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Tree-like structures and Freak waves generation induced by quintic-nonlinearity and cubic-Raman effect in a nonlinear metamaterial

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Abstract

The goal of this investigation is based on electromagnetic wave behavior which is modeled by a nonlinear Schrödinger equation in a nonlinear metamaterial optical system. Two regimes are under consideration known as negative index regime and absorption regime. During the propagation, second-order dispersion, cubic-quintic nonlinearities and cubic-Raman effect are taken into account. The characterization of electromagnetic wave is assured by collective coordinates method and conventional ansatz function. The soliton light pulse is progressively modified by quintic-nonlinearity and cubic-Raman effect in order to provoke freak waves, tree structures and soliton self-frequency shift generation. Hence, the coefficients of the Schrödinger equation are plotted in order to improve the explanation of these strange phenomena. Moreover, all those above mentioned phenomena are strongly influenced by frequency and the regime under consideration. Thus, some specific freak waves such as Kuznetsov–Ma, Peregrine and Riemann waves are found. It is clearly pointed out the key role of internal and external disturbances due to cubic-Raman effects. That leads to freak waves, tree structures and soliton self frequency shift generation for both regimes. The so-called self-compensation process as technique of cancellation of these phenomena is also investigated.

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References

  • Agrawal, G.P.: Nonlinear Fiber Optics, 2nd edn. Academic Press, San Diego (1995)

    MATH  Google Scholar 

  • Akhmediev, N., Kibler, B., Baronio, F., Belia, M., Zhong, W.P., Zhang, Y., Chang, W., Soto-Crespo, J.M., Vouzas, P., Grelu, P., Lecaplain, C., Hammani, K., Rica, S., Picozzi, A., Tlidi, M., Panajotov, K., Mussot, A., Bendahmane, A., Szriftgiser, P., Genty, G., Dudley, J., Kudlinski, A., Demircan, A., Morgner, U., Amiraranashvili, S., Bree, C., Steinmeyer, G., Masoller, C., Broderick, N.G.R., Runge, A.F.J., Erkintalo, M., Residori, S., Bortolozzo, U., Arecchi, F.T., Wabnitz, S., Tiofack, C.G., Coulibaly, S., Taki, M.: Roadmap on optical rogue waves and extreme events. J. Opt. 18, 063001 (2016)

    ADS  Google Scholar 

  • Atangana, J., Essama, B.G., Mokhtari, B., Kofane, T.C.: Cubic-quintic saturable nonlinearity effects on a light pulse strongly distorted by the fourth-order dispersion. J. Mod. Opt. 60, 292–300 (2013)

    ADS  MathSciNet  MATH  Google Scholar 

  • Atangana, J., Essama, B.G.O., Motto, F.B., Mokhtari, B., Eddeqaqi, N.C., Kofane, T.C.: Kuznetsov-Ma waves train generation in a left-handed material. J. Mod. Opt. 62, 392–402 (2014)

    ADS  Google Scholar 

  • Atangana, J., Ndi Nnanga, B.M., Onana Essama, B.G., Mokthari, B., Cherkaoui Eddeqaqi, N., Kofane, T.C.: Efficient method of calculation of Raman soliton self-frequency shift in nonlinear optical media. Opt. Commun. 339, 194–208 (2015)

    Google Scholar 

  • Baizakov, B.B., Bouketir, A., Al-Marzoug, S.M., Bahlouli, H.: Effect of quintic nonlinearity on modulation instability in coupled nonlinear Schrödinger systems. Nonlinear Sci. 180, 792–798 (2018)

    Google Scholar 

  • Bayindir, C.: Rogue waves of the Kundu–Eckhaus equation in a chaotic wave field. Phys. Rev. E 93, 062215 (2016)

    ADS  MathSciNet  Google Scholar 

  • Beckwitt, K., Wise, F.W., Qian, L., Walker II, L.A., Canto-Said, E.: Compensation for self-focusing by use of cascade quadratic nonlinearity. Opt. Lett. 26, 1696–1698 (2001)

    ADS  Google Scholar 

  • Benjamin, T.B., Feir, J.E.: The disintegration of Wa-vetrains on deep water. Fluid Mech. 27, 417–430 (1967)

    ADS  MATH  Google Scholar 

  • Blow, K.J., Doran, N.J., Wood, D.: Suppression of the soliton self-frequency shift by bandwidth-limited amplification. J. Opt. Soc. Am. B 5, 1301–1304 (1988)

    ADS  Google Scholar 

  • Boardman, A.D., Alberucci, A., Assanto, G., Grimalsky, V.V., Kibler, B., McNiff, J., Nefedov, I.S., Rapoport, Y.G., Valagiannopoulos, C.A.: Waves in hyperbolic and double negative metamaterials including rogues and solitons. Nanotechnology 28, 444001 (2017)

    ADS  Google Scholar 

  • Brunetti, M., Marchiando, N., Berti, N., Kasparian, J.: Nonlinear fast growth of water waves under wing forcing. Phys. Lett. A. 378, 1025–1030 (2014)

    ADS  MathSciNet  MATH  Google Scholar 

  • Charalampidis, E.G., Lee, J., Kevrekidis, P.G.: Photonic rogue waves. Phys. Rev. E 98, 032903 (2018a)

    ADS  Google Scholar 

  • Charalampidis, E.G., Maraver, J.C., Frantzeskakis, D.J., Kevrekidis, P.G.: Rogue waves in ultrecold bosonic seas. Rom. Rep. Phys. 70, 504 (2018b)

    Google Scholar 

  • Chen, W.C., Cardin, A., Koirala, M., Liu, X., Tyler, T., West, K.G., Bingham, C.M., Starr, T., Starr, A.F., Jokerst, N.M., Padilla, W.J.: Role of surface electromagnetic waves in metamaterial absorbers. Opt. Express 24, 6783–6792 (2016)

    ADS  Google Scholar 

  • Cui, Y., He, Y., Jin, Y., Ding, F., Yang, L., Ye, Y., Zhong, S., Lin, Y., He, S.: Plasmonic and metamaterial structures as electromagnetic absorbers. Laser Photonics Rev. 8(4), 495–520 (2014)

    ADS  Google Scholar 

  • Dinda, P., Moubissi, A.B., Nakkeeran, K.: A collective variable approach for dispersion-managed solitons. J. Phys. A Math. Gen. 34, 103–110 (2001)

    MathSciNet  MATH  Google Scholar 

  • Dinda, P., Nakkeeran, K., Labruyere, A.: Suppression of soliton self-frequency shift by upshifted filtering. Opt. Lett. 27, 382–384 (2002)

    ADS  Google Scholar 

  • Dudley, J.M., Genty, G., Coen, S.: Supercontinuum generation in photonic crystal fiber. Rev. Mod. Phys. 78, 1135–1184 (2006)

    ADS  Google Scholar 

  • Dudley, J.M., Dias, F., Erkintalo, M., Genty, G.: Instabilities, breathers and rogue waves in optics. Nat. Photonics 8, 755–764 (2014)

    ADS  Google Scholar 

  • Gordon, J.P.: Theory of the soliton self-frequency shift. Opt. Lett. 11, 662–664 (1986)

    ADS  Google Scholar 

  • Gouadjio Dontsop, P.Y., Onana Essama, B.G., Dongo, J.M., Mbou Dedzo, M., Atangana, J., Yemele, D., Kofane, T.C.: Akhmediev–Peregrine rogue waves generation in a composite right/left-handed transmission line. Opt. Quant. Electron. 48, 59–77 (2016)

    Google Scholar 

  • Gouveia-Neto, A.S., Gomes, A.S.L., Taylor, J.R.: Suppression and manipulation of the soliton self-frequency shift. Opt. Lett. 14, 514–516 (1989)

    ADS  Google Scholar 

  • Huang, Y., Zhou, Z., Zhang, Y., Chen, G., Xiao, H.: A temperature self-compensated LPFG sensor for large strain measurements at high temperature. IEEE Trans. Instrum. Meas. 59, 2997–3004 (2010)

    Google Scholar 

  • Janssen, P.A.E.M.: Wave-induced stress and drag of air flow over sea waves. J. Phys. Oceanogr. 19, 745–754 (1989)

    ADS  Google Scholar 

  • Judge, A.C., Bang, O., Eggleton, B.J., Kuhlmey, B.T., Mägi, E.C., Pant, R., de Sterke, C.M.: Optimization of the soliton self-frequency shift in a tapered photonic crystal fiber. J. Opt. Soc. Am. B 26, 2064–2071 (2009)

    ADS  Google Scholar 

  • Kato, M., Fujiura, K., Kurihara, T.: Asynchronous all-optical bit-by-bit self-signal recognition and demultiplexing from overlapped signals achieved by self-frequency shift of raman soliton. Electron. Lett. 40, 381–382 (2004)

    ADS  Google Scholar 

  • Kharif, C., Pelinovsky, E.: Physical mechanisms of the rogue wave phenomenon. Eur. J. Mech. B/Fluid 21, 561–577 (2003)

    MathSciNet  MATH  Google Scholar 

  • 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, 463–467 (2012)

    ADS  Google Scholar 

  • Kuse, N., Nomura, Y., Ozawa, A., Kuwata-Gonokami, M., Watanabe, S., Kobayashi, Y.: Self-compensation of third-order dispersion for ultrashort pulse generation demonstrated in an Yb fiber oscillator. Opt. Lett. 35, 3868–3870 (2010)

    ADS  Google Scholar 

  • Kuznetsov, E.A.: Solitons in a parametrically unstable plasma. Akademiia Nauk SSSR Doklady 236, 575–577 (1977)

    ADS  Google Scholar 

  • Landy, N.I., Sajuyigbe, S., Mock, J.J., Smith, D.R., Padilla, W.J.: A perfect metamaterial absorber. Phys. Rev. Lett. 100, 207402 (2008)

    ADS  Google Scholar 

  • Lee, J.H., Howe, J.V., Xu, C., Liu, X.: Soliton self-frequency shift: experimental demonstrations and applications. IEEE J. Sel. Top. Quantum Electron. 14, 713–723 (2008)

    ADS  Google Scholar 

  • Li, P., Yang, R., Zhiyong, X.: Gray solitary-wave solutions in nonlinear negative-index materials. Phys. Rev. E 82, 046603 (2010)

    ADS  Google Scholar 

  • Li, W.X., Guo, Z.W., Guo, Z.B., Qiang, Z.H.: Rogue waves of the higher-order dispersive nonlinear Schrödinger equation. Commun. Theor. Phys. 58, 531–538 (2012)

    ADS  MathSciNet  MATH  Google Scholar 

  • Litchinitser, N.M., Shalaev, V.M.: Loss as a route to transparency. Nat. Photonics 3, 75–79 (2009)

    ADS  Google Scholar 

  • Liu, W.: The study status and development of metamaterial absorber. In: International Conference on Logistics Engineering, Management and Computer Science (LEMCS), pp. 1318–1322 (2015)

  • Liu, G., Hou, W., Qiao, W., Han, M.: Fast-response fiber-optic anemometer with temperature self-compensation. Opt. Expr. 23, 13562–13570 (2015a)

    ADS  Google Scholar 

  • Liu, C., van der Wel, R.E.C., Rotenberg, N., Kuipers, L., Krauss, T.F., Falco, A.D., Fratalocchi, A.: Triggering extreme events at the nanoscale in photonic seas. Nat. Phys. 11, 358–363 (2015b)

    Google Scholar 

  • Liu, B., Li, L., Malomed, B.A.: Stability band structure for periodic states in periodic potentials. Appl. Numer. Math. 141, 44–53 (2018)

    MathSciNet  MATH  Google Scholar 

  • Lobet, M., Lard, M., Sarrazin, M., Deparis, O., Henrard, L.: Plasmon hybridization in pyramidal metamaterials: a route towards ultra-broadband absorption. Opt. Express 22, 12678–12690 (2014)

    ADS  Google Scholar 

  • Ma, Y.: The perturbed plane-wave solutions of the cubic Schrödinger equation. Stud. Appl. Math. 60, 43–58 (1979)

    ADS  MathSciNet  MATH  Google Scholar 

  • Mitschke, F.M., Mollenauer, L.F.: Discovery of the soliton self-frequency shift. Opt. Lett. 11, 659–661 (1986)

    ADS  Google Scholar 

  • Mokhtari, B., Cherkaoui Eddeqaqi, N., Atangana, J., Onana Essama, B.G., Kofane, T.C.: Nonlinear dispersion equation and guided modes in a slab waveguide composed of a negative-index medium. Opt. Quant. Electron. 46, 155–163 (2013)

    Google Scholar 

  • Mokhtari, B., Cherkaoui Eddeqaqi, N., Atangana, J., Onana Essama, B.G., Kofane, T.C.: Influence of refractive index cladding on guided modes in a negative-index slab waveguide, multimedia computing and systems. In: 2014 International Conference On-IEEE (ICMCS), Marrakech, 14–16 April, pp. 1358–1362 (2014a)

  • Mokhtari, B., Cherkaoui Eddeqaqi, N., Atangana, J., Onana Essama, B.G., Kofane, T.C.: Impact of cladding permeability on guided modes in a negative index slab waveguide. Int. J. Emerg. Technol. Adv. Eng. 4, 721–726 (2014b)

    Google Scholar 

  • Mokhtari, B., Cherkaoui Eddeqaqi, N., Atangana, J., Onana Essama, B.G., Kofane, T.C.: Possible behaviors of TE modes in a left-handed slab waveguide. Int. J. Mater. Eng. 5, 274–288 (2014c)

    Google Scholar 

  • Nishizawa, N., Goto, T.: Compact system of wavelength-tunable femtosecond soliton pulse generation using optical fibers. Photonics Technol. Lett. 11, 325–327 (1999)

    ADS  Google Scholar 

  • Onana Essama, B.G., Atangana, J., Mokhtari, B., Eddeqaqi, N.C., Kofane, T.C.: Theoretical model for electromagnetic wave propagation in negative index material induced by cubic-quintic nonlinearities and third-order dispersion effects. Opt. Quant. Electron. 46, 911–924 (2013)

    Google Scholar 

  • Onana Essama, B.G., Atangana, J., Motto, F.B., Mokhtari, B., Eddeqaqi, N.C., Kofane, T.C.: Rogue waves generation in a left-handed nonlinear transmission line with series varactor diodes. J. Mod. Opt. 61, 1002–1008 (2014a)

    ADS  Google Scholar 

  • Onana Essama, B.G., Atangana, J., Motto, F.B., Mokhtari, B., Cherkaoui Eddeqaqi, N., Kofane, T.C.: Optical rogue waves generation in a nonlinear metamaterial. Opt. Commun. 331, 334–347 (2014b)

    ADS  Google Scholar 

  • Onana Essama, B.G., Atangana, J., Motto, F.B., Mokhtari, B., Cherkaoui Eddeqaqi, N., Kofane, T.C.: Rogue wave trains generation in a metamaterial induced by cubic-quintic-nonlinearities and second-order dispersion. Phys. Rev. E 90, 032911–12 (2014c)

    ADS  Google Scholar 

  • Onorato, M., Proment, D.: Approximate rogue wave solutions of the forced and damped nonlinear Schrödinger equation for water waves. Phys. Lett. A. 376, 3057–3059 (2012)

    ADS  Google Scholar 

  • Onorato, M., Residori, S., Bortolozzo, U., Montina, A., Arecchi, F.T.: Rogue waves and their generating mechanisms in different physical contexts. Phys. Rep. 528, 47–89 (2013a)

    ADS  MathSciNet  Google Scholar 

  • Onorato, M., Proment, D., Clauss, G., Klein, M.: Rogue waves: from nonlinear Schrödinger breather solutions to sea-keeping test. PLoS ONE 8, 54625–54629 (2013b)

    ADS  Google Scholar 

  • Osborne, A.R.: Nonlinear Ocean Waves and the Inverse Scattering Transform. Academic Press, Amsterdam (2010)

    MATH  Google Scholar 

  • Pant, R., Judge, A.C., Magi, E.C., Kuhlmey, B.T., de Sterke, M., Eggleton, B.J.: Characterization and optimization of photonic cristal fibers for enhanced soliton self-frequency shift. J. Opt. Soc. Am. B 27, 1894–1901 (2010)

    ADS  Google Scholar 

  • Pelinovsky, E., Kharif, C. (eds.): Extreme Ocean Waves. Springer, New York (2008)

    MATH  Google Scholar 

  • Podolskiy, V.A., Kuhta, N.A., Milton, G.W.: Optimizing the superlens: manipulating geometry to enhance the resolution. Appl. Phys. Lett. 87, 231113 (2005)

    ADS  Google Scholar 

  • Popov, A.K., Shalaev, V.M.: Merging Nonlinear Optics and Negative-index Metamaterials (2011). arXiv:1108.0867v1 [Physics.Optics]

  • Popov, A.K., Shalaev, V.M.: Compensating losses in negative-index metamaterials by optical parametric amplification. Opt. Lett. 31, 2169–2171 (2006)

    ADS  Google Scholar 

  • Scalora, M., Syrchin, M.S., Akozbek, N., Poliakov, E.Y., aguano, S.G.D., Mattiucci, N., Blomer, M.J., Zheltikov, A.M.: Generalized nonlinear Schrödinger equation for dispersive susceptibility and permeability : application to negative index materials. Phys. Rev. Lett. 95, 013902 (2005)

    ADS  Google Scholar 

  • Schadt, D., Jaskorzynska, B.: Suppression of the Raman self-frequency shift by cross-phase modulation. J. Opt. Soc. Am. B 5, 2374–2378 (1988)

    ADS  Google Scholar 

  • Serebryannikov, E.E., Hu, M.L., Li, Y.F., Wang, C.Y., Wang, Z., Chai, L., Zheltikov, A.M.: Enhanced soliton self-frequency shift of ultrashort light pulses. JETP Lett. 81, 487–490 (2005)

    ADS  Google Scholar 

  • Skryabin, D.V., Luan, F., Knight, C., Russell, J.: Soliton self-frequency shift cancellation in photonic crystal fibers. Science 301, 1705–1708 (2003)

    ADS  Google Scholar 

  • Snee, D.D., Ma, Y.-P.: Edge solitons in a nonlinear mechanical topological insulator. Nonlinear Sci. 30, 100487 (2018)

    Google Scholar 

  • Song, G., Zhang, C., Cheng, Q., Jing, Y., Qiu, C., Cui, T.: Transparent coupled membrane metamaterials with simultaneous microwave absorption and sound reduction. Opt. Express 26, 22916–22925 (2018)

    ADS  Google Scholar 

  • Soto-Crespo, J.M., Grelu, P., Akhmediev, N.: Dissipative rogue waves: extreme pulses generated by passively mode-locked lasers. Phys. Rev. E 84, 016604 (2011)

    ADS  Google Scholar 

  • Sulem, C., Sulem, P.L.: The Nonlinear Schrödinger Equation: Self-focusing and Wave Collapse. Springer, New York (1999)

    MATH  Google Scholar 

  • Tan, J., Deng, Z.H., Wu, T., Tang, B.: Propagation and interaction of magnetic solitons in a ferromagnetic thin film with the interfacial Dzyaloshinskii - Moriya interaction. J. Magn. Magn. Mater. 475, 445–452 (2019)

    ADS  Google Scholar 

  • Toffoli, A., Fernandez, L., Monbaliu, J., Benoit, M., Gagnaire-Renou, E., Lefèvre, J.M., Cavaleri, L., Proment, D., Pakozdi, C., Stansberg, C.T., Waseda, T., Onorato, M.: Experimental evidence of the modulation of a plane wave to oblique perturbations and generation of rogue waves in finite water depth. Phys. Fluids 25, 091701–091707 (2013)

    ADS  Google Scholar 

  • Tütüncü, B., Torpi, H., Urul, B.: A comparative study on different types of metamaterials for enhancement of microstrip patch antenna directivity at the Ku-band (12 GHz), Turk. J. Electr. Eng. Comp. Sci. 26, 1171–1179 (2018)

    Google Scholar 

  • Uzunov, I.M.: Description of the suppression of the soliton self-frequency shift by bandwidth-limited amplification. Phys. Rev. E 82, 066603 (2010)

    ADS  Google Scholar 

  • Van Gorder, R.A., Krause, A.L., Kwiecinski, J.A.: Amplitude death criteria for coupled complex Ginzburg–Landau systems. Nonlinear Sci. 97(1), 151–159 (2019)

    MATH  Google Scholar 

  • Veselago, V.G.: The electrodynamics of substances with simultaneously negative values of \(\varepsilon \) and \(\mu \). Sov. Phys. Uspekhi 10, 509–514 (1968)

    ADS  Google Scholar 

  • Voronin, A.A., Zheltikov, A.M.: Soliton self-frequency shift decelerated by self-steepening. Opt. Lett. 33, 1723–1725 (2008)

    ADS  Google Scholar 

  • Watts, C.M., Liu, X., Padilla, W.J.: Metamaterial electromagnetic wave absorbers. Adv. Mater. 24, 98–120 (2012)

    Google Scholar 

  • Webb, K.J., Yang, M., Ward, D.W., Nelson, K.A.: Metrics for negative-refractive-index materials. Phys. Rev. E 70, 035602R (2004)

    ADS  Google Scholar 

  • Xiang, Y., Dai, X., Wen, S., Guo, J., Fan, D.: Controllable Raman Soliton self frequency shift in nonlinear metamaterial. Phys. Rev. A 84, 033815 (2011)

    ADS  Google Scholar 

  • Xiang, Y., Wu, J., Dai, X., Wen, S., Guo, J., Wang, Q.: Manipulating dispersive wave generation by anomalous self-steepening effect in metamaterials. Opt. Exp. 20, 26828–26836 (2012)

    ADS  Google Scholar 

  • Xu, C., Liu, X.: Photonic analog-to-digital converter using soliton self-frequency shift and interleaving spectral filters. Opt. Lett. 28, 986–988 (2003)

    ADS  Google Scholar 

  • Zakharov, V.: Stability of periodic waves of finite amplitude on the surface of a deep fluid. J. Appl. Mech. Tech. Phys. 9, 190–194 (1968)

    ADS  Google Scholar 

  • Zakharov, V., Ostrovsky, L.: Modulation instability: the beginning. Phys. D. Nonlinear Phenom. 238, 540–548 (2009)

    ADS  MathSciNet  MATH  Google Scholar 

  • Ziolkowski, R.W.: Superluminal transmission of information through an electromagnetic metamaterial. Phys. Rev. E 63, 046604 (2001)

    ADS  Google Scholar 

  • Ziolkowski, R.W.: Pulsed and CW Gaussian beam interactions with double negative metamaterial slabs. Opt. Exp. 11, 662–681 (2003)

    ADS  Google Scholar 

  • Ziolkowski, R.W., Kipple, A.D.: Causality and double-negative metamaterials. Phys. Rev. E 68, 026615 (2003)

    ADS  Google Scholar 

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Authors and Affiliations

  1. Laboratory of Mechanics, Department of Physics, Faculty of Science, University of Yaounde I, P.O. Box 812, Yaounde, Cameroon

    Bibiane Mireille Ndi Nnanga

  2. Laboratory of Electronics, Department of Physics, Faculty of Science, University of Dschang, P.O. Box 67, Dschang, Cameroon

    Paul Yannick Gouadjio Dontsop & David Yemele

  3. Applied Biotechnology and Engineering Laboratory, Department of Electrical Engineering, Higher Technical Teachers, Training College (HTTTC) of Ebolowa, University of Yaounde I, P.O. Box 886, Ebolowa, Cameroon

    Bedel Giscard Onana Essama

  4. Department of Physics, Islamic University of Gaza, P.O. Box 108, Gaza City, Palestine

    Mohammed Shabat

  5. Higher Teacher Training College of Yaounde, University of Yaounde I, P.O. Box 47, Yaounde, Cameroon

    Jacques Atangana

  6. Centre d’Excellence Africain des Technologies de l’Information et de la Communication (CETIC), Université de Yaoundé I, Yaounde, Cameroon

    Bibiane Mireille Ndi Nnanga, Bedel Giscard Onana Essama & Jacques Atangana

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  1. Bibiane Mireille Ndi Nnanga
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  2. Paul Yannick Gouadjio Dontsop
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Ndi Nnanga, B.M., Gouadjio Dontsop, P.Y., Onana Essama, B.G. et al. Tree-like structures and Freak waves generation induced by quintic-nonlinearity and cubic-Raman effect in a nonlinear metamaterial. Opt Quant Electron 52, 356 (2020). https://doi.org/10.1007/s11082-020-02469-4

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