72] | Wang, C. J.; Fang, H.; Tang, X. X., State transition of lump-type waves for the (2+1)-dimensional generalized KdV equation, Nonlinear Dynam., 95, 2943 (2019) · Zbl 1437.35613 |
| [73] | Duan, L.; Zhao, L. C.; Xu, W. H.; Liu, C.; Yang, Z. Y.; Yang, W. L., Soliton excitations on a continuous-wave background in the modulational instability regime with fourth-order effects, Phys. Rev. E, 95, Article 042212 pp. (2017) |
| [74] | Duan, L.; Yang, Z. Y.; Gao, P.; Yang, W. L., Excitation conditions of several fundamental nonlinear waves on continuous-wave background, Phys. Rev. E, 99, Article 012216 pp. (2019) |
| [75] | Wang, L.; Zhu, Y. J.; Wang, Z. Q.; Xu, T.; Qi, F. H.; Xue, Y. S., Asymmetric rogue waves breather-to-soliton conversion, and nonlinear wave interactions in the Hirota-Maxwell-Bloch system, J. Phys. Soc. Japan, 85, Article 024001 pp. (2016) |
| [76] | Liu, C.; Yang, Z. Y.; Zhao, L. C.; Yang, W. L., Transition, coexistence, and interaction of vector localized waves arising from higher-order effects, Ann. Phys., 362, 130 (2015) |
| [77] | Ren, Y.; Yang, Z. Y.; Liu, C.; Yang, W. L., Different types of nonlinear localized and periodic waves in an erbium-doped fiber system, Phys. Lett. A, 379, 2291 (2015) |
| [78] | Wang, L.; Wang, Z. Q.; Zhang, J. H.; Qi, F. H.; Li, M., Stationary nonlinear waves, superposition modes and modulational instability characteristics in the AB system, Nonlinear Dynam., 86, 185 (2016) |
| [79] | Zhang, H. S.; Wang, L.; Wang, X.; Xie, X. Y., Transformed nonlinear waves, state transitions and modulation instability in a three-component AB model for the geophysical flows, Nonlinear Dynam., 102, 349 (2020) |
| [80] | Li, P.; Wang, L.; Kong, L. Q.; Wang, X.; Xie, Z. Y., Nonlinear waves in the modulation instability regime for the fifth-order nonlinear Schrödinger equation, Appl. Math. Lett., 85, 110 (2018) · Zbl 1405.35197 |
| [81] | Geng, X. G.; Li, R. M.; Xue, B., A vector general nonlinear Schrödinger equation with (m + n) components, J. Nonlinear Sci., 30, 991 (2020) · Zbl 1437.35627 |
| [82] | Li, R. M.; Geng, X. G., Rogue periodic waves of the sine-Gordon equation, Appl. Math. Lett., 102, Article 106147 pp. (2020) · Zbl 1440.35030 |
| [83] | Li, R. M.; Geng, X. G., On a vector long wave-short wave-type model, Stud. Appl. Math., 144, 164 (2020) · Zbl 1454.35314 |
| [84] | Wei, J.; Geng, X. G.; Zeng, X., The Riemann theta function solutions for the hierarchy of Bogoyavlensky lattices, Trans. Amer. Math. Soc., 371, 1483 (2019) · Zbl 1403.37078 |
| [85] | Geng, X. G.; Liu, H., The nonlinear steepest descent method to long-time asymptotics of the coupled nonlinear Schrödinger equation, J. Nonlinear Sci., 28, 739 (2018) · Zbl 1390.35325 |
| [86] | Wei, J.; Wang, X.; Geng, X. G., Periodic and rational solutions of the reduced Maxwell-Bloch equations, Comm. Nonlinear Sci. Numer. Simul., 59, 1 (2018) · Zbl 1510.78011 |
| [87] | Sun, W. R.; Wang, L., Solitons, breathers and rogue waves of the coupled Hirota system with 4 ×4 lax pair, Commun. Nonlinear Sci. Numer. Simul., 82, Article 105055 pp. (2020) · Zbl 1451.35198 |
| [88] | Sukhorukov, A. A.; Akhmediev, N. N., Intensity limits for stationary and interacting multi-soliton complexes, Phys. Lett. A, 305, 160 (2002) · Zbl 1001.35098 |
| [89] | Akhmediev, N.; Ankewicz, A., Multi-soliton complexes, Chaos, 10, 600 (2000) · Zbl 0971.78011 |
| [90] | Bashkin, E. P.; Vagov, A. V., Instability and stratification of a two-component Bose-Einstein condensate in a trapped ultracold gas, Phys. Rev. B, 56, 10 (1997) |
| [91] | Buryak, A. V.; Akhmediev, N. N., Stationary pulse propagation in N-core nonlinear fiber arrays, IEEE J. Quantum Electron., 31, 682 (1995) |
| [92] | Segev, M.; Christodoulides, D. N., Incoherent Solitons: Self-Trapping of Weakly-Correlated Wave-Packets (2001), Springer: Springer Berlin, Heidelberg |
| [93] | de Sterke, C. M.; Sipe, J. E., Gap solitons, Prog. Opt., 33, 203 (1994) |
| [94] | Manakov, S. V., On the theory of two-dimensional stationary self-focusing of electromagnetic waves, Sov. Phys.—JETP, 38, 248 (1974) |
| [95] | Christodoulides, D. N.; Joseph, R. I., Vector solitons in birefringent nonlinear dispersive media, Opt. Lett., 13, 53 (1988) |
| [96] | Tratnik, M. V.; Sipe, J. E., Bound solitary waves in a birefringent optical fiber, Phys. Rev. A, 38, 2011 (1988) |
| [97] | Davydov, A. S., Solitons in Molecular Systems (1991), Reidel: Reidel Dordrecht · Zbl 1063.81686 |
| [98] | Scott, A., Davydov’s soliton, Phys. Rep., 217, 1 (1992) · Zbl 0644.92005 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
