×

Modulational instability and sister chirped femtosecond modulated waves in a nonlinear Schrödinger equation with self-steepening and self-frequency shift. (English) Zbl 07474638

Summary: We investigate the modulational instability phenomenon and demonstrate that the competing cubic-quintic nonlinearity induces propagating solitonlike bright (dark) solitons, first-order rogue waves embedded on a continuous wave background, and two sister modulated waves in the nonlinear Schrödinger equation with self-steepening and self-frequency shift. We show that the nonlinear chirp associated with each optical pulse propagating on a continuous wave background is directly proportional to the intensity of the wave. We also show that the frequency chirp associated with each of two sister modulated waves is simultaneously directly and inversely proportional to the intensity of the wave. Our investigations show that chirping features and behavior depend on both the self-steepening term and self-frequency shift, while its amplitude can be controlled by varying the parameters of the group velocity dispersion and cubic nonlinearity.

MSC:

35Qxx Partial differential equations of mathematical physics and other areas of application
37Kxx Dynamical system aspects of infinite-dimensional Hamiltonian and Lagrangian systems
78Axx General topics in optics and electromagnetic theory
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Fibich, G., (The nonlinear Schrödinger equation: Singular solutions and optical collapse. The nonlinear Schrödinger equation: Singular solutions and optical collapse, Applied mathematical sciences, vol. 192 (2015), Springer, Cham: Springer, Cham Switzerland) · Zbl 1351.35001
[2] Chen, Shihua; Baronio, Fabio; Soto-Crespo, Jose M.; Grelu, Philippe; Mihalache, Dumitru, Versatile rogue waves in scalar, vector, and multidimensional nonlinear systems, J Phys A, 50, Article 463001 pp. (2017) · Zbl 1386.76097
[3] Williams, F.; Tsitoura, F.; Horikis, T. P.; Kevrekidis, P. G., Solitary waves in the resonant nonlinear Schrödinger equation: Stability and dynamical properties, Phys Lett A, 384, Article 126441 pp. (2020) · Zbl 1448.35484
[4] Chabchoub, Amin; Grimshaw, Roger H. J., The hydrodynamic nonlinear Schrödinger equation: Space and time, Fluids, 1, 23 (2016)
[5] Nauman, Raza; Asad, Zubair, Optical dark and singular solitons of generalized nonlinear Schrödinger’s equation with anti-cubic law of nonlinearity, Modern Phys Lett B, 33, Article 1950158 pp. (2019)
[6] Raza, N.; Osman, M. S.; Abdel-Aty, A. H.; Abdel-Khalek, Sayed; Besbes, H. R., Optical solitons of space-time fractional Fokas-Lenells equation with two versatile integration architectures, Adv Differential Equations, 2020, 517 (2020)
[7] Zhanga, Hai-Qiang; Chen, Fa, Rogue waves for the fourth-order nonlinear Schrödinger equation on the periodic background, Chaos, 31, Article 023129 pp. (2021) · Zbl 1467.35273
[8] Colliander, J.; Keel, M.; Staffilani, G.; Takaoka, H.; Tao, T., Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in \(\mathbf{R}^3\), Ann of Math, 167, 767 (2008) · Zbl 1178.35345
[9] Copie, Fran CÇcois; Randoux, Stéphane; Suret, Pierre, The physics of the one-dimensional nonlinear Schrödinger equation in fiber optics: Rogue waves, modulation instability and self-focusing phenomena, Rev Phys, 5, Article 100037 pp. (2020)
[10] Biswas, A.; Milovic, D.; Edwards, M., Nonlinear Schrödinger’s equation, (Mathematical theory of dispersion-managed optical solitons. Nonlinear physical science (2010), Springer: Springer Berlin, Heidelberg)
[11] Bonetti, J.; Linale, N.; Sánchez, A. D.; Hernandez, S. M.; Fierens, P. I.; Grosz, D. F., Modified nonlinear Schrödinger equation for frequency-dependent nonlinear profiles of arbitrary sign, J Opt Soc Amer B, 36, 3139 (2019)
[12] Deb BM, Chattaraj PK. Generalized nonlinear Schrö dinger equations in quantum fluid dynamics. In: Lakshmanan M. (Ed.) Solitons. Springer series in nonlinear dynamics. Berlin, Heidelberg: Springer. · Zbl 0642.76138
[13] Yamgoué, S. B.; Deffo, G. R.; Tala-Tebue, E.; Pelap, F. B., Exact solitary wave solutions of a nonlinear Schrödinger equation model with saturable-like nonlinearities governing modulated waves in a discrete electrical lattice, Chin Phys B, 27, 12, Article 126303 pp. (2018)
[14] Liu, Wu-Ming; Kengne, Emmanuel, Schrödinger Equations in Nonlinear Systems (2019), Springer · Zbl 1436.81006
[15] Antoine, Xavier; Geuzaine, Christophe; Tang, Qinglin, Perfectly matched layer for computing the dynamics of nonlinear Schrödinger equations by pseudospectral methods. Application to rotating Bose-Einstein condensates, Commun Nonlinear Sci Numer Simul, 90, Article 105406 pp. (2020) · Zbl 1453.65353
[16] Liu, Wu-Ming; Wu, Biao; Niu, Qian, Nonlinear effects in interference of Bose-Einstein condensates, Phys Rev Lett, 84, 11, 2294 (2000)
[17] Kengne, E.; Lakhssassi, A., Compensation process and generation of chirped femtosecond solitons and double-kink solitons in Bose-Einstein condensates with time-dependent atomic scattering length in a time-varying complex potential, Nonlinear Dyn, 104, 4221 (2021)
[18] Kengne, E.; Malomed, Boris A.; Liu, WuMing, Phase engineering of chirped rogue waves in Bose-Einstein condensates with a variable scattering length in an expulsive potential, Commun Nonlinear Sci Numer Simul, 103, Article 105983 pp. (2021) · Zbl 1477.35243
[19] Yemélé, D.; Talla, P. K.; Kofané, T. C., Dynamics of modulated waves in anonlinear discrete LC transmission line: dissipative effects, J Phys D: Appl Phys, 36, 1429 (2003)
[20] Kengne, E.; Liu, W. M., Engineering rogue waves with quintic nonlinearity and nonlinear dispersion effects in a modified Nogochi nonlinear electric transmission network, Phys Rev E, 102, Article 012203 pp. (2020)
[21] Marquié, R.; Bilbault, J. M.; Remoissenet, M., Nonlinear Schrödinger models and modulational instability in real electrical lattices, Physica D, 87, 371 (1995)
[22] Kengne, Emmanuel, Engineering chirped Lambert W-kink signals in a nonlinear electrical transmission network with dissipative elements, Eur Phys J Plus, 136, 266 (2021)
[23] Goyal, Alka Amit; Gupta, Rama; Kumar, C. N., Chirped femtosecond solitons and double-kink solitons in the cubic-quintic nonlinear Schrödinger equation with self-steepening and self-frequency shift, Phys Rev A, 84, Article 063830 pp. (2011)
[24] Singer, F.; Potasek, Mary.; Fang, J. M.; Teich, Malvin. Carl., Femtosecond solitons in nonlinear optical fibers: Classical and quantum effects, Phys Rev A, 46, 7, 4192 (1992)
[25] Trofimov, V. A.; Stepanenko, S.; Razgulin, A., Generalized nonlinear Schrödinger equations describing the Second Harmonic Generation of femtosecond pulse, containing a few cycles, and their integrals of motion, PLoS ONE, 14, 12, Article e0226119 pp. (2019)
[26] Bai, Juan; Rongcao, Y.; Jinping, Tian, Femtosecond quasi-bright soliton solution and its properties under influence of higher-order effects in metamaterials, Acta Opt Sin, 40, 152 (2020)
[27] Zayed, Elsayed M. E.; Alngar, Mohamed E. M.; Biswas, Anjan; Triki, Mehmet EkiciHouria; Alzahrani, Abdullah Kamis; Belic, Milivoj R., Chirped and chirp-free optical solitons infiber Bragg gratingshaving dispersive reflectivity with polynomial form of nonlinearityusing sub-ODE approach, Optik, 204, Article 164096 pp. (2020)
[28] Rizvi, Syed Tahir Raza; Ali, Kashif; Ahmad, Marwa, Optical solitons for Biswas-Milovic equation by new extended auxiliary equation method, Optik, 204, Article 164181 pp. (2020)
[29] Rizvi, Syed Tahir Raza; Ahmad, Sarfraz; Faisal Nadeem, M.; Awais, Muhammad, Optical dromions for perturbed nonlinear Schrödinger equation with cubic quintic septic media, Optik, 226, Article 165955 pp. (2021)
[30] Biswas, A.; Milovic, D.; Edwards, M. E., Mathematical theory of dispersion-managed optical solitons (2010), Springer Verlag: Springer Verlag New York, NY, USA · Zbl 1201.81002
[31] Brown, C. T.A.; Cataluna1, M. A.; Lagatsky, A. A.; Rafailov, E. U.; Agate, M. B.; Leburn, C. G.; Sibbett, W., Compact laser-diode-based femtosecond sources, New J Phys, 6, 175 (2004)
[32] Margiolakis, A.; Tsibidis, G. D.; Dani, K. M.; Tsironis, G. P., Ultrafast dynamics and subwavelength periodic structure formation following irradiation of GaAs with femtosecond laser pulses, Phys Rev B, 98, Article 224103 pp. (2018)
[33] Groot, M. L.; Van Grondelle, R., Femtosecond time-resolved infrared spectroscopy, (Aartsma, T. J.; Matysik, J., Biophysical techniques in photosynthesis. Biophysical techniques in photosynthesis, Advances in photosynthesis and respiration, vol 26 (2008), Springer)
[34] Lu, Qiming; Shen, Qi; Guan, Jianyu; Li, Min; Chen, Jiupeng; Liao, Shengkai; Zhang, Qiang; Peng, Chengzhi, Sensitive linear optical sampling system with femtosecond precision, Rev Sci Instrum, 91, Article 035113 pp. (2020)
[35] Ablowitz, M. J.; Clarkson, P. A., Soliton, nonlinear evolution equations and inverse scattering (1991), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0762.35001
[36] Agrawal, G. P., Nonlinear fiber optics (2001), Academic: Academic San Diego
[37] Mohamadou, A.; Wamba, E.; Doka, S. Y.; Ekogo, T. B.; Kofane, T. C., Generation of matter-wave solitons of the Gross-Pitaevskii equation with a time-dependent complicated potential, Phys Rev A, 84, Article 023602 pp. (2011)
[38] Agrawal, G. P., Nonlinear fiber optics (2007), Academic
[39] Daoui, A. K.; Triki, H.; Biswas, A.; Zhou, Q.; Moshokoa, S. P.; Belic, M., Chirped bright and double-kinked quasi-solitons in optical metamaterials with self-steepening nonlinearity, J Modern Opt, 66, 192 (2019)
[40] Kundu, A., LandauCLifshitz and higher-order nonlinear systems gauge generated from nonlinear Schrödinger-type equations, J Math Phys, 25, 3433 (1984)
[41] Calogero, F.; Eckhaus, W., Nonlinear evolution equations, rescalings, model PDEs and their integrability: I, Inv Prob, 3, 229 (1987) · Zbl 0645.35087
[42] Levi, D.; Scimiterna, C., The Kundu-Eckhaus equation and its discretizations, J Phys A, 42, Article 465203 pp. (2009) · Zbl 1181.35243
[43] Kodama, Y., Optical solitons in a monomode fiber, J Stat Phys, 39, 597 (1985)
[44] Clarkson, P. A.; Tuszynski, J. A., Exact solutions of the multidimensional derivative nonlinear Schrödinger equation for many-body systems of criticality, J Phys A, 23, 4269 (1990) · Zbl 0738.35090
[45] Johnson, R. S., On the modulation of water waves in the neighbourhood of kh \(\approx 1 . 363\), Proc R Soc London A, 357, 131 (1977)
[46] Scalora, M.; Syrchin, M. S.; Akozbek, N.; Poliakov, E. Y.; D’Aguanno, G.; Mattiucci, N.; Bloemer, M. J.; Zheltikov, A. M., Generalized nonlinear Schrödinger equation for dispersive susceptibility and permeability: Application to negative index materials, Phys Rev Lett, 95, Article 013902 pp. (2005)
[47] Ankiewicz, A.; Kedziora, D. J.; Chowdury, A.; Bandelow, U.; Akhmediev, N., Infinite hierarchy of nonlinear Schrödinger equations and their solutions, Phys Rev E, 93, Article 012206 pp. (2016)
[48] de Oliveira, J. R.; Marco A. de Moura, D. J.; Miguel Hickmann, J.; Gomes, A. S.L., Self-steepening of optical pulses in dispersive media, J Opt Soc Amer B, 9, 2025 (1992)
[49] Han, S.-H., Effect of self-steepening on optical solitons in a continuous wave background, Phys Rev E, 83, Article 066601 pp. (2011)
[50] Lucek, J. K.; Blow, K. J., Soliton self-frequency shift in telecommunications fiber, Phys Rev A, 45, 6666 (1992)
[51] Li, Z.; Li, L.; Tian, H.; Zhou, G., New types of solitary wave solutions for the higher order nonlinear Schrödinger equation, Phys Rev Lett, 84, 4096 (2000)
[52] Palacios, S. L.; Guinea, A.; Fernandez-Diaz, J. M.; Crespo, R. D., Dark solitary waves in the nonlinear Schrödinger equation with third order dispersion, self-steepening, and self-frequency shift, Phys Rev E, 60, R45 (1999)
[53] Scalora, M.; Syrchin, M. S.; Akozbek, N.; Poliakov, E. Y.; D’Aguanno, G.; Mattiucci, N.; Bloemer, M. J.; Zheltikov, A. M., Generalized nonlinear Schrödinger equation for dispersive susceptibility and permeability: Application to negative index materials, Phys Rev Lett, 95, Article 013902 pp. (2005)
[54] Whitham, G. B., Linear and nonlinear waves (1999), John Wiley: John Wiley New York · Zbl 0940.76002
[55] Veveakis, E.; Regenauer-Lieb, K., Cnoidal waves in solids, J Mech Phys Solids, 78, 231 (2015) · Zbl 1349.74212
[56] Boyd, J. P., The double cnoidal wave of the Korteweg-de Vries equation: An overview, J Math Phys, 25, 3390 (1984) · Zbl 0558.35065
[57] Wilson, M.; Aboites, V.; Pisarchik, A. N.; Ruiz-Oliveras, F.; Taki, M., Stable cnoidal wave formation in an erbium-doped fiber laser, Appl Phys Express, 4, Article 112701 pp. (2011)
[58] Clarke, R. A., Solitary and cnoidal planetary waves, Geophys Astrophys Fluid Dyn, 2, 343 (1971)
[59] Tala-Tebue, E.; Djoufack, Z. I.; Kamdoum-Tamo, P. H.; Kenfack-Jiotsa, A., Cnoidal and solitary waves of a nonlinear Schrödinger equation in an optical fiber, Optik, 174, 508-512 (2018)
[60] Mahmood, S.; Haas, F., Ion-acoustic cnoidal waves in a quantum plasma, Phys Plasmas, 21, Article 102308 pp. (2014)
[61] Kengne, E.; Liu, W. M., Dissipative ion-acoustic solitons in ion-beam plasma obeying a \(\kappa \)-distribution, AIP Adv, 10, Article 045218 pp. (2020)
[62] Kaur, imardeep; Kaur, Rupinder; Saini, N. S., On-acoustic cnoidal waves with the density effect of spin-up and spin-down degenerate electrons in a dense astrophysical plasma, De Gruyter Naturforsch, 75, 103 (2020)
[63] Osborne, A. R., Shallow water cnoidal wave interactions, Nonlinear Processes Geophys, 1, 241 (1994)
[64] Deng, Bolei; Li, Jian; Tournat, Vincent; Purohit, Prashant K.; Bertoldi, Katia, Dynamics of mechanical metamaterials: A framework to connect phonons, nonlinear periodic waves and solitons, J Mech Phys Solids, 147, Article 104233 pp. (2021)
[65] Giardetti, Nickolas; Shapiro, Amy; Windle, Stephen; Douglas Wright, J., Metastability of solitary waves in diatomic FPUT lattices, Math Eng, 1, 419 (2019) · Zbl 1432.37102
[66] Mo, Chengyang; Singh, Jaspreet; Raney, Jordan R.; Purohit, Prashant K., Cnoidal wave propagation in an elastic metamaterial, Phys Rev E, 100, Article 013001 pp. (2019)
[67] Kengne, E.; Liu, W. M., Exact solutions of the derivative nonlinear Schrödinger equation for a nonlinear transmission line, Phys Rev E, 73, Article 026603 pp. (2006)
[68] Kengne, Emmanuel; Liu, WuMing, Phase engineering chirped super rogue waves in a nonlinear transmission network with dispersive elements, Adv Theory Simul, Article 2100062 pp. (2021) · Zbl 1477.35243
[69] Younis, Muhammad; Ali, Safdar; Rizvi, Syed Tahir Raza; Tantawy, Mohammad; Tariq, Kalim U.; Bekir, Ahmet, Investigation of solitons and mixed lump wave solutions with (3 + 1) -dimensional potential-YTSF equation, Commun Nonlinear Sci Numer Simulat, 94, Article 105544 pp. (2021) · Zbl 1456.37078
[70] Rizvi, Syed Tahir Raza; Seadawy, Aly R.; Ali, Ijaz; Bibi, Ishrat; Younis, Muhammad, Chirp-free optical dromions for the presence of higher order spatio-temporal dispersions and absence of self-phase modulation in birefringent fibers, Modern Phys Lett B, 34, 35, Article 2050399 pp. (2020)
[71] Madelung, E., Eine anschauliche Deutung der Gleichung von Schrödinger, Naturwissenschaften, 14, 1004 (1926)
[72] Kumar, V. R.; Radha, R.; Wadati, M., Phase engineering and solitons of Bose-Einstein condensates with two- and three-body interactions, J Phys Soc Japan, 79, Article 074005 pp. (2010)
[73] Marquié, R.; Bilbault, J. M.; Remoissenet, M., Nonlinear Schrödinger models and modulational instability in real electrical lattices, Physica D, 87, 371 (1995)
[74] Liang, Z. X.; Zhang, Z. D.; Liu, W. M., Dynamics of a bright soliton in Bose-Einstein condensates with time-dependent atomic scattering length in an expulsive parabolic potential, Phys Rev Lett, 94, Article 050402 pp. (2005)
[75] Xu, Z. Y.; Li, L.; Li, Z.; Zhou, G., Modulation instability and solitons on a cw background in an optical fiber with higher-order effects, Phys Rev E, 67, Article 026603 pp. (2003)
[76] Weierstrass, K., Mathematische werke V, 4-16 (1915), Johnson: Johnson New York
[77] Whittaker, E. T.; Watson, G. N., A course of modern analysis, 454 (1927), Cambridge University Press: Cambridge University Press Cambridge
[78] Chandrasekharan, K., Elliptic functions, 44 (1985), Springer: Springer Berlin · Zbl 0575.33001
[79] Schürmann, H. W.; Serov, V. S., Traveling wave solutions of a generalized modified Kadomtsev-Petviashvili equation, J Math Phys, 45, 2181 (2004) · Zbl 1071.35115
[80] Schürmann, H. W., Traveling-wave solutions of the cubic-quintic nonlinear Schrödinger equation, Phys Rev E, 54, 4312 (1996)
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.