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Breathers-like rogue wave trains induced by nonlinear dynamics of DNA breathing

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Abstract

We introduce in the deoxyribonucleic acid (DNA) molecular chain at the first time the collective coordinates’ theory to measure the exact signature of internal excitation coming from particular effects. We use Type I Ansatz function with seven coordinates to improve the comprehension of particular rogue waves mechanism of generation in DNA system. Further, we establish a cubic-quintic nonlinear Schrödinger equation (CQNLSE) with third-order dispersion (TOD) and fourth-order dispersion (FOD). Some interesting results are obtained. Among them we have the propagation of the dark soliton which generates the semi-Gaussian, the Gaussian and the distorted periodical waves trains. In addition, the combined action of quintic-nonlinearity and TOD induces the Sasa–Satsuma waves field, the first-and second-forms of Sasa–Satsuma wave trains’ fields. The decrease of quintic-nonlinearity leads to the Peregrine soliton, the third-form of Sasa–Satsuma wave trains’ field, the Akhmediev wave trains’ field, Akhmediev Triangular rogue waves, and the Akhmediev–Peregrine Triangular rogue waves. Further, when FOD comes into play it generates the shifted soliton, the Peregrine soliton associated with twin parallel solitons. Another increase of frequency induces the multi-Sasa–Satsuma wave trains’field, and the fourth-form of Sasa–Satsuma wave trains’ field. This last structure contains several hills of Sasa–Satsuma wave trains. Furthermore, the stabilization conditions of the dark soliton are presented. Moreover, the comprehension of internal behavior due to the action of all effects above mentioned is also investigated in detail.

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References

  1. Alain Mvogo, Germain H. Ben-Bolie, Timoleon C. Kofane, Fractional nonlinear dynamics of DNA breathing. Commun. Nonl. Sci. Numer. Simul. 48, 258–269 (2017)

    MathSciNet  MATH  Google Scholar 

  2. M. Peyrard, A.R. Bishop, Statistical mechanics of a nonlinear model for DNA denaturation. Phys. Rev. Lett. 62, 2755 (1989)

    ADS  Google Scholar 

  3. T. Dauxois, Dynamics of breather modes in a nonlinear “helicoidal’’ model of DNA. Phys. Lett. A. 159, 390 (1991)

    ADS  Google Scholar 

  4. T. Dauxois, M. Peyrard, A.R. Bishop, Dynamics and thermodynamics of a nonlinear model for DNA denaturation. Phys. Rev. E 47, 684 (1993)

    ADS  MATH  Google Scholar 

  5. T. Dauxois, M. Peyrard, Entropy-driven transition in a one-dimensional system. Phys. Rev. E 51, 4027 (1995)

    ADS  Google Scholar 

  6. C.R. Calladine, H.R. Drew, R.F. Luisi, A.A. Travers, Understanding DNA (Elsevier, Amsterdam, 1997)

    Google Scholar 

  7. S.W. Englander, N.R. Kallenbach, A.J. Heeger, J.A. Krumhansl, S. Litwin, Nature of the open state in long polynucleotide double helices: possibility of soliton excitations. Proc. Natl. Acad. Sci. USA 77, 7222–6 (1980)

    ADS  Google Scholar 

  8. S. Yomosa, Soliton excitations in deoxyribonucleic acid (DNA) double helices. Phys. Rev. A 27, 2120 (1983)

    ADS  MathSciNet  Google Scholar 

  9. S. Takeno, S. Homma, Topological solitons and modulated structure of bases in DNA double helices: a dynamic plane base-rotator model. Prog. Theor. Phys. 70, 308 (1983)

    ADS  Google Scholar 

  10. I. Podlubny, Geometric and physical interpretation of fractional integration and fractional differentiation. Fract. Calc. Appl. Anal. 5, 367 (2002)

    MathSciNet  MATH  Google Scholar 

  11. R. Caponetto, R. Dongola, L. Fortuna, I. Petr’as, Fractional Order system: modeling and control applications. World Scientific Series on Nonlinear Science (Series A 72) (2010)

  12. J.A. Tenreiro Machado, A.C. Costa, M.D. Quelhas, Fractional dynamics in DNA. Commun. Nonlinear Sci. Numer. Simul. 16, 2963–2969 (2011)

    ADS  MATH  Google Scholar 

  13. T.T. Hartley, C.F. Lorenzo, H.K. Qammer, Chaos in a fractional order Chua’s system. IEEE Trans. Circ. Syst. I 42(8), 485–490 (1995)

    Google Scholar 

  14. I. Grigorenko, E. Grigorenko, Chaotic dynamics of the fractional lorenz system. Phys. Rev. Lett. 91(485), 034101 (2003)

    ADS  Google Scholar 

  15. A. Mvogo, A. Tambue, G.H. Ben-Bolie, T.C. Kofane, Commun. Nonlinear Sci. Numer. Simulat. 39, 396 (2016)

    ADS  Google Scholar 

  16. N. Engheta, On the role of fractional calculus in electromagnetic theory. IEEE Ant. Prop. Mag. 39, 35–46 (1997)

    Google Scholar 

  17. A. Iomin, Fractional-time quantum dynamics. Phys. Rev. E 80, 022103 (2009)

    ADS  Google Scholar 

  18. B.A. Carreras, V.E. Lynch, G.M. Zaslavsky, Anomalous diffusion and exit time distribution of particle tracers in plasma turbulence model. Phys. Plasmas 8, 5096–8 (2001)

    ADS  Google Scholar 

  19. J.C. Gutierrez-Vega, Fractionalization of optical beams: II. Elegant Laguerre–Gaussian modes. Opt. Express 15, 6300–6313 (2007)

    ADS  Google Scholar 

  20. A. Carpinteri, F. Mainardi, Fractals and fractional calculus in continuum mechanics (Springer, Berlin, 1997)

    MATH  Google Scholar 

  21. S. Chen, F. Baronio, J.M.S. Crespo, P. Grelu, D. Mihalache, Versatile rogue waves in scalar, vector, and multidimensional nonlinear systems. J. Phys. A: Math. Theor. 50, 463001–78 (2017)

    MathSciNet  MATH  Google Scholar 

  22. H. Chen, Z. Xu and Z. Dai, Rogue Wave for the (3+1)-Dimensional Yu-Toda-Sasa-Fukuyama Equation, Abstr. Appl. Anal. (2014)

  23. S. Chen, Twisted rogue-wave pairs in the Sasa–Satsuma equation. Phys. Rev. E 88, 023202–11 (2013)

    ADS  Google Scholar 

  24. S. Chen, Y. Ye, F. Baronio, Y. Liu, X.M. Cai, P. Grelu, Optical Peregrine rogue waves of self-induced transparency in a resonant erbium-doped fiber. Opt. Express 25, 29687–29712 (2017)

    ADS  Google Scholar 

  25. J.M. Soto-Crespo, N. Devine, N.P. Hoffmann, N. Akhmediev, Rogue waves of the Sasa–Satsuma equation in a chaotic wave field. Phys. Rev. E 90, 032902–7 (2014)

    ADS  Google Scholar 

  26. A. Ankiewicz, J.M. Soto - Crespo, M. A. Chowdhury, N. Akhmediev, Rogue waves in optical fibers in presence of third-order dispersion, self-steepening, and self-frequency shift. J. Opt. Soc. Am. B 30, 87–94 (2013)

    ADS  Google Scholar 

  27. N. Sasa, J. Satsuma, New-type of soliton solutions for a higher-order nonlinear Schrödinger equation. J. Phys. Soc. Jpn. 60, 409–417 (1991)

    ADS  MATH  Google Scholar 

  28. L. Guo, Y. Cheng, D. Mihalache, J. He, Darboux transformation and higher-order solutions of the Sasa–Satsuma equation. Rom. J. Phys. 63, 205–225 (2018)

    Google Scholar 

  29. B.G. Onana Essama, S. Ndjakomo Essiane, Jacques Atangana, Multi-Sasa Satsuma rogue events and multi-wave trains generation in a nonlinear left-handed transmission line. Eur. Phys. J. Plus 136, 49–68 (2021)

    Google Scholar 

  30. B.G. Onana Essama, S. Ndjakomo Essiane, F. Biya - Motto, M. Shabat, J. Atangana, Multi-wave trains and Sasa–Satsuma freak events generation in an optical metamaterial. China. J. Phys. 69, 50–69 (2021)

    MathSciNet  Google Scholar 

  31. B.G. Onana Essama, S. Ndjakomo Essiane, F. Biya-Motto, B.M. Ndi Nnanga, M. Shabat, J. Atangana, Dynamical evolution of Sasa-Satsuma rogue waves, breather solutions, and new special wave phenomena in a nonlinear metamaterial. J. Phys. Stat. Sol. B 258, 2000316–24 (2021)

    ADS  Google Scholar 

  32. Y. Ohta, J. Yang, General high-order rogue waves and their dynamics in the nonlinear Schrödinger equation. R. Soc. Lond. Proc. Ser. A 468, 1716 (2012)

    ADS  MathSciNet  MATH  Google Scholar 

  33. P. Gaillard, Families of quasi-rational solutions of the NLS equation and multi-rogue waves. J. Phys. A 44, 5204 (2011)

    ADS  MATH  Google Scholar 

  34. P. Dubard, V.B. Matveev, Multi-rogue waves solutions to the focusing NLS equation and the KP-I equation. Nat. Haz. Earth Syst. 11, 667–672 (2011)

    Google Scholar 

  35. V.B. Matveev, M. Salle, Darboux transformations and solitons (Springer, Berlin, 1991)

    MATH  Google Scholar 

  36. N. Akhmediev, N.V. Mitskevich, Extremely high degree of n-soliton pulse compression in an optical fiber. IEEE J. Quantum Electron. 27, 849–857 (1991)

    ADS  Google Scholar 

  37. D.J. Kedziora, A. Ankiewicz, N. Akhmediev, Triangular rogue wave cascades. Phys. Rev. E 86, 056602–9 (2012)

    ADS  Google Scholar 

  38. P.T. Dinda, A.B. Moubissi, K. Nakkeeran, Collective variable theory for optical solitons in fibers. Phys. Rev. E 64, 016608 (2001)

    ADS  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  40. J. Atangana, A. Kamagate, P. Tchofo Dinda, A. Labruyère, T.C. Kofane, Effective characterization of the phase and intensity profiles of asymmetrically distorted light pulses in optical fiber systems. J. Opt. Soc. Am. B 26, 371–383 (2009)

    ADS  MathSciNet  Google Scholar 

  41. S. Fewo, H. Moussambi, T.C. Kofane, Analysis of non-paraxial solitons using collective variable approach. Phys. Scr. 84, 035002 (2011)

    ADS  MATH  Google Scholar 

  42. S. Shwetanshumala, A. Biswas, Femtosecond pulse propagaton in optical fibers under higher order effects: a collective variable approach. J. Theor. Phys. 47, 1699–1708 (2008)

    MATH  Google Scholar 

  43. Shwetanshumala, Temporal solitons in nonlinear media modeled by modified complex Ginzburg Landau Equation under collective variable approach. S. J. Theor. Phys. 48, 1122–1131 (2009)

    MathSciNet  MATH  Google Scholar 

  44. D. Chevizovich, D. Michieletto, A. Mvogo, F. Zakiryanov, S. Zdravkovic, A review on nonlinear DNA physics. R. Soc. Open Sci. 7, 200774–23 (2020)

    ADS  Google Scholar 

  45. T.A. Davydova, Y.A. Zalinznyak, Schrodinger ordinary solitons and chirped solitons: fourth-order dispersive effects and cubic-quintic nonlinearity. Phys D. 156, 260–82 (2001)

    MathSciNet  MATH  Google Scholar 

  46. B.G. Onana Essama, J.T. Ngo Bisse, S. Ndjakomo Essiane, J. Atangana, M-shaped and other exotic solitons generated by cubic-quintic saturable nonlinearities in a nonlinear electrical transmission network with higher-order dispersion effects. Chaos Solit. Fract. 161, 112320–11314 (2022)

    MathSciNet  Google Scholar 

  47. J.B. Okaly, Fabien II. Ndzana, R.L. Woulache, T.C. Kofane, Solitary wavelike solutions in nonlinear dynamics of damped DNA systems. Eur. Phys. J. Plus. 134, 598–22 (2019)

    Google Scholar 

  48. G.P. Agrawal, Nonlinear fiber optics, 2nd edn. (Academic Press, San Diego, 1995)

    MATH  Google Scholar 

  49. Z. Wang, Y. Feng, Z. Bo, J. Zhao, T. Jiang, Dark Schrodinger solitons and harmonic generation in left-handed nonlinear transmission line. J Appl Phys. 107, 094907 (2010)

    ADS  Google Scholar 

  50. B.G. Onana Essama, S. Ndjakomo Essiane, J. Atangana, Multi-Sasa Satsuma rogue events and multiwave trains generation in a nonlinear left-handed transmission line. Eur. Phys. J. Plus. 136, 49–68 (2021)

    Google Scholar 

  51. E. Kengne, Engineering chirped LambertW-kink signals in a nonlinear electrical transmission network with dissipative elements. Eur. Phys. J. Plus. 136, 266–93 (2021)

    Google Scholar 

  52. D. Pushkarov, S. Tanev, Bright and dark solitary wave propagation and bistability in the anomalous dispersion region of optical waveguides with third- and fifth - order nonlinearities. J. Opt. Commun. 124, 354–364 (1996)

    ADS  Google Scholar 

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

    Google Scholar 

  54. S. Zdravkovic, M.V. Sataric, Nonlinear Schrödinger equation and DNA dynamics. Phys. Lett. A 373, 126–132 (2008)

    ADS  MATH  Google Scholar 

  55. S. Zdravkovic, Helicoidal Peyrard–Bishop model Of DNA dynamics. J. Nonl. Math. Phys. 18, 463–484 (2011)

    MathSciNet  MATH  Google Scholar 

  56. J. Atangana, B.G. Onana Essama, B. Mokhtari, T.C. Kofane, 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 

  57. B.G. Onana Essama, S. Ndjakomo Essiane, F. Biya - Motto, M. Shabat, J. Atangana, Peregrine soliton and Akhmediev breathers in a chameleon electrical transmission line. J. Appl. Math. Phys. 8, 2775–2792 (2020)

    Google Scholar 

  58. B.G. Onana Essama, S. Ndjakomo Essiane, F. Biya - Motto, M. Shabat, Jacques Atangana, Triangular rogue waves and multi- wave trains generation in a chameleon electrical transmission line. Am. J. Opt. Photon. 8, 61–73 (2020)

    Google Scholar 

  59. S. Ndjakomo Essiane, B.G. Onana Essama, M.M. Shabat, J. Atangana, Multi-Raman soliton self-frequency shifts and dissipative rogue wave trains induced by cubic-quintic-Raman contributions in a double-negativematerial. Physica B. 632, 413731 (2022)

    Google Scholar 

  60. D. Peregrine, Water waves, nonlinear Schrödinger equations and their solutions. J. Austr. Math. Soc. Ser. 25, 16–43 (1983)

    MATH  Google Scholar 

  61. J.M. Dudley, G. Genty, F. Dias, B. Kibler, N. Akhmediev, Modulation instability, Akhmediev breathers and continuous wave supercontinuum generation. Opt. Express 17, 21497–21508 (2009)

    ADS  Google Scholar 

  62. N. Akhmediev, A. Ankiewicz, M. Taki, Waves that appear from nowhere and disappear without a trace. Phys. Lett. A 373, 675–678 (2009)

    ADS  MATH  Google Scholar 

  63. B.M. Ndi Nnanga, P.Y. Gouadjio Dontsop, B.G. Onana Essama, Mohammed Shabat, David Yemele, Jacques Atangana, Tree-like structures and Freak waves generation induced by quintic-nonlinearity and cubic-Raman efect in a nonlinear metamaterial. Opt. Quan. Electron. 52, 356–386 (2020)

    Google Scholar 

  64. E.G. Charalampidis, J. Cuevas - Maraver, D.J. Frantzeskakis, P.G. Kevrekidis, Rogue waves in ultracold bosonic seas. Roman. Repor. Phys. 70, 504–25 (2018)

    Google Scholar 

  65. R.H.J. Grimshaw, A. Tovbis, Rogue waves: analytical predictions. Proc. R. Soc. A Math. Phys. Eng. Sci. 469, 2157–2212 (2013)

    Google Scholar 

  66. L. Wei, Q. De-Qin, H. Jing-Song, Localized properties of rogue wave for a higher-order nonlinear Schrödinger equation. Commun. Theor. Phys. 63, 525–534 (2015)

    ADS  MathSciNet  MATH  Google Scholar 

  67. A. Ankiewicz, A.P. Bassom, P.A. Clarkson, E. Dowie, Conservation laws and integral relations for the Boussinesq equation. Stud. Appl. Math. 139, 104–128 (2017)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors thank the anonymous reviewers, and the associate editor in charge of this manuscript. Special thanks to Professor Salome Ndjakomo Essiane, chairman of ACRITEE (Association Camerounaise pour la Recherche et l’Innovation en Technologie de l’Energie et de l’Environnement) for the specific contribution of its scientific group.

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Authors and Affiliations

  1. Applied Biotechnology and Engineering Laboratory, Department of Electrical Engineering, Higher Technical Teacher’s Training College (HTTTC) of Ebolowa, ACRITEE, University of Ebolowa, P.O. Box 886, Ebolowa, Cameroon

    Bedel Giscard Onana Essama & Salome Ndjakomo Essiane

  2. Applied Biotechnology and Engineering Laboratory, Department of Maintenance, Higher Technical Teacher’s Training College (ENSET) of Ebolowa, ACRITEE, University of Ebolowa, P.O. Box 886, Ebolowa, Cameroon

    Jacquie Therese Ngo Bisse

  3. Laboratory of Electronics, Department of Physics, Higher Teacher Training College (ENS) of Yaounde, ACRITEE, University of Yaounde I, P.O. Box 47, Yaounde, Cameroon

    Jacques Atangana

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  1. Bedel Giscard Onana Essama
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  2. Jacquie Therese Ngo Bisse
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  3. Salome Ndjakomo Essiane
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Correspondence to Bedel Giscard Onana Essama.

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Appendix

Appendix

1.1 Quantities related to the coefficients of Eq.(4) in Sect. 2.3

The quantities \(M_1\), \(M_2\), \(M_3\), \(M_4\) and \(M_5\) linked to \(A_2\), \(A_3\), \(A_4\) and \(B_1\) are all given as follows [1]:

$$\begin{aligned} M_{1}= & {} -18\,{\frac{ \left( 2\,\gamma +2 \right) \left( 2\,\gamma -1 \right) ^ {2}{\omega }^{2\,\gamma -2}}{ \left( 2\,\gamma +1 \right) ^{2}\gamma \, \left( {\omega }^{2\,\gamma -1} \right) ^{2}} \left( \frac{6.28}{N}- \frac{41.3}{N^3} \right) ^{2}}+4- \frac{78.9}{N^2} \end{aligned}$$
(15)
$$\begin{aligned} M_{2}= & {} {\frac{ \left( 2\,\gamma +2 \right) \left( 2\,\gamma -1 \right) ^{3}{ \omega }^{2\,\gamma -2}}{{\omega }^{2} \left( 2\,\gamma +1 \right) ^{2} \gamma \, \left( {\omega }^{2\,\gamma -1} \right) ^{2}} \left( \frac{6.28}{N}- \frac{41.3}{N^3}\right) ^{2}} \end{aligned}$$
(16)
$$\begin{aligned} M_{3}= & {} {\frac{ \left( 2\,\gamma +2 \right) \left( 2\,\gamma -1 \right) ^{2}{ \omega }^{2\,\gamma -2}}{\omega \, \left( 2\,\gamma +1 \right) ^{2}\gamma \, \left( {\omega }^{2\,\gamma -1} \right) ^{2}} \left( \frac{6.28}{N}- \frac{41.3}{N^3} \right) ^{2}} \end{aligned}$$
(17)
$$\begin{aligned} M_{4}= & {} {\frac{ \left( 2\,\gamma +2 \right) \left( 2\,\gamma -1 \right) ^{2}{ \omega }^{2\,\gamma -2} \left( 2\,\gamma -2 \right) }{{\omega }^{2} \left( 2\,\gamma +1 \right) ^{2}\gamma \, \left( {\omega }^{2\,\gamma -1} \right) ^{2}} \left( \frac{6.28}{N}- \frac{41.3}{N^3} \right) ^ {2}} \end{aligned}$$
(18)
$$\begin{aligned} M_{5}= & {} {\omega }^{2\,\gamma }-16\, \left( \frac{3.14}{N}- \frac{5.16}{N^3}\right) ^{2} \end{aligned}$$
(19)

1.2 Collective coordinate’s equations of motion developed in Sect. 2.4

$$\begin{aligned} {\dot{Z}}_1(\zeta , \tau )= & {} \left( \frac{15}{256}Z_{{1}}\,{Z_{{3}}}^{4}Z_{{4}}{Z_{{7}}}^{2}+\frac{1}{16}Z_{{1}}\,{Z_{{3}} }^{2}{Z_{{4}}}^{3}\nonumber \right. \\{} & {} \left. +\frac{3}{16}Z_{{1}}\,{Z_{{3}}}^{2}Z_{{4}}Z_{{5}}Z_{{7}}\right. \nonumber \\{} & {} +\left. \frac{1}{4}Z_{{1}}Z_{{4}}\,{Z_{{5}}}^{2} +\frac{1}{4}\frac{Z_{{1}}Z_{{4}}}{Z_3^2}\,\right) A_{{4}}(\zeta , \tau )\nonumber \\{} & {} + \left( -\frac{3}{16}Z_{{1}}{Z_{{3}}}^{2}Z_{{4}}Z_{{7}}-\frac{1}{2}Z_{{1}}Z_{{4}}Z_{{5}}\right) A_{{3}}(\zeta , \tau )\nonumber \\{} & {} +\frac{Z_{{1}}Z_{{4}}}{2}A_{{2}}(\zeta , \tau ) \end{aligned}$$
(20)
$$\begin{aligned} {\dot{Z}}_2(\zeta , \tau )= & {} -\left( \frac{15}{1024}\,{Z_{{3}}}^{6}{Z_{{7}}}^{3}+\frac{3}{63}\,{Z_{{3}}}^{4}{ Z_{{4}}}^{2}Z_{{7}}+\frac{1}{8}\,{Z_{{3}}}^{2}{Z_{{4}}}^{2}Z_{{5}}\right) \nonumber \\{} & {} A_{{4}}(\zeta , \tau ) \nonumber \\{} & {} - \left( \frac{1}{16}\,{Z_{{3}}}^{2}{Z_{{5}}}^{2}Z_{{7}}+ \frac{1}{6}\,{Z_{{5}}}^{3}+\frac{1}{48}\,Z_{{7}}+\frac{1}{2}\,\frac{Z_{{5}}}{Z_3^2}\right) \nonumber \\{} & {} A_{{4}}(\zeta , \tau ) \nonumber \\{} & {} - \left( -\frac{3}{128}\,{Z_{{3}}}^{4}{Z_{{7}}}^{2}-\frac{1}{8}\,{Z_{{3}}}^{2}{Z_{{4}}}^{2}-\frac{1}{8}\,{Z_{{3}} }^{2}Z_{{5}}Z_{{7}}\right. \nonumber \\{} & {} \left. -\frac{1}{2}\,{Z_{{5}}}^{2}-\frac{1}{2}\frac{1}{Z_3^2}\,\right) A_{{3}}(\zeta , \tau ) \nonumber \\{} & {} - \left( \frac{1}{8}\,{Z_{{3}}}^{2}Z_{{7}}+\,Z_{{5}}\right) \nonumber \\{} & {} A_{{2}}(\zeta , \tau )\end{aligned}$$
(21)
$$\begin{aligned} {\dot{Z}}_3(\zeta , \tau )= & {} \left( \frac{15}{128}\,{Z_{{3}}}^{5}Z_{{4}}{Z_{{7}}}^{2}+\frac{1}{8}\,{Z_{{3} }}^{3}{Z_{{4}}}^{3}\right. \nonumber \\{} & {} \left. +\frac{3}{8}\,{Z_{{3}}}^{3}Z_{{4}}Z_{{5}}Z_{{7}}+\frac{1}{2}\,{Z_{{3}}}Z_{{4}}{Z_{{5}}}^{2} +\frac{1}{2}\frac{Z_{{4}}}{Z_{{3}}}\,\right) A_{{4}}(\zeta , \tau )\nonumber \\{} & {} - \left( -\frac{3}{8}\,{Z_{{3}}}^{3}Z_{{4}}Z_{{7}}-\,{Z_{{3}}}Z_{{4}}Z_{{5}}\right) A_{{3}}(\zeta , \tau )\nonumber \\{} & {} -\left( {Z_{{3}}}^{2}Z_{{4}} \right) A_{{2}}(\zeta , \tau )\end{aligned}$$
(22)
$$\begin{aligned} {\dot{Z}}_4(\zeta , \tau )= & {} \left( \frac{15}{512}{Z_{{3}}}^{6}{Z_{{7}}}^{4} +\frac{39}{128}{Z_{{3}}}^{4}{Z_{{4}}}^{2}{Z_{{7}}}^{2}\right. \nonumber \\{} & {} \left. +\frac{3}{32}Z_{{3}}^{4}Z_{{5}}{Z_{{7}}}^{3}-\frac{2}{Z_3^6}+\frac{1}{2} {Z_{{4}}}^{2}{Z_{{5}}}^{2}+\frac{1}{32}{Z_{{7}}}^{2}\right) A_{{4}}(\zeta , \tau ) \nonumber \\{} & {} + \left( \frac{1}{8}\,{Z_{{3}}}^{2}{Z_{{4}}}^{4}+\frac{15}{24}{Z_{{3}}}^{2}{Z_{{4}}}^{2}Z_{{5}}Z_{{7}}\right. \nonumber \\{} & {} \left. +\frac{1}{8}\,{Z_{{3}}}^{2}{Z_{{5}}}^{2}{Z_{{7}}}^{2}-\frac{1}{2}\frac{Z_{{5}}Z_{{7}}}{Z_3^2}-2\frac{{Z_{{5}}}^{2}}{Z_3^4} \right) A_{{4}}(\zeta , \tau ) \nonumber \\{} & {} + \left( -\frac{27}{288}{Z_{{3}}}^{4}{Z_{{7}}}^{3}-\frac{15}{24}\, {Z_{{3}}}^{2}{Z_{{4}}}^{2} Z_{{7}}\right. \nonumber \\{} & {} \left. -\frac{1}{4}\, {Z_{{3}}}^{2}Z_{{5}}{Z_{{7}}}^{2}-{Z_{{4}}}^{2}Z_{{5}}+\frac{1}{2}\frac{Z_{{7}}}{{Z_{{3}}}^{2}}+4\frac{Z_{{5}}}{{Z_{{3}}}^{4}} \right) A_{{3}}(\zeta , \tau ) \nonumber \\{} & {} + \left( \frac{1}{4}\,{Z_{{3}}}^{2}{Z_{{7}}}^{2}+{Z_{{4}}}^{2}\right) A_{{2}}(\zeta , \tau )\nonumber \\{} & {} + \left( -\frac{8\sqrt{3}}{9}\frac{{Z_{{1}}}^{4}}{{Z_{{3}}}^{2}}\right) B_2(\zeta , \tau ) +\left( -\sqrt{2}\,\frac{{Z_{{1}}}^{2}}{{Z_{{3}}}^{2}}\right) B_{{1}}(\zeta , \tau )\end{aligned}$$
(23)
$$\begin{aligned} {\dot{Z}}_5(\zeta , \tau )= & {} -\left( \frac{1}{16}\,{Z_{{3}}}^{2}Z_{{4}}{Z_{{5}}}^{2}Z_{{7}} -\frac{1}{2}Z_{{4}}Z_{{7}}\right. \nonumber \\{} & {} \left. -\frac{3}{2}\frac{Z_{{4}}\,Z_{{5}}}{Z_3^2}+\frac{75}{1024}\,{Z_{{3}}}^{6}Z_{{4}}{Z_{{7}}}^{3} \right) A_{{4}}(\zeta , \tau )\nonumber \\{} & {} -\left( \frac{1}{8}\,{Z_{{3}}}^{4}{Z_{{4}}}^{3}Z_{{7}}+\frac{9}{64}\,{Z_{{3}}}^{4}Z_{{4}}Z_{{5}}{Z_{{7}}}^{2}\right. \nonumber \\{} & {} \left. +\frac{1}{8}\,{Z_ {{3}}}^{2}{Z_{{4}}}^{3}Z_{{5}}\right) A_{{4}}(\zeta , \tau )\nonumber \\{} & {} - \left( -\frac{9}{64}\,{Z_{{3}}}^{4}Z_{{4}}{Z_{{7}}}^{2}-\frac{1}{8}\,{Z_{{3}}}^{2}{Z_{{4}}}^{3}-\frac{1}{8}\,{Z_{{3}}}^{2} Z_{{4}}Z_{{5}}Z_{{7}}\right. \nonumber \\{} & {} \left. + \frac{3}{2}Z_{{4}}\right) A_{{3}}(\zeta , \tau )\nonumber \\{} & {} - \left( \frac{1}{8}\,{Z_{{3}}}^{2}Z_{{4}}Z_{{7}}\right) A_{{2}}(\zeta , \tau ) \end{aligned}$$
(24)
$$\begin{aligned} {\dot{Z}}_6(\zeta , \tau )= & {} -\left( \frac{105}{32768}{Z_{{3} }}^{8}{Z_{{7}}}^{4}+\frac{1}{16}{Z_{{3}}}^{2}{Z_{{5}}}^{3}Z_ {{7}}- \frac{3}{8}\frac{1}{Z_3^4}\right. \nonumber \\{} & {} \left. +\frac{15}{512}{Z_{{3}}}^{6}{Z_{{4}}}^{2 }{Z_{{7}}}^{2}+\frac{15}{1024}{Z_{{3}}}^{6}Z_{{5}}{Z_{{7}} }^{3}\right) A_{{4}}(\zeta , \tau ) \nonumber \\{} & {} - \left( \frac{1}{128}{Z_{{3}}}^{4}{Z_{{4}}}^{4}-\frac{1}{16}{Z_{{4}}}^{2}+\frac{1}{8}{Z_{{5}}}^{4}\right. \nonumber \\{} & {} \left. +\frac{3}{32}\, {Z_{{3}}}^{2}{Z_{{4}}}^{2}Z_{{5}}Z_{{7}}+\frac{9}{256}{Z_{{3}}}^{4}{Z_{{5}}}^{2}{Z_{{7}}}^{2}\right. \nonumber \\{} & {} \left. + \frac{1}{8}{Z_{{3}}}^{2}{Z_{{4}}}^{2}{Z_{{5}}}^{2}\right) A_{{4}}(\zeta , \tau ) \nonumber \\{} & {} -\left( -\frac{1}{8}{Z_{{3}}}^{2}{Z_{{5}}}^{2}Z_{{7}} -\frac{3}{64}{Z_{{3}}}^{4}{Z_{{4}}}^{2}Z_{{7}}\right. \nonumber \\{} & {} \left. -\frac{3}{64}{Z_{{3}}}^{4}Z_{{5}}{Z_{{7}}}^{2}-\frac{1}{3}{Z_{{5}}}^{3}\right) A_{{3}}(\zeta , \tau ) -\left( -\frac{5\sqrt{2}}{8}\,{Z_{{1}}}^{2} \right) \nonumber \\{} & {} B_{{1}}(\zeta , \tau )\nonumber \\{} & {} - \left( -\frac{1}{8}{Z_{{3}}}^{2}{Z_{{4}}}^{2}Z_{{5}}-\frac{5}{512}{Z_{{3}}}^{6}{Z_{{7}}}^{3} +\frac{1}{48}Z_{{7}}\right. \nonumber \\{} & {} \left. + \frac{1}{2}\frac{Z_{{5}}}{Z_3^2}\right) \nonumber \\{} & {} A_{{3}}(\zeta , \tau )-\left( -\frac{4\sqrt{3}}{9} {Z_{{1}}}^{4} \right) B_2(\zeta , \tau ) \nonumber \\{} & {} - \left( \frac{1}{8}{Z_{{3}}}^{2}Z_{{5}}Z_{{7}}-\frac{1}{Z_3^2}+\frac{3}{128}{Z_{{3}}}^{4}{Z_{{7}}}^{2}+ \frac{1}{2}{Z_{{5}}}^{2}\right) A_{{2}}(\zeta , \tau )\end{aligned}$$
(25)
$$\begin{aligned} {\dot{Z}}_7(\zeta , \tau )= & {} \left( \frac{105}{128}\,{Z_{{3}}}^{4}Z_{{4}}{Z_{{7}}}^{3}+\frac{5}{4}\,{Z_{{3 }}}^{2}{Z_{{4}}}^{3}Z_{{7}}\right. \nonumber \\{} & {} \left. +\frac{15}{8}\,{Z_{{3}}}^{2}Z_{{4}}Z_{{5}}{Z_{{7 }}}^{2}+{Z_ {{4}}}^{3}Z_{{5}}+\frac{3}{2}\,{Z_{{4}}}{Z_{{5}}}^ {2}Z_{{7}}\right. \nonumber \\{} & {} \left. -3\frac{Z_{{4}}Z_{{7}}}{Z_3^2}-12\frac{Z_{{4}}Z_{{5}}}{Z_3^4}\right) A_{{4}}(\zeta , \tau )\nonumber \\{} & {} + \left( -\frac{15}{8}\,{Z_{{3}}}^{2}Z_{{4}}{Z_{{7}}}^{2}-{Z_{{4}}}^{3}-3Z_{{4}}Z_{{5}}Z_{{7}}+ 12\frac{Z_{{4}}}{Z_3^4}\right) \nonumber \\{} & {} A_{{3}}(\zeta , \tau ) +\left( 3Z_{4}Z_{{7}}\right) A_ {{2}}(\zeta , \tau ) \end{aligned}$$
(26)

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Essama, B.G.O., Bisse, J.T.N., Essiane, S.N. et al. Breathers-like rogue wave trains induced by nonlinear dynamics of DNA breathing. Appl. Phys. A 128, 1069 (2022). https://doi.org/10.1007/s00339-022-06206-4

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