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Solving the modified complex short pulse equation of focusing type: a Riemann–Hilbert approach

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Abstract

For the modified complex short pulse equation on the line with zero boundary conditions, a Riemann–Hilbert approach is presented. A parametric representation of the solution to the related Cauchy problem is obtained. The explicit formulae are worked for the one-soliton solutions and the two-soliton solutions, which, depending on the parameters, may be either smooth solitons, cuspons or breathers.

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References

  1. Bo, G.Q., Zhang, W.G.: Initial value problem and soliton solutions of the single-cycle short pulse equation via the Riemann–Hilbert approach. J. Phys. Commun. 2, 115004 (2018)

    Article  Google Scholar 

  2. Boutet de Monvel, A., Shepelsky, D.: Riemann–Hilbert approach for the Camassa–Holm equation on the line. C. R. Math. Acad. Sci. Paris 343, 627–632 (2006)

  3. Boutet de Monvel, A., Shepelsky, D.: A Riemann–Hilbert approach for the Degasperis–Procesi equation. Nonlinearity 26, 2081–2107 (2013)

  4. Boutet de Monvel, A., Shepelsky, D.: The Ostrovsky-Vakhnenko equation by a Riemann-Hilbert approach. J. Phys. A: Math. Theor. 48, 035204 (2015)

  5. Boutet de Monvel, A., Shepelsky, D., Zielinski, L.: The short-wave model for the Camassa–Holm equation: a Riemann–Hilbert approach. Inverse Probl. 27, 105006 (2011)

  6. Boutet de Monvel, A., Shepelsky, D., Zielinski, L.: The short pulse equation by a Riemann–Hilbert approach. Lett. Math. Phys. 107, 1345–1373 (2017)

  7. Feng, B.F.: Complex short pulse and coupled complex short pulse equations. Phys. D 297, 62–75 (2015)

    Article  MathSciNet  Google Scholar 

  8. Feng, B.F., Ling, L., Zhu, Z.: A defocusing complex short pulse equation and its multi-dark soliton solution by Darboux transformation. Phys. Rev. E 93, 052227 (2016)

    Article  MathSciNet  Google Scholar 

  9. Guo, B., Liu, N.: A Riemann–Hilbert approach for the modified short pulse equation. Appl. Anal. 98, 1646–1659 (2019)

    Article  MathSciNet  Google Scholar 

  10. Kuetche, V.K., Youssoufa, S., Kofane, T.C.: Ultrashort optical waveguide excitations in uniaxial silica fibers: elastic collision scenarios. Phys. Rev. E 90, 063203 (2014)

    Article  Google Scholar 

  11. Ling, L., Feng, B.F., Zhu, Z.: Multi-soliton, multi-breather and higher-order rogue wave solutions to the complex short pulse equation. Phys. D 327, 13–29 (2015)

    Article  MathSciNet  Google Scholar 

  12. Mao, H., Liu, Q.P.: The short pulse equation: Bäcklund transformations and applications. Stud. Appl. Math. 145, 791–811 (2020)

    Article  MathSciNet  Google Scholar 

  13. Matsuno, Y.: Multiloop soliton and multibreather solutions of the short pulse model equation. J. Phys. Soc. Jpn. 76, 084003 (2007)

    Article  Google Scholar 

  14. Matsuno, Y.: Integrable multi-component generalization of a modified short pulse equation. J. Math. Phys. 57, 111507 (2016)

    Article  MathSciNet  Google Scholar 

  15. Prinari, B., Trubatch, A.D., Feng, B.F.: Inverse scattering transform for the complex short-pulse equation by a Riemann–Hilbert approach. Eur. Phys. J. Plus. 135, 717 (2020)

    Article  Google Scholar 

  16. Sakovich, A., Sakovich, S.: The short pulse equation is integrable. J. Phys. Soc. Jpn. 74, 239–241 (2005)

    Article  Google Scholar 

  17. Sakovich, S.: Transformation and integrability of a generalized short pulse equation. Commun. Nonlinear Sci. Numer. Simul. 39, 21–28 (2016)

    Article  MathSciNet  Google Scholar 

  18. Schäfer, T., Wayne, C.E.: Propagation of ultra-short optical pulses in cubic nonlinear media. Phys. D 196, 90–105 (2004)

    Article  MathSciNet  Google Scholar 

  19. Shen, S.F., Feng, B.F., Ohta, Y.: A modified complex short pulse equation of defocusing type. J. Nonlinear Math. Phys. 24, 195–209 (2017)

    Article  MathSciNet  Google Scholar 

  20. Xu, J.: Long-time asymptotics for the short pulse equation. J. Differ. Equ. 265, 3494–3532 (2018)

    Article  MathSciNet  Google Scholar 

  21. Xu, J., Fan, E.: Long-time asymptotic behavior for the complex short pulse equation. J. Differ. Equ. 269, 10322–10349 (2020)

    Article  MathSciNet  Google Scholar 

  22. Zhao, D.: Zhaqilao: on two new types of modified short pulse equation. Nonlinear Dyn. 100, 615–627 (2020)

    Article  Google Scholar 

  23. Zhaqilao, Z., Hu, Q., Qiao, Z.: Multi-soliton solutions and the Cauchy problem for a two-component short pulse system. Nonlinearity 30, 3773–3798 (2017)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

This work is supported by the National Natural Science Foundation of China (Grant Nos. 11871471 and 11931017) and the Yue Qi Outstanding Scholar Project, China University of Mining & Technology, Beijing (Grant No. 00-800015Z1177).

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  1. Department of Mathematics, China University of Mining and Technology, Beijing, 100083, People’s Republic of China

    Cong Lv & Q. P. Liu

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  1. Cong Lv
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  2. Q. P. Liu
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Correspondence to Q. P. Liu.

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Lv, C., Liu, Q.P. Solving the modified complex short pulse equation of focusing type: a Riemann–Hilbert approach. Anal.Math.Phys. 12, 27 (2022). https://doi.org/10.1007/s13324-021-00637-7

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  • DOI: https://doi.org/10.1007/s13324-021-00637-7

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