The modified coupled Hirota equation: Riemann-Hilbert approach and N-soliton solutions. (English) Zbl 1474.35590

Xu, Siqi

Abstr. Appl. Anal. 2019, Article ID 8342876, 10 p. (2019).

Summary: The Cauchy initial value problem of the modified coupled Hirota equation is studied in the framework of Riemann-Hilbert approach. The N-soliton solutions are given in a compact form as a ratio of \((N + 1) \times(N + 1)\) determinant and \(N \times N\) determinant, and the dynamical behaviors of the single-soliton solution are displayed graphically.

MSC:

35Q55	NLS equations (nonlinear Schrödinger equations)
35C08	Soliton solutions
30E25	Boundary value problems in the complex plane

References:

[1]	Hirota, R., Exact envelope-soliton solutions of a nonlinear wave equation, Journal of Mathematical Physics, 14, 7, 805-809 (1973) · Zbl 0257.35052
[2]	Ankiewicz, A.; Soto-Crespo, J. M.; Akhmediev, N., Rogue waves and rational solutions of the Hirota equation, Physical Review E: Statistical, Nonlinear, and Soft Matter Physics, 81, 4 (2010)
[3]	Tao, Y.; He, J., Multisolitons, breathers, and rogue waves for the Hirota equation generated by the Darboux transformation, Physical Review E: Statistical, Nonlinear, and Soft Matter Physics, 85, 2 (2012)
[4]	Li, L.; Wu, Z.; Wang, L.; He, J., High-order rogue waves for the Hirota equation, Annals of Physics, 334, 198-211 (2013) · Zbl 1284.35405
[5]	Guo, B.; Ling, L.; Liu, Q. P., Nonlinear schrödinger equation: generalized darboux transformation and rogue wave solutions, Physical Review E: Statistical, Nonlinear, and Soft Matter Physics, 85, 2 (2012)
[6]	Dai, C.; Zhang, J., New solitons for the Hirota equation and generalized higher-order nonlinear Schrodinger equation with variable coefficients, Journal of Physics A: Mathematical and General, 39, 4, 723-737 (2006) · Zbl 1088.35538
[7]	Bhrawy, A. H.; Alshaery, A. A.; Hilal, E. M.; Manrakhan, W. N.; Savescu, M.; Biswas, A., Dispersive optical solitons with Schrödinger-Hirota equation, Journal of Nonlinear Optical Physics & Materials, 23, 1 (2014)
[8]	Faddeev, L.; Volkov, A. Y., Hirota equation as an example of an integrable symplectic map, Fifty Years of Mathematical Physics: Selected Works of Ludwig Faddeev, 252-262 (2016)
[9]	Eslami, M.; Mirzazadeh, M. A.; Neirameh, A., New exact wave solutions for hirota equation, Pramana—Journal of Physics, 84, 1, 3-8 (2015)
[10]	Wang, X.; Liu, C.; Wang, L., Darboux transformation and rogue wave solutions for the variable-coefficients coupled Hirota equations, Journal of Mathematical Analysis and Applications, 449, 2, 1534-1552 (2017) · Zbl 1361.35160
[11]	Tasgal, R. S.; Potasek, M. J., Soliton solutions to coupled higher-order nonlinear Schrödinger equations, Journal of Mathematical Physics, 33, 3, 1208-1215 (1992)
[12]	Nakkeeran, K., Exact soliton solutions for a family of N coupled nonliear Schrödinger equations in optical fiber media, Physical Review E: Statistical, Nonlinear, and Soft Matter Physics, 62, 1, part B, 1313-1321 (2000)
[13]	Bindu, S. G.; Mahalingam, A.; Porsezian, K., Dark soliton solutions of the coupled Hirota equation in nonlinear fiber, Physics Letters A, 286, 5, 321-331 (2001) · Zbl 0971.78019
[14]	Chen, S., Dark and composite rogue waves in the coupled Hirota equations, Physics Letters A, 378, 38-39, 2851-2856 (2014) · Zbl 1298.35027
[15]	Yang, J.; Kaup, D. J., Squared eigenfunctions for the Sasa-SATsuma equation, Journal of Mathematical Physics, 50, 2 (2009) · Zbl 1202.35275
[16]	Yang, J., Nonlinear Waves in Integrable and Nonintegrable Systems. Nonlinear Waves in Integrable and Nonintegrable Systems, Mathematical Modeling and Computation, 16 (2010), Philadelphia, Pa, USA: Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa, USA · Zbl 1234.35006
[17]	Wang, D.-S.; Zhang, D.-J.; Yang, J., Integrable properties of the general coupled nonlinear Schrödinger equation, Journal of Mathematical Physics, 51, 2, 133-148 (2010) · Zbl 1309.35145
[18]	Guo, B.; Ling, L., Riemann-Hilbert approach and N-soliton formula for coupled derivative Schrödinger equation, Journal of Mathematical Physics, 53, 7 (2012) · Zbl 1276.81068
[19]	Geng, X.; Wu, J. P., Riemann-Hilbert approach and \(N\)-soliton solutions for a generalized Sasa-Satsuma equation, Wave Motion. An International Journal Reporting Research on Wave Phenomena, 60, 62-72 (2016) · Zbl 1467.35282
[20]	Wu, J.; Geng, X., Inverse scattering transform and soliton classification of the coupled modified Korteweg-de Vries equation, Communications in Nonlinear Science and Numerical Simulation, 53, 83-93 (2017) · Zbl 1510.35281
[21]	Eslami, M.; Mirzazadeh, M., First integral method to look for exact solutions of a variety of Boussinesq-like equations, Ocean Engineering, 83, 133-137 (2014)
[22]	Biswas, A.; Mirzazadeh, M.; Eslami, M., Dispersive dark optical soliton with Schödinger-Hirota equation by G′/G-expansion approach in power law medium, Optik - International Journal for Light and Electron Optics, 125, 16, 4215-4218 (2014)
[23]	Ekici, M.; Mirzazadeh, M.; Sonmezoglu, A.; Ullah, M. Z.; Asma, M.; Zhou, Q.; Moshokoa, S. P.; Biswas, A.; Belic, M., Dispersive optical solitons with Schrödinger-Hirota equation by extended trial equation method, Optik - International Journal for Light and Electron Optics, 136, 451-461 (2017)
[24]	Kilic, B.; Inc, M., Optical solitons for the Schrödinger-Hirota equation with power law nonlinearity by the Bäcklund transformation, Optik - International Journal for Light and Electron Optics, 138, 64-67 (2017)
[25]	Xu, S.; Geng, X., \(N\) -soliton solutions for the nonlocal two-wave interaction system via the Riemann-Hilbert method, Chinese Physics B, 27, 12 (2018)
[26]	Ablowitz, M. J.; Fokas, A. S., Complex Variables, Introduction and Applications. Complex Variables, Introduction and Applications, Cambridge University Press (2003), New York, NY, USA · Zbl 1088.30001