The Riemann–Hilbert approach to the generalized second-order flow of three-wave hierarchy

References

• Gardner CS, Greene JM, Kruskal MD, et al. Method for solving the Korteweg-de Vries equation. Phys Rev Lett. 1967;19:1095–1097.
   Web of Science ®Google Scholar
• Zakharov VE, Shabat AB. Exact theory of two-dimensional self-focusing and onedimensional self-modulation of waves in non linear media. Zh Exp Teor Fiz. 1971;61:118. Russian. English transl. in Sov Phys JEPT. 1972;34:62.
   Google Scholar
• Ablowitz M, Kaup D, Newell A, et al. The inverse scattering transform Fourier analysis for nonlinear problems. Stud Appl Math. 1974;53:249–315.
   Web of Science ®Google Scholar
• Gerdjikov VS, Kulish PP. The generating operator for the n×n linear system. Phys D. 1981;3:549–564.
   Web of Science ®Google Scholar
• Lax PD. Integrals of nonlinear equations of evolution and solitary waves. Commun Pure Appl Math. 1968;21:467–490.
   Web of Science ®Google Scholar
• Beals R, Coifman RR. Scattering and inverse scattering for first order systems. Commun Pure Appl Math. 1984;37:39.
   Web of Science ®Google Scholar
• Deift P, Zhou X. A steepest descent method for oscillatory Riemann-Hilbert problems. Ann Math (2). 1993;137:295–368.
   Web of Science ®Google Scholar
• Zakharov VE, Manakov SV, Novikov SP, et al. The theory of solitons: the inverse scattering method. New York (NY): Consultants Bureau; 1984.
   Google Scholar
• Gerdjikov VS. Basic aspects of soliton theory. In: Mladenov IM, Hirshfeld AC, editors, Sixth International Conference on Geometry, Integrability and Quantization; 2004 June 3–10; Varna, Bulgaria. Sofia; 2005. p. 1–48.
   Google Scholar
• Yang J. Nonlinear waves in integrable and non-integrable systems. Society for Industrial and Applied Mathematics, Philadelphia; 2010.
   Google Scholar
• Wang DS, Zhang DJ, Yang J. Integrable properties of the general coupled nonlinear Schrödinger equations. J Math Phys. 2010;51:023510.
   Web of Science ®Google Scholar
• Ma WX. Riemann-Hilbert problems and n-soliton solutions for a coupled mKdV system. J Geom Phys. 2018;132:45–54.
   Web of Science ®Google Scholar
• Ma WX. Application of the Riemann-Hilbert approach to the multicomponent AKNS integrable hierarchies. Nonlinear Anal Real World Appl. 2019;47:1–17.
   Web of Science ®Google Scholar
• Ma WX. Riemann-Hilbert problems of a six-component mKdV system and its soliton solutions. Acta Math Sci. 2019;39B:509–523.
   Web of Science ®Google Scholar
• Ma WX. The inverse scattering transform and soliton solutions of a combined modified Korteweg-de Vries equation. J Math Anal Appl. 2019;471:796–811.
   Web of Science ®Google Scholar
• Xu J, Fan EG. Long-time asymptotics for the Fokas-Lenells equation with decaying initial value problem: without solitons. J Differ Equ. 2015;259:1098–1148.
   Web of Science ®Google Scholar
• Zhang XE, Chen Y. Inverse scattering transformation for generalized nonlinear Schrödinger equation. Appl Math Lett. 2019;98:306–313.
   Web of Science ®Google Scholar
• Geng XG, Wu JP. Riemann-Hilbert approach and n-soliton solutions for a generalized Sasa-Satsuma equation. Wave Motion. 2016;60:62–72.
   Web of Science ®Google Scholar
• Zhang Y, Dong HH, Wang DS. Riemann-Hilbert problems and soliton solutions for a multi-component cubic-quintic nonlinear Schrödinger equation. J Geom Phys. 2020;149:103569.
   Web of Science ®Google Scholar
• Kharif C, Pelinovsky E, Slunyaev A. Rogue waves in the ocean. Berlin: Springer; 2009. (Advances in Geophysical and Environmental Mechanics and Mathematics; 14).
   Google Scholar
• Osborne AR. Nonlinear ocean waves, the inverse scattering transform. Amsterdam: Elsevier; 2010.
   Google Scholar
• Gao XY. Mathematical view with observational/experimental consideration on certain (2+1)-dimensional waves in the cosmic/laboratory dusty plasmas. Appl Math Lett. 2019;91:165–172.
   Web of Science ®Google Scholar
• Gao XY, Guo YJ, Shan WR. Water-wave symbolic computation for the Earth, Enceladus and Titan: the higher-order Boussinesq-Burgers system auto- and non-auto-Bäcklund transformations. Appl Math Lett. 2020;104:106170.
   Web of Science ®Google Scholar
• Wang DS, Shi YR, Feng WX, et al. Dynamical and energetic instabilities of F=2 spinor Bose-Einstein condensates in an optical lattice. Phys D. 2017;30:351–352.
   Google Scholar
• Yin HM, Tian B, Zhao XC. Chaotic breathers and breather fission/fusion for a vector nonlinear Schrödinger equation in a birefringent optical fiber or wavelength division multiplexed system. Appl Math Comput. 2020;368:124768.
   Web of Science ®Google Scholar
• Chen SS, Tian B, Sun Y, et al. Generalized Darboux transformations rogue waves and modulation instability for the coherently coupled nonlinear Schrödinger equations in nonlinear optics. Ann Phys (Berlin). 2019;531:1900011.
   Web of Science ®Google Scholar
• Du Z, Tian B, Chai HP, et al. Dark-bright semi-rational solitons and breathers for a higher-order coupled nonlinear Schrödinger system in an optical fiber. Appl Math Lett. 2020;102:106110.
   Web of Science ®Google Scholar
• Fordy AP, Kulish PP. Nonlinear Schrödinger equations and simple lie algebras. Commun Math Phys. 1983;89:427–443.
   Web of Science ®Google Scholar
• Wahlquist HD, Estabrook FB. Prolongation structures of nonlinear evolution equations. J Math Phys. 1975;16:1–7.
   Web of Science ®Google Scholar
• Choudhury SR, Russo M. The extended Estabrook-Wahlquist method. Phys D. 2016;327:58–72.
   Web of Science ®Google Scholar
• Yan ZW, Li CZ. Fermionic covariant prolongation structure theory for a (2+1)-dimensional super nonlinear evolution equation. Int J Geom Methods Mod Phys. 2018;15:1850114.
   Web of Science ®Google Scholar
• Igonin S, Manno G. Lie algebras responsible for zero-curvature representations of scalar evolution equations. J Geom Phys. 2019;138:297–316.
   Web of Science ®Google Scholar
• Wang DS, Yin SJ, Tian Y, et al. Integrability and bright soliton solutions to the coupled nonlinear Schrödinger equation with higher-order effects. Appl Math Comput. 2014;229:296–309.
   Web of Science ®Google Scholar
• Humphreys J. Introduction to lie algebras and representation theory GTM 9. Springer-Verlag, New York; 1972.
   Google Scholar
• Wadati M, Sanuki H, Konno K. Relationships among inverse method, Backlund transformation and an infinite number of conservation laws. Progr Theoret Phys. 1975;53:419–436.
   Web of Science ®Google Scholar
• Doktorov EV, Leble SB. A dressing method in mathematical physics.  Springer, Dordrecht; 2007.
   Google Scholar
• Shabat AB. An inverse scattering problem. Differ Equ. 1979;15:1299–1307.
   Google Scholar
• Zakharov VE, Shabat AB. A scheme for integrating nonlinear equations of mathematical physics by the method of the inverse scattering transform. I. Funct Anal Appl. 1974;8:43–53.
   Google Scholar
• Zakharov VE, Shabat AB. A scheme for integrating nonlinear equations of mathematical physics by the method of the inverse scattering transform. II. Funct Anal Appl. 1979;13:166–174.
   Google Scholar
• Gerdjikov VS. Algebraic and analytic aspects of soliton type equations. Contemp Math. 2002;301:35–68.
   Google Scholar