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Riemann–Hilbert approach and N double-pole solutions for the third-order flow equation of nonlinear derivative Schrödinger-type equation

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Abstract

This paper is concerned with the application of the inverse scattering transforms to a third-order flow equation of nonlinear derivative Schrödinger-type equation with zero boundary conditions and double zeros of the analytic scattering coefficients. First of all, we give the third-order flow equation of nonlinear derivative Schrödinger-type equation and its Lax pair by using the zero curvature equation, which are firstly given to our knowledge. Then, in the case of double zeros for the scattering coefficient and the reflectionless, we discuss in detail the direct problem and the inverse problem of this equation, including Jost solutions, scattering coefficient, the matrix Riemann–Hilbert (RH) problem (analyticity, symmetries and asymptotic behaviors) and trace formulae. Thus, N double-pole solutions of the third-order flow equation are obtained by solving the matrix RH problem. Finally, some examples are given.

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References

  1. Lax, P.D.: Integrals of nonlinear equations of evolution and solitary waves. Commun. Pure. Appl. Math. 21, 467–490 (1968). https://doi.org/10.1002/cpa.3160210503

    Article  MathSciNet  MATH  Google Scholar 

  2. Ablowitz, M.J., Fokas, A.S.: Complex Variables: Introduction and Applications. Cambridge University Press, Cambridge (2003)

    Book  MATH  Google Scholar 

  3. Hasegawa, A., Tappert, F.: Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion. Appl. Phys. Lett. 23, 142–144 (1973). https://doi.org/10.1063/1.1654836

  4. Hasegawa, A., Kodama, Y.: Signal transmission by optical solitons in monomode fiber. In: Proceedings of the IEEE, Vol. 69, pp. 1145–1150 (1981) https://doi.org/10.1109-/PROC.1981.12129

  5. Hasegawa, A., Kodama, Y.: Solitons in Optical communication. Oxford University Press, Oxford (1995)

    MATH  Google Scholar 

  6. Turitsyn, S.K., Prilepsky, J.E., Thai, L.S., et al.: Nonlinear Fourier transform for optical data processing and transmission: advances and perspectives. Optica 4, 307–322 (2017). https://doi.org/10.1364/OPTICA.4.000307

    Article  Google Scholar 

  7. Biswas, A.: Stochastic perturbation of dispersion-managed optical solitons. Opt. Quant. Electron. 37, 649–659 (2005). https://doi.org/10.1007/s11082-005-5688-x

    Article  Google Scholar 

  8. 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). https://doi.org/10.1007/s11071-014-1678-5

    Article  MathSciNet  MATH  Google Scholar 

  9. Agalarov, A., Zhulego, V., Gadzhimuradov, T.: Bright, dark, and mixed vector soliton solutions of the general coupled nonlinear Schrödinger equations. Phys. Rev. E 91, 042909 (2015). https://doi.org/10.1103/PhysRevE.91.042909

    Article  MathSciNet  Google Scholar 

  10. Jiwari, R., Kumar, S., Mittal, R.C., Awrejcewicz, J.: A meshfree approach for analysis and computational modeling of non-linear Schrödinger equation. Comput. Appl. Math. 39, 95 (2020). https://doi.org/10.1007/s40314-020-1113-0

    Article  MATH  Google Scholar 

  11. Yadav, O.P., Jiwari, R.: Some soliton-type analytical solutions and numerical simulation of nonlinear Schrödinger equation. Nonlinear Dyn. 95, 2825–2836 (2019). https://doi.org/10.1007/s11071-018-4724-x

    Article  MATH  Google Scholar 

  12. Novikov, S., Manakov, S.V., Pitaeskii, L.P., Zakharov, V.E.: Theory of Soliton: The Inverse Scattering Method. Springer, Berlin (1984)

    Google Scholar 

  13. Coifman, R.R., Beals, R.: Scattering and inverse scattering for first order systems. Commun. Pure Appl. Math. 37, 39–90 (1984). https://doi.org/10.1002/cpa.316

    Article  MathSciNet  MATH  Google Scholar 

  14. Zhang, G.Q., Yan, Z.Y.: Focusing and defocusing mKdV equations with nonzero boundary conditions: inverse scattering transforms and soliton interactions. Physica D 410, 132521 (2020). https://doi.org/10.1016/j.physd.2020.132521

    Article  MathSciNet  MATH  Google Scholar 

  15. Peng, W.Q., Chen, Y.: Double poles soliton solutions for the Gerdjikov–Ivanov type of derivative nonlinear schrödinger equation with zero/nonzero boundary couditions. J. Math. Phys. 63(3), 033502 (2022). https://doi.org/10.1063/5.006180710.1063/5.0061807

    Article  Google Scholar 

  16. Zhang, G.Q., Chen, S.Y., Yan, Z.Y.: Focusing and defocusing Hirota equations with non-zero boundary conditions: Inverse scattering transforms and soliton solutions. Commun. Nonlinear Sci. Numer. Simul. 80, 104927 (2020). https://doi.org/10.1016/j.cnsns.2019.104927

    Article  MathSciNet  MATH  Google Scholar 

  17. Pu, J.C., Chen, Y.: Double and Triple-Pole Solutions for the third-order flow equation of the Kaup–Newell system with zero/nonzero boundary conditions. arXiv:2105.06098v3 [nlin.SI]

  18. Zhu, J.Y., Chen, Y.: A new form of general soliton solutions and multiple zeros solutions for a higher-order Kaup–Newell equation. J. Math. Phys. 62(12), 123501 (2022). https://doi.org/10.1063/5.0064411

    Article  MathSciNet  MATH  Google Scholar 

  19. Zhang, G.Q., Yan, Z.Y.: The derivative nonlinear Schrödinger equation with zero/nonzero boundary conditions: inverse scattering transforms and N-double-pole solutions. J. Nonlinear Sci. 30, 3089–3127 (2020). https://doi.org/10.1007/s00332-020-09645-6

    Article  MathSciNet  MATH  Google Scholar 

  20. Zhang, X.F., Tian, S.F., Yang, J.J.: The Riemann–Hilbert approach for the focusing Hirota equation with single and double poles. Anal. Math. Phys. 11(86), 1–18 (2021). https://doi.org/10.1007/s13324-021-00522-3

    Article  MathSciNet  MATH  Google Scholar 

  21. Wadati, M., Konno, K., Ichikawa, Y.H.: A generalization of inverse scattering method. J. Phys. Soc. Jpn. 46, 1965–1966 (1979). https://doi.org/10.1143/JPSJ.46.1965

    Article  MathSciNet  MATH  Google Scholar 

  22. Lin, Y., Fang, Y., Dong, H.: Prolongation structures and \(N\)-soliton solutions for a new nonlinear Schrödinger-type equation via Riemann–Hilbert approach. Math. Probl. Eng. 2019, 4058041 (2019). https://doi.org/10.1155/2019/4058041

    Article  MATH  Google Scholar 

  23. Zhang, B., Fan, E.G.: Riemann–Hilbert approach for a Schrödinger-type equation with nonzero boundary conditions. Mod. Phys. Lett. B 35(12), 2150208 (2021). https://doi.org/10.1142/S0217984921502080

    Article  Google Scholar 

  24. Zhang, G.F., He, J.S., Cheng, Y.: Riemann–Hilbert approach and N double-pole solutions for a nonlinear Schrödinger-type equation. Chin. Phys. B 31, 110201 (2022). https://doi.org/10.1088/1674-1056/ac7a1b

    Article  Google Scholar 

  25. Li, Y.S.: Soliton and Integrable System. Shanghai Scientific and Technological Education Publishing House, Shanghai (1999). (in Chinese)

    Google Scholar 

  26. Moses, J., Malomed, B.A., Wise, F.W.: Self-steepening of ultrashort optical pulses without self-phase-modulation. Phys. Rev. A 76, 021802 (2007). https://doi.org/10.1103/PhysRevA.76.021802

    Article  Google Scholar 

  27. Trippenbach, M., Band, Y.B.: Effects of self-steepening and self-frequency shifting on short-pulse splitting in dispersive nonlinear media. Phys. Rev. A 57, 4791–4803 (1998). https://doi.org/10.1103/PhysRevA.57.4791

    Article  Google Scholar 

  28. Mohamadou, A., Latchio-Tiofack, C.G., Kofane, T.C.: Wave train generation of solitons in systems with higher-order nonlinearities. Phys. Rev. E 82, 016601 (2010). https://doi.org/10.1103/PhysRevE.82.016601

    Article  Google Scholar 

  29. Choudhuri, A., Porsezian, K.: Impact of dispersion and non-Kerr nonlinearity on the modulational instability of the higher-order nonlinear Schrödinger equation. Phys. Rev. A 85(3), 1431–1435 (2012). https://doi.org/10.1103/PhysRevA.85.033820

    Article  Google Scholar 

  30. Renninger, W.H., Chong, A., Wise, F.W.: Dissipative solitons in normal-dispersion fiber lasers. Phys. Rev. A 77, 023814 (2008). https://doi.org/10.1103/PhysRevA.77.023814

    Article  Google Scholar 

  31. Peng, J.S., Zhan, L., Gu, Z.C., Qian, K., Luo, S.Y., Shen, Q.S.: Experimental observation of transitions of different pulse solutions of the Ginzburg–Landau equation in a modelocked fiber laser. Phys. Rev. A 86, 033808 (2012). https://doi.org/10.1103/PhysRevA.86.033808

    Article  Google Scholar 

  32. Akhmediev, N., Afanasjev, V.V.: Novel arbitrary-amplitude soliton solutions of the cubic-quintic complex Ginzburg–Landau equation. Phys. Rev. Lett. 75, 2320–2323 (1995). https://doi.org/10.1103/PhysRevLett.75.2320

    Article  Google Scholar 

  33. Akhmediev, N., Afanasjev, V.V., Soto-Crespo, J.M.: Singularities and special soliton solutions of the cubic-quintic complex Ginzburg-Landau equation. Phys. Rev. E 53, 1190–1200 (1996). https://doi.org/10.1103/PhysRevE.53.1190

    Article  Google Scholar 

  34. Soto-Crespo, J.M., Akhmediev, N., Afanasjev, V.V.: Stability of the pulselike solutions of the quintic complex Ginzburg–Landau equation. J. Opt. Soc. Am. B 13, 1439–1449 (1996). https://doi.org/10.1364/josab.13.001439

    Article  Google Scholar 

  35. Kharif, C., Pelinovsky, E.: Physical mechanisms of the rogue wave phenomenon. Eur. J. Mech. B. Fluids 22, 603–634 (2003). https://doi.org/10.1016/j.euromechflu.2003.09.002

    Article  MathSciNet  MATH  Google Scholar 

  36. Ma, X.: Riemann–Hilbert approach for a higher-order Chen–Lee–Liu equation with high-order poles. Commun. Nonlinear Sci. Numer. Simul. 114, 106606 (2022). https://doi.org/10.1016/j.cnsns.2022.106606

    Article  MathSciNet  MATH  Google Scholar 

  37. Lin, H., He, J., Wang, L., Mihalache, D.: Several categories of exact solutions of the third-order flow equation of the Kaup–Newell system. Nonlinear Dyn. 100, 2839–2858 (2020). https://doi.org/10.1007/s11071-020-05650-2

    Article  Google Scholar 

  38. Cheng, Q., Fan, E.: Long-time asymptotics for a mixed nonlinear Schrödinger equation with the Schwartz initial data. J. Math. Anal. Appl. 489, 124188 (2020). https://doi.org/10.1016/j.jmaa.2020.124188

    Article  MathSciNet  MATH  Google Scholar 

  39. Zhou, X.: Direct and inverse scattering transforms with arbitrary spectral singularities. Commun. Pure Appl. Math. 42, 895–938 (1989). https://doi.org/10.1002/cpa.3160420702

    Article  MathSciNet  MATH  Google Scholar 

  40. Biondini, G., Kovačič, G.: Inverse scattering transform for the focusing nonlinear Schrödinger equation with nonzero boundary conditions. J. Math. Phys. 55, 031506 (2014). https://doi.org/10.1063/1.4868483

    Article  MathSciNet  MATH  Google Scholar 

  41. 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, 1–18 (2020). https://doi.org/10.1140/epjp/s13360-020-00714-z

    Article  Google Scholar 

  42. Zhao, Y., Fan, E.G.: Inverse scattering transformation for the fokas lenells equation with nonzero boundary conditions. J. Nonlinear Math. Phys. 28, 38–52 (2021). https://doi.org/10.2991/JNMP.K.200922.003

    Article  MathSciNet  MATH  Google Scholar 

  43. Weng, W., Yan, Z.: Inverse scattering and N-triple-pole soliton and breather solutions of the focusing nonlinear Schrödinger hierarchy with nonzero boundary conditions. Phys. Lett. A 407, 127472 (2021). https://doi.org/10.1016/j.physleta.2021.127472

  44. Liu, N., Xuan, Z.X., Sun, J.Y.: Triple-pole soliton solutions of the derivative nonlinear Schrödinger equation via inverse scattering transform. Appl. Math. Lett. 125, 107741 (2022). https://doi.org/10.1016/j.aml.2021.107741

    Article  MATH  Google Scholar 

  45. Weng, W.F., Yan, Z.Y.: The multi-triple-pole solitons for the focusing mKdV hierarchy with nonzero boundary conditions. Mod. Phys. Lett. B 35, 2150483 (2021). https://doi.org/10.1142/S0217984921504832

    Article  MathSciNet  Google Scholar 

  46. Wang, X.B., Han, B.: Inverse scattering transform of an extended nonlinear Schrödinger equation with nonzero boundary conditions and its multisoliton solutions. J. Math. Anal. Appl. 487(1), 123968 (2020). https://doi.org/10.1016/j.jmaa.2020.123968

    Article  MathSciNet  MATH  Google Scholar 

  47. Zhang, Z.Z., Fan, E.G.: Inverse scattering transform and multiple high-order pole solutions for the Gerdjikov–Ivanov equation under the zero/nonzero background. Angew. Math. Phys. 72, 153 (2021). https://doi.org/10.1007/s00033-021-01583-x

    Article  MathSciNet  MATH  Google Scholar 

  48. Mao, J.J., Tian, S.F., Xu, T.Z., Shi, L.F.: The bound-state soliton solutions of a higher-order nonlinear Schrödinger equation for inhomogeneous Heisenberg ferromagnetic system. Nonlinear Dyn. 104, 2639–2652 (2021). https://doi.org/10.1007/s11071-021-06425-z

    Article  Google Scholar 

  49. Zhang, Y.S., Tao, X.X., Yao, T.T., He, J.S.: The regularity of the multiple higher-order poles solitons of the NLS equation. Stud. Appl. Math. 145, 812–827 (2020). https://doi.org/10.1111/sapm.12338

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

This work is supported by the National Natural Science Foundation of China (Grant Nos. 12071304 and 11871446) and the Guangdong Basic and Applied Basic Research Foundation (Grant No. 2022A15 15012554).

Funding

This work is is funded by the National Natural Science Foundation of China (Grant Nos. 12071304 and 11871446) and the Guangdong Basic and Applied Basic Research Foundation (Grant No. 2022A15 15012554).

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Authors and Affiliations

  1. Institute for Advanced Study, Shenzhen University, Shenzhen, 518060, Guangdong, People’s Republic of China

    Guofei Zhang & Jingsong He

  2. School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, Anhui, People’s Republic of China

    Guofei Zhang & Yi Cheng

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  1. Guofei Zhang
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Zhang, G., He, J. & Cheng, Y. Riemann–Hilbert approach and N double-pole solutions for the third-order flow equation of nonlinear derivative Schrödinger-type equation. Nonlinear Dyn 111, 6677–6687 (2023). https://doi.org/10.1007/s11071-022-08194-9

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