nach oben

Optical and Quantum Electronics

Erschienen in:

01.05.2024

Unraveling the dynamic complexity: exploring the (3+1)-dimensional conformable mKdV-ZK equation

verfasst von: Xiaoye Ding, Salah Mahmoud Boulaaras, Hamood Ur Rehman, Ifrah Iqbal, Aziz Ullah Awan, Iffat Sabir

Erschienen in: Optical and Quantum Electronics | Ausgabe 5/2024

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config

KI-gestützte Suche

Aus

Abstract

The primary objective of this article is to construct novel solitary wave solutions for the nonlinear (3+1)-dimensional modified Korteweg-de Vries-Zakharov-Kuznetsov equation with a conformable derivative by employing the subsidiary ordinary differential equation method and \(\left( \frac{1}{\varphi (\varsigma )},\frac{ \varphi '(\varsigma )}{\varphi (\varsigma )}\right) \) method. The solutions obtained encompass diverse bright, dark, singular, and periodic solitary waves. Additionally, the article thoroughly discusses the sufficient conditions for the existence of these solutions. To visually represent the obtained solutions, we include 2D and 3D graphical images using the computational software Maple 18. We also included a comparison of fractional derivatives at different values of fractional order and conducted an analysis to understand the influence of nonlinear parameters on wave behaviors. The significance of these solutions lies in their wide-ranging applications in applied sciences and mathematical physics. Furthermore, the proposed methods are invaluable in efficiently solving nonlinear partial differential equations prevalent in engineering and physical science.

Vorheriger Artikel Design of ultra-broadband and highly coherent MIR supercontinuum generation using a photonic crystal fiber

Nächster Artikel Stability improvement of high-power semiconductor laser diode regulators used in infrared solid-state laser applications

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Abdoon, M.A., Hasan, F.L., Taha, N.E.: Computational technique to study analytical solutions to the fractional modified kdv-zakharov-kuznetsov equation. Abstract Appl. Anal. 2022, 2162356 (2022)MathSciNet

Abdulkareem, H.H., Ismael, H.F., Panakhov, E.S., Bulut, H.: Some novel solutions of the coupled Whitham-Broer-Kaup equations. Int. Conf. Computat. Math. Eng. Sci. 4, 200–208 (2020)

Akram, G., Sadaf, M., Arshed, S., Sameen, F.: Traveling wave solutions of conformable time-fractional Klien-Fock-Gordon equation by the improved tan \(( \psi /2)\)-expansion method. J King Saud Univ.-Sci. 34(3), 101822 (2022a)

Akram, G., Sadaf, M., Khan, M.A.U.: Soliton solutions of Lakshmanan-Porsezian-Daniel model using modified auxiliary equation method with parabolic and anti-cubic law of nonlinearities. Optik 252, 168372 (2022b)ADS

Akram, G., Sadaf, M., Sameen, F.: Optical solitons for the complex Ginzburg-Landau equation with Kerr law and non-Kerr law nonlinearity. Opt. Quant. Electron. 54(10), 630 (2022c)

Akram, G., Sadaf, M., Sarfraz, M., Anum, N.: Dynamics investigation of (1+ 1)-dimensional time-fractional potential Korteweg-de Vries equation. Alex. Eng. J. 61(1), 501–509 (2022d)

Al-Ghafri, K.S., Rezazadeh, H.: Solitons and other solutions of (3+ 1)-dimensional space-time fractional modified KdV-Zakharov-Kuznetsov equation. Appl. Math. Nonlinear Sci. 4(2), 289–304 (2019)MathSciNet

Alam, M.N., Hafez, M.G., Akbar, M.A.: Exact traveling wave solutions to the (3+ 1)-dimensional mKdV-ZK and the (2+ 1)-dimensional Burgers equations via exp \((- \phi (\eta ))\)-expansion method. Alex. Eng. J. 54(3), 635–644 (2015)

Alquran, M.: New interesting optical solutions to the quadratic-cubic Schrodinger equation by using the Kudryashov-expansion method and the updated rational sine-cosine functions. Opt. Quant. Electron. 54(10), 666 (2022)

Alquran, M. (2023). Classi?cation of single-wave and bi-wave motion through fourth-order equations generated from the Ito model. Physica Scripta

Arnous, A. H., Nofal, T. A., Biswas, A., Yildirim, Y., and Asiri, A. (2023). Cubic-quartic optical solitons of the complex Ginzburg-Landau equation: A novel approach. Nonlinear Dynamics, 1-16

Baskonus, H.M., Bulut, H., Sulaiman, T.A.: New complex hyperbolic structures to the lonngren-wave equation by using sine-gordon expansion method. Appl. Math. Nonlinear Sci. 4(1), 129–138 (2019)MathSciNet

Baskonus, H.M., Sulaiman, T.A., Bulut, H.: On the novel wave behaviors to the coupled nonlinear Maccari’s system with complex structure. Optik 131, 1036–1043 (2017)ADS

Biswas, A., Jawad, A.J.A.M., Zhou, Q.: Resonant optical solitons with anti-cubic nonlinearity. Optik 157, 525–531 (2018)ADS

Bulut, H., Sulaiman, T.A., Baskonus, H.M., Aktürk, T.: On the bright and singular optical solitons to the (2+ 1)-dimensional NLS and the Hirota equations. Opt. Quant. Electron. 50, 1–12 (2018)

Ekici, M., Mirzazadeh, M., Eslami, M., Zhou, Q., Moshokoa, S.P., Biswas, A., Belic, M.: Optical soliton perturbation with fractional-temporal evolution by first integral method with conformable fractional derivatives. Optik 127(22), 10659–10669 (2016)ADS

El-Labany, S.K., Moslem, W.M., El-Awady, E.I., Shukla, P.K.: Nonlinear dynamics associated with rotating magnetized electron-positron-ion plasmas. Phys. Lett. A 375(2), 159–164 (2010)ADS

Ghanbari, B., Inc, M.: A new generalized exponential rational function method to find exact special solutions for the resonance nonlinear Schrödinger equation. Europ. Phys. J. Plus 133(4), 142 (2018)ADS

Hilfer, R. (Ed.). (2000). Applications of fractional calculus in physics. World scientific

Hosseini, K., Alizadeh, F., Hinçal, E., Baleanu, D., Akgül, A., Hassan, A.M.: Lie symmetries, bifurcation analysis, and Jacobi elliptic function solutions to the nonlinear Kodama equation. Results Phys. 54, 107129 (2023a)

Hosseini, K., Sadri, K., Hinçal, E., Sirisubtawee, S., Mirzazadeh, M.: A generalized nonlinear Schrödinger involving the weak nonlocality: its Jacobi elliptic function solutions and modulational instability. Optik 288, 171176 (2023b)ADS

Hosseini, K., Hinçal, E., Ilie, M.: Bifurcation analysis, chaotic behaviors, sensitivity analysis, and soliton solutions of a generalized Schrödinger equation. Nonlinear Dyn. 111(18), 17455–17462 (2023c)

Huang, D.J., Zhang, H.Q.: Extended hyperbolic function method and new exact solitary wave solutions of Zakharov equations. Appl. Math. Computat. 200(1), 110–122 (2005)

Ionescu, C., Lopes, A., Copot, D., Machado, J.T., Bates, J.H.: The role of fractional calculus in modeling biological phenomena: a review. Commun. Nonlinear Sci. Numer. Simulat. 51, 141–159 (2017)ADSMathSciNet

Iqbal, M., Seadawy, A.R., Althobaiti, S.: Mixed soliton solutions for the (2+ 1)-dimensional generalized breaking soliton system via new analytical mathematical method. Results Phys. 32, 105030 (2022)

Iqbal, M., Seadawy, A.R., Lu, D., Zhang, Z.: Physical structure and multiple solitary wave solutions for the nonlinear Jaulent-Miodek hierarchy equation. Modern Phys. Lett. B 14, 2341016 (2023)MathSciNet

Iqbal, M., Seadawy, A.R., Lu, D., Zhang, Z.: Multiple optical soliton solutions for wave propagation in nonlinear low-pass electrical transmission lines under analytical approach. Opt. Quant. Electron. 56(1), 35 (2024)ADS

Iqbal, M., Seadawy, A. R., Lu, D., & Zhang, Z. (2023). Structure of analytical and symbolic computational approach of multiple solitary wave solutions for nonlinear Zakharov-Kuznetsov modified equal width equation. Numerical Methods for Partial Differential Equations

Ismael, H. F., and Bulut, H. (2020). On the solitary wave solutions to the (2+ 1)-dimensional Davey-Stewartson equations. In: 4th International Conference on Computational Mathematics and Engineering Sciences (CMES-2019) 4, 156-165. Springer International Publishing

Jaradat, I., Alquran, M.: Construction of solitary two-wave solutions for a new two-mode version of the Zakharov-Kuznetsov equation. Mathematics 8(7), 1127 (2020)

Jaradat, I., Alquran, M.: A variety of physical structures to the generalized equal-width equation derived from Wazwaz-Benjamin-Bona-Mahony model. J. Ocean Eng. Sci. 7(3), 244–247 (2022)

Javeed, S., Baleanu, D., Waheed, A., Shaukat Khan, M., Affan, H.: Analysis of homotopy perturbation method for solving fractional order differential equations. Mathematics 7(1), 40 (2019)

Javidi, M., Golbabai, A.: Exact and numerical solitary wave solutions of generalized Zakharov equation by the variational iteration method. Chaos Solitons Fract. 36(2), 309–313 (2008)ADS

Javidi, M., Golbabai, A.: Method. Chaos Solitons Fract. 36(2), 309–313 (2008)ADS

Jin, Q., Xia, T., Wang, J.: The exact solution of the space-time fractional modified Kdv-Zakharov-Kuznetsov equation. J. Appl. Math. Phys. 5(4), 844–852 (2017)

Khalil, R., Al Horani, M., Yousef, A., Sababheh, M.: A new definition of fractional derivative. J. Computat. Appl. Math. 264, 65–70 (2014)MathSciNet

Khan, M., Gondal, M.A., Hussain, I., Vanani, S.K.: A new comparative study between homotopy analysis transform method and homotopy perturbation transform method on a semi infinite domain. Math. Comput. Modell. 55(3–4), 1143–1150 (2012)MathSciNet

Khater, A.H., Callebaut, D.K., Seadawy, A.R.: Nonlinear dispersive instabilities in Kelvin-Helmholtz magnetohydrodynamic flows. Phys. Scr. 67(4), 340 (2003)ADS

Lu, D., Seadawy, A.R., Arshad, M., Wang, J.: New solitary wave solutions of (3+ 1)-dimensional nonlinear extended Zakharov-Kuznetsov and modified KdV-Zakharov-Kuznetsov equations and their applications. Results Phys. 7, 899–909 (2017)ADS

Lu, D., Seadawy, A.R., Iqbal, M.: Construction of new solitary wave solutions of generalized Zakharov-Kuznetsov-Benjamin-Bona-Mahony and simplified modified form of Camassa-Holm equations. Open Phys. 16(1), 896–909 (2018)

Magin, R.: Fractional calculus in bioengineering, part 1. Crit. Rev. Biomed. Eng. 32(1), 1004 (2004)

Rehman, H.U., Asjad Imran, M., Bibi, M., Riaz, M., Akgül, A.: New soliton solutions of the 2D-chiral nonlinear Schrodinger equation using two integration schemes. Math. Methods Appl. Sci. 44(7), 5663–5682 (2021b)MathSciNet

Rehman, H.U., Hassan, M.U., Saleem, M.S., Nasri, R., Santina, D., Mlaiki, N.: Soliton solutions of Zakhrov equation in ionized plasma using new extended direct algebraic method. Results Phys. 46, 106325 (2023)

Rehman, H.U., Iqbal, I., Hashemi, M.S., Mirzazadeh, M., Eslami, M.: Analysis of cubic-quartic-nonlinear Schrödinger’s equation with cubic-quintic-septic-nonic form of self-phase modulation through different techniques. Optik 287(25), 171028 (2023)ADS

Rehman, H.U., Saleem, M.S., Zubair, M., Jafar, S., Latif, I.: Optical solitons with Biswas-Arshed model using mapping method. Optik 194, 163091 (2019)ADS

Rehman, H.U., Saleem, M.S., Zubair, M., Jafar, S., Latif, I.: Mathematics 7(1), 40 (2019)

Rehman, H.U., Seadawy, A.R., Younis, M., Rizvi, S.T.R., Anwar, I., Baber, M.Z., Althobaiti, A.: Weakly nonlinear electron-acoustic waves in the fluid ions propagated via a (3+ 1)-dimensional generalized Korteweg-de-Vries-Zakharov-Kuznetsov equation in plasma physics. Results Phys. 33, 105069 (2022)

Rehman, H.U., Ullah, N., Imran, M.A.: Optical solitons of Biswas-Arshed equation in birefringent fibers using extended direct algebraic method. Optik 226, 165378 (2021a)ADS

Sadaf, M., Akram, G., Dawood, M.: An investigation of fractional complex Ginzburg-Landau equation with Kerr law nonlinearity in the sense of conformable, beta and M-truncated derivatives. Opt. Quant. Electron. 54(4), 248 (2022)

Sadaf, M., Akram, G., Mariyam, H.: Abundant solitary wave solutions of Gardner’s equation using new \(\phi ^6\)-model expansion method. Alex. Eng. J. 61(7), 5253–5267 (2022)

Sahoo, S., Ray, S.S.: Analysis of Lie symmetries with conservation laws for the (3+ 1) dimensional time-fractional mKdV-ZK equation in ion-acoustic waves. Nonlinear Dyn. 90, 1105–1113 (2017)MathSciNet

Sahoo, S., Ray, S.S., Abdou, M.A.: New exact solutions for time-fractional Kaup-Kupershmidt equation using improved \((G^{\prime }/G)\)-expansion and extended \((G^{\prime }/G)\)-expansion methods. Alex. Eng. J. 59(5), 3105–3110 (2020)

Sajid, N., Perveen, Z., Sadaf, M., Akram, G., Abbas, M., Abdeljawad, T., Alqudah, M.A.: Implementation of the Exp-function approach for the solution of KdV equation with dual power law nonlinearity. Computat. Appl. Math. 41(8), 338 (2022)MathSciNet

Seadawy, A.R.: Stability analysis for Zakharov-Kuznetsov equation of weakly nonlinear ion-acoustic waves in a plasma. Comput. Math. Appl. 67(1), 172–180 (2014)MathSciNet

Seadawy, A.R., Iqbal, M., Baleanu, D.: Construction of traveling and solitary wave solutions for wave propagation in nonlinear low-pass electrical transmission lines. J. King Saud Univ. Sci. 32(6), 2752–2761 (2020)

Seadawy, A.R., Iqbal, M., Lu, D.: Nonlinear wave solutions of the Kudryashov-Sinelshchikov dynamical equation in mixtures liquid-gas bubbles under the consideration of heat transfer and viscosity. J. Taibah Univ. Sci. 13(1), 1060–1072 (2019)

Seadawy, A.R., Iqbal, M., Lu, D.: Propagation of kink and anti-kink wave solitons for the nonlinear damped modified Korteweg-de Vries equation arising in ion-acoustic wave in an unmagnetized collisional dusty plasma. Phys. A Statist. Mechan. Appl.cations 544, 123560 (2020)

Uddin, M.S., Begum, M., Ullah, M.S., Abdeljabbar, A.: Soliton solutions of a (2+ 1)-dimensional nonlinear time-fractional Bogoyavlenskii equation model. Partial Diff. Equ. Appl. Math. 8, 100591 (2023)

Ullah, M.S.: Interaction solution to the (3+ 1)-D negative-order KdV first structure. Part. Differen. Equ. Appl. Math. 8, 100566 (2023)

Ullah, M.S., Ali, M.Z., Roshid, H.O., Hoque, M.F.: Collision phenomena among lump, periodic and stripe soliton solutions to a (2+ 1)-dimensional Benjamin-Bona-Mahony-Burgers Model. Europ. Phys. J. Plus 136, 1–9 (2021)

Ullah, M.S., Abdeljabbar, A., Roshid, H.O., Ali, M.Z.: Application of the unified method to solve the Biswas-Arshed model. Results Phys. 42, 105946 (2022a)

Ullah, M.S., Ahmed, O., Mahbub, M.A.: Collision phenomena between lump and kink wave solutions to a (3+ 1)-dimensional Jimbo-Miwa-like model. Partial Differ. Equat. Appl. Math. 5, 100324 (2022b)

Ullah, M.S., Mostafa, M., Ali, M.Z., Roshid, H.O., Akter, M.: Soliton solutions for the Zoomeron model applying three analytical techniques. PLoS One 18(7), e0283594 (2023a)

Ullah, M.S., Baleanu, D., Ali, M.Z.: Novel dynamics of the Zoomeron model via different analytical methods. Chaos Solitons Fractals 174, 113856 (2023b)MathSciNet

Ullah, M.S., Seadawy, A.R., Ali, M.Z.: Optical soliton solutions to the Fokas-Lenells model applying the \(\phi ^{6}\)-model expansion approach. Opt. Quant. Electron. 55(6), 495 (2023)

Ullah, M.S., Roshid, H.O., Ali, M.Z.: New wave behaviors and stability analysis for the (2+ 1)-dimensional Zoomeron model. Opt. Quant. Electron. 56(2), 240 (2024)

Vakhnenko, V.O., Parkes, E.J., Morrison, A.J.: A Bäcklund transformation and the inverse scattering transform method for the generalised Vakhnenko equation. Chaos Solitons Fractals 17(4), 683–692 (2003)ADSMathSciNet

Verheest, F., Mace, R.L., Pillay, S.R., Hellberg, M.A.: Unified derivation of Korteweg-de Vries-Zakharov-Kuznetsov equations in multispecies plasmas. J. Phys. A Math. General 35(3), 795 (2002)ADSMathSciNet

Wang, J., Kamran Jamal, A., Li, X.: Numerical solution of fractional-order Fredholm integro-differential equation in the sense of Atangana-Baleanu. Math. Probl. Eng. 12, 1–8 (2021)

Wu, G.C.: A fractional variational iteration method for solving fractional nonlinear differential equations. Comput. Math. Appl. 61(8), 2186–2190 (2011)MathSciNet

Yan, L., Yel, G., Kumar, A., Baskonus, H.M., Gao, W.: Newly developed analytical scheme and its applications to the some nonlinear partial differential equations with the conformable derivative. Fract. Fract. 5(4), 238 (2021)

Yang, D.Y., Tian, B., Qu, Q.X., Zhang, C.R., Chen, S.S., Wei, C.C.: Lax pair, conservation laws, Darboux transformation and localized waves of a variable-coefficient coupled Hirota system in an inhomogeneous optical fiber. Chaos Solitons Fract. 150, 110487 (2021)MathSciNet

Yel, G., Sulaiman, T.A., Baskonus, H.M.: On the complex solutions to the (3+ 1)-dimensional conformable fractional modified KdV-Zakharov-Kuznetsov equation. Modern Phys. Lett. B 34(05), 2050069 (2020)ADSMathSciNet

Yin, X., Yang, L., and Liu, Q. (2020). The evolution equation of non-linear waves and its exact solutions by subsidiary ordinary differential equation method. Modern Phys.

Younas, U., Ren, J., Baber, M.Z., Yasin, M.W., Shahzad, T.: Ion-acoustic wave structures in the fluid ions modeled by higher dimensional generalized Korteweg-de Vries-Zakharov-Kuznetsov equation. J. Ocean Eng. Sci. 8(6), 623–635 (2023)

Younis, M., Rehman, H.U., Rizvi, S.T.R., Mahmood, S.A.: Dark and singular optical solitons perturbation with fractional temporal evolution. Superlatt. Microstruct. 104, 525–531 (2017)ADS

Yusuf, A., Sulaiman, T.A., Abdeljabbar, A., Alquran, M.: Breather waves, analytical solutions and conservation laws using Lie-Bäcklund symmetries to the (2+ 1)-dimensional Chaffee-Infante equation. J. Ocean Eng. Sci. 8(2), 145–151 (2023)

Zhang, J.L., Wang, M.L., Li, X.Z.: The subsidiary ordinary differential equations and the exact solutions of the higher order dispersive nonlinear Schrödinger equation. Phys. Lett. A 357(3), 188–195 (2006)ADS

Zhou, T.Y., Tian, B., Zhang, C.R., Liu, S.H.: Auto-Bäcklund transformations, bilinear forms, multiple-soliton, quasi-soliton and hybrid solutions of a (3+ 1)-dimensional modified Korteweg-de Vries-Zakharov-Kuznetsov equation in an electron-positron plasma. Europ. Phys. J. Plus 137(8), 912 (2022)ADS

Zhou, Q., Yao, D.Z., Cui, Z.: Exact solutions of the cubic-quintic nonlinear optical transmission equation with higher-order dispersion terms and self-steepening term. J. Modern Opt. 59(1), 57–60 (2012)ADS

Titel: Unraveling the dynamic complexity: exploring the (3+1)-dimensional conformable mKdV-ZK equation
verfasst von: Xiaoye Ding
Salah Mahmoud Boulaaras
Hamood Ur Rehman
Ifrah Iqbal
Aziz Ullah Awan
Iffat Sabir
Publikationsdatum: 01.05.2024
Verlag: Springer US
Erschienen in: Optical and Quantum Electronics / Ausgabe 5/2024
Print ISSN: 0306-8919
Elektronische ISSN: 1572-817X
DOI: https://doi.org/10.1007/s11082-024-06465-w

Neuer Inhalt

Bildnachweise

VDI-Icon, Profil Icon, inhalt2, Springer Professional Modul/© Springer Fachmedien Wiesbaden GmbH, Internationaler Motorenkongress/© [M] ATZlive | Chisnikov / Fotolia.com, Search Icon, Banner Hanser, Gardiner von Trapp/© Alpega Group, Benny Hahn/© ZEP GmbH, Customer Experience/© © oatawa / Getty Images / iStock, Zeitschrift Wissensmanagement Cover, PatentFit-Logo/© Springer Fachmedien Wiesbaden GmbH, 2023_Antrieb/© supervisuell, ATZ-Webinar: Prototypenfreie Entwicklung durch Offline- und Driver-in-the-Loop-HiL-Tests /© (c) VI-grade, chassis.tech plus 2023/© [M] ATZlive / TÜV SÜD PRODUCT SERVICE GMBH

Springer Professional

Abstract

Bitte loggen Sie sich ein, um Zugang zu Ihrer Lizenz zu erhalten.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Technik"

Springer Professional "Wirtschaft+Technik"

Springer Professional "Wirtschaft"

Weitere Artikel der Ausgabe 5/2024

Comments on “Numerical investigation and circuit analysis of interdigitated photoconductive antenna for terahertz applications”

Flexible multi-band dual-chirped microwave waveform generator based on a recirculating frequency shifting loop

On the super solitonic structures for the fractional nonlinear Schrödinger equation

Novel computational technique for fractional Kudryashov–Sinelshchikov equation associated with regularized Hilfer–Prabhakar derivative

Chirped dark soliton propagation in optical fiber under a self phase modulation and a self-steepening effect for higher order nonlinear Schrödinger equation

The coupling study between multi-channel laser diodes and multimode fiber in a fiber pump laser

Neuer Inhalt

Bitte loggen Sie sich ein, um Zugang zu Ihrer Lizenz zu erhalten.

Bitte loggen Sie sich ein, um Zugang zu Ihrer Lizenz zu erhalten.