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On two new types of modified short pulse equation

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Abstract

A complex modified short pulse (mSP) equation and a nonlocal mSP equation are presented. The integrability and the covariant property of both new equations are shown by constructing their corresponding Lax pair and Darboux transformation, respectively. For the complex mSP equation, some multiple smooth soliton, cuspon soliton, loop soliton, breather and rogue wave solutions are constructed by N-fold Darboux transformation. For the nonlocal mSP equation, some unstable soliton and stable soliton solutions are obtained. The dynamical characteristics of the obtained solutions are shown through some figures.

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References

  1. 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 

  2. Robelo, M.L.: On equations which describe pseudospherical surface. Stud. Appl. Math. 81, 221–248 (1989)

    Article  MathSciNet  Google Scholar 

  3. Chung, Y., Jones, C.K.R.T., Schäfer, T., Wayne, C.E.: Ultra-short pulse in linear and nonlinear media. Nonlinearity 18, 1374 (2005)

    MathSciNet  MATH  Google Scholar 

  4. Brunelli, J.C., Sakovich, S.: On integrability of the Yao–Zeng two-component short-pulse equation. Phys. Lett. A 377, 80–82 (2012)

    Article  MathSciNet  Google Scholar 

  5. Brunelli, J.C.: The bi-hamiltonian structure of the short pulse equation. Phys. Lett. A 353, 475–478 (2006)

    Article  MathSciNet  Google Scholar 

  6. Brunelli, J.C., Sakovich, S.: Hamiltonian integrability of two-component short pulse equations. J. Math. Phys. 54, 012701 (2013)

    Article  MathSciNet  Google Scholar 

  7. Feng, B.F., Maruno, K., Ohta, Y.: Integrable semi-discretization of a multi-component short pulse equation. J. Math. Phys. 56, 043502 (2015)

    Article  MathSciNet  Google Scholar 

  8. Zhang, Z.Y., Chen, Y.F.: Conservation laws of the generalized short pulse equation. Chin. Phys. B 24, 020201 (2015)

    Article  Google Scholar 

  9. Matsuno, Y.: A novel multi-component generalization of the short pulse equation and its multisoliton solutions. J. Math. Phys. 52, 123702 (2011)

    Article  MathSciNet  Google Scholar 

  10. Rui, W.G.: Different kinds of exact solutions with two-loop character of the two-component short pulse equations of the first kind. Commun. Nonlinear Sci. Numer. Simul. 18, 2667–2678 (2013)

    Article  MathSciNet  Google Scholar 

  11. Parkes, E.J.: A note on IOP-soliton solutions of the short-pulse equation. Phys. Lett. A 374, 4321–4323 (2010)

    Article  MathSciNet  Google Scholar 

  12. Xie, S.L., Hong, X.C., Gao, B.: The periodic traveling-wave solutions of the short-pulse equation. Appl. Math. Comput. 218, 2542–2548 (2011)

    MathSciNet  MATH  Google Scholar 

  13. Fakhar, K., Wang, G.W., Kara, A.H.: Symmetry reductions and conservation laws of the short pulse equation. Optik 127, 10201–10207 (2016)

    Article  Google Scholar 

  14. Gao, B., Zhang, Y.: Symmetries and conservation laws of the Yao–Zeng two-component short-pulse equation. Bound. Value Probl. 2019, 45–61 (2019)

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  16. Feng, B.F.: An integrable coupled short pulse equation. J. Phys. A 45, 085202 (2012)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  18. Hirota, R.: The Direct Method in Soliton Theory. Cambridge University Press, Cambridge (2004)

    Book  Google Scholar 

  19. Zhaqilao: The interaction solitons for the complex short pulse equation. Commun. Nonlinear Sci. Numer. Simul. 47, 379–393 (2017)

  20. van der Mee, C.: Complex short-pulse solutions by gague transformation. J. Geom. Phys. 148, 103539 (2020)

    Article  MathSciNet  Google Scholar 

  21. Ling, L.M., Feng, B.F., Zhu, Z.N.: Multi-soliton, multi-breather and higher order rogue wave solutions to the complex short pulse equation. Phys. D 327, 13–29 (2016)

    Article  MathSciNet  Google Scholar 

  22. Gupta, R.K., Kumar, V., Jiwari, R.: Exact and numerical solutions of coupled short pulse equation with time-dependent coefficients. Nonlinear Dyn. 79, 455–464 (2015)

    Article  MathSciNet  Google Scholar 

  23. Gao, B., He, C.F.: Analysis of a coupled short pulse system via symmetry method. Nonlinear Dyn. 90, 2627–2636 (2017)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  25. Matsuno, Y.: Integrable multi-component generalization of modified short pulse equation. J. Math. Phys. 57, 11507 (2016)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  27. Gu, C.H., Hu, H.S., Zhou, Z.X.: Darboux Transformation in Integrable Systems: Theory and Their Applications to Geometry. Springer, Dordrecht (2005)

    Book  Google Scholar 

  28. Matveev, V.B., Salle, M.A.: Daboux Transformation and Soliton. Springer, Berlin (1991)

    Book  Google Scholar 

  29. Zhaqilao, Qiao, Z.J.: Darboux transformation and explicit solutions for two integrable equations. J. Math. Anal. Appl. 380, 794–806 (2011)

  30. Zhaqilao: Darboux transformation and N-soliton solutions for a more general set of coupled integrable dispersionless system. Commun. Nonlinear Sci. Numer. Simulat. 16, 3949–3955 (2011)

  31. Hu, H.C., Tong, B., Lou, S.Y.: Nonsingular positon and complexiton solutions for the coupled KdV system. Phys. Lett. A 351, 403–412 (2006)

    Article  MathSciNet  Google Scholar 

  32. Xu, S.W., He, J.S., Wang, L.H.: The Darboux transformation of the derivative nonlinear Schrodinger equation. J. Phys. A: Math. Theor. 44, 305203 (2011)

    Article  MathSciNet  Google Scholar 

  33. Ablowitz, J.M., Musslimani, Z.H.: Integrable nonlocal nonlinear Schrödinger equation. Phys. Rev. Lett. 110, 064105 (2013)

    Article  Google Scholar 

  34. Brunelli, J.C.: Nonlocal short pulse equations. Braz. J. Phys. 48, 421–425 (2018)

    Article  Google Scholar 

  35. Li, L., Duan, C.N., Yu, F.J.: An improved Hirota bilinear method and new application for a nonlocal integrable complex modified Korteweg–de Vries (MKdV) equation. Phys. Lett. A 383, 1578–1582 (2019)

    Article  MathSciNet  Google Scholar 

  36. Zuo, D.W., Zhang, G.F.: Exact solutions of the nonlocal Hirota equations. Appl. Math. Lett. 93, 66–71 (2019)

    Article  MathSciNet  Google Scholar 

  37. Zhang, Q.Y., Zhang, Y., Ye, R.S.: Exact solutions of nonlocal Fokas–Lenells equation. Appl. Math. Lett. 98, 336–343 (2019)

    Article  MathSciNet  Google Scholar 

  38. Wu, W.B., Lou, S.Y.: Exact solutions of an Alice–Bob KP equation. Commun. Theor. Phys. 71, 629–632 (2019)

    Article  MathSciNet  Google Scholar 

  39. Wen, Z.C., Yan, Z.Y.: Solitons and their stability in the nonlocal nonlinear Schrödinger equation with PT-symmetric potentials. Chaos 27, 053105 (2017)

    Article  MathSciNet  Google Scholar 

  40. Shi, Y., Shen, S.F., Zhao, S.L.: Solutions abd connections of nonlocal derivative nonlinear Schrödinger equations. Nonlinear Dyn. 95, 1257–1267 (2019)

    Article  Google Scholar 

  41. Wang, Y.Y., Dai, C.Q., Xu, Y.Q., Zheng, J., Fan, Y.: Dynamics of nonlocal and localized spatiotemporal solitons for a partially nonlocal nonlinear Schrödinger equation. Nonlinear Dyn. 92, 1261–1269 (2018)

    Article  Google Scholar 

  42. Yu, F.J., Li, L.: Dynamics of some novel breather solutions and rogue waves for the PT-symmetric nonlocal soliton equations. Nonlinear Dyn. 95, 1867–1877 (2019)

    Article  Google Scholar 

  43. Ma, L.Y., Zhao, H.Q., Gu, H.: Integrability and gauge equivalence of the reverse space-time nonlocal Sasa–Satsuma equation. Nonlinear Dyn. 91, 1909–1920 (2018)

    Article  Google Scholar 

  44. Zhaqilao: Nonlinear Wave Equations in Integrable System. Science Press, Beijing (2018)

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Acknowledgements

This work is supported by the National Natural Science Foundation of China under (Grant Nos. 11861050, 11261037), the Caoyuan Yingcai Program of Inner Mongolia Autonomous Region, China under (Grant No. CYYC2011050) and the Graduate Students’ Scientific Research Innovation Fund Program of Inner Mongolia Normal University (Grant No. CXJJS19099).

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  1. College of Mathematics Science, Inner Mongolia Normal University, Hohhot, 010022, People’s Republic of China

    Dan Zhao &  Zhaqilao

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  1. Dan Zhao
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  2. Zhaqilao
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Correspondence to Zhaqilao.

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Zhao, D., Zhaqilao On two new types of modified short pulse equation. Nonlinear Dyn 100, 615–627 (2020). https://doi.org/10.1007/s11071-020-05530-9

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  • DOI: https://doi.org/10.1007/s11071-020-05530-9

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