Riemann-Hilbert Problem and Multi-soliton Solutions to a Matrix Gerdjikov-Ivanov Equation

Abstract

In this paper, the authors investigate a new matrix Gerdjikov-Ivanov equation with an n₁×n₂ complex-valued potential matrix function, and study its integrability via the Riemann-Hilbert method. Based on the Lax pair of the matrix Gerdjikov-Ivanov equation, the analytical and symmetric properties of eigenfunctions are derived in detail. Meanwhile, a Riemann-Hilbert problem is successfully formulated. By solving the Riemann-Hilbert problem with reflectionless case, they systematically derive the multi-soliton solutions to the matrix Gerdjikov-Ivanov equation. As examples of the N-soliton solution formula, the localized structures and dynamic behaviors of one- and two-soliton solutions are analyzed and graphically illustrated. These results not only show the effectiveness of the Riemann-Hilbert method, but also reveal the complex spectral structure of the proposed equation.

Keywords

Riemann-Hilbert problem / Matrix Gerdjikov-Ivanov equation / Lax pair / Multi-soliton solutions / 35Q51 / 35Q15 / 35Q55

References

[1] Clarkson P A, Tuszynski J A. Exact solutions of the multidimensional derivative nonlinear Schrödinger equation for many-body systems of criticality. J. Phys. A: Gen. Phys., 1990, 23(19): 4269-4288

[2] Kundu A. Exact solutions to higher-order nonlinear equations through gauge transformation. Phys. D., 1987, 25(1–3): 399-406

[3] Fan E G. A family of completely integrable multi-Hamiltonian systems explicitly related to some celebrated equations. J. Math. Phys., 2001, 42(9): 4327-4344

[4] Mio K, Ogino T, Minami K, et al. . Modulational instability and envelope-solitons for nonlinear Alfvén waves propagating along the magnetic field in plasmas. J. Phys. Soc. Japan, 1976, 41(2): 667-673

[5] Kodama Y. Optical solitons in a monomode fiber. J. Stat. Phys., 1985, 39(5): 597-614

[6] Kaup D J, Newell A C. An exact solution for a derivative nonlinear Schrödinger equation. J. Math. Phys., 1978, 19(4): 798-801

[7] Chen H H, Lee Y C, Liu C S. Integrability of nonlinear Hamiltonian systems by inverse scattering method. Phys. Scr., 1979, 20(3–4): 490-492

[8] Gerdzhikov V S, Ivanov M. A quadratic pencil of general type and nonlinear evolution equations. II. Hierarchies of Hamiltonian structures. Bulg. J. phys., 1983, 10(2): 130-143

[9] Wadati M, Sogo K. Gauge transformations in soliton theory. J. Phys. Soc. Japan, 1983, 52(2): 394-398

[10] Fan E G. Darboux transformation and soliton-like solutions for the Gerdjikov-Ivanov equation. J. Phys. A: Math. Gen., 2000, 33(39): 6925-6933

[11] Fan E G. Integrable evolution systems based on Gerdjikov-Ivanov equations, bi-Hamiltonian structure, finite-dimensional integrable systems and N-fold Darboux transformation. J. Math. Phys., 2000, 41(11): 7769-7782

[12] Dai H H, Fan E G. Variable separation and algebro-geometric solutions of the Gerdjikov-Ivanov equation. Chaos Solitons Fractals, 2004, 22(1): 93-101

[13] Guo L J, Zhang Y S, Xu S W, et al. . The higher order rogue wave solutions of the Gerdjikov-Ivanov equation. Phys. Scr., 2014, 89(3): 035501

[14] Xu J, Fan E G, Chen Y. Long-time asymptotic for the derivative nonlinear Schrödinger equation with step-like initial value. Math. Phys. Anal. Geom., 2013, 16: 253-288

[15] Tian S F, Zhang T T. Long-time asymptotic behavior for the Gerdjikov-Ivanov type of derivative nonlinear Schrödinger equation with time-periodic boundary condition. Proc. Amer. Math. Soc., 2018, 1461713-1729

[16] Guo B L, Liu N. The Gerdjikov-Ivanov-type derivative nonlinear Schrödinger equation: Long-time dynamics of nonzero boundary conditions. Math. Methods. Appl. Sci., 2019, 42: 4839-4861

[17] Nie H, Zhu J Y, Geng X G. Trace formula and new form of N-soliton to the Gerdjikov-Ivanov equation. Anal. Math. Phys., 2018, 8(3): 415-426

[18] Ieda J, Uchiyama M, Wadati M. Inverse scattering method for square matrix nonlinear Schrödinger equation under nonvanishing boundary conditions. J. Math. Phys., 2007, 48(1): 013507

[19] Tsuchida T, Wadati M. Complete integrability of derivative nonlinear Schrödinger type equations. Inverse Probl., 1999, 1551363-1373

[20] Zhang Y, Dong H H. Multi-component Gerdjikov-Ivanov system and its Riemann-Hilbert problem under zero boundary conditions. Nonlinear Anal. Real World Appl., 2021, 60: 103279

[21] Zhang Y, Cheng Y, He J S. Riemann-Hilbert method and N-soliton for two-component Gerdjikov-Ivanov equation. J. Nonlinear Math. Phys., 2017, 24(2): 210-223

[22] Wu J P. Integrability aspects and multi-soliton solutions of a new coupled Gerdjikov-Ivanov derivative nonlinear Schrödinger equation. Nonlinear Dyn., 2019, 96(1): 789-800

[23] Ablowitz M J, Clarkson P A. Solitons, Nonlinear Evolution Equations and Inverse Scattering. 1991, Cambridge, Cambridge University Press

[24] Ablowitz M J, Fokas A S. Complex Variables: Introduction and Applications. 2003, Cambridge, Cambridge University Press

[25] Yang J K. Nonlinear Waves in Integrable and Nonintegrable Systems. 2010, Philadelphia, SIAM

[26] Wang D S, Zhang D J, Yang J K. Integrable properties of the general coupled nonlinear Schrödinger equations. J. Math. Phys., 2010, 51023510

[27] Kaup D J, Yang J K. The inverse scattering transform and squared eigenfunctions for a degenerate 3 × 3 operator. Inverse Probl., 2009, 25(10): 105010

[28] Yang J K, Kaup D J. Squared eigenfunctions for the Sasa-Satsuma equation. J. Math. Phys., 2009, 502023504

[29] Ma W X. Application of the Riemann-Hilbert approach to the multicomponent AKNS integrable hierarchies. Nonlinear Anal., 2019, 471-17

[30] Ma W X. Riemann-Hilbert problems and inverse scattering of nonlocal real reverse-spacetime matrix AKNS hierarchies. Phys. D., 2022, 430133078

[31] Ma W X. Riemann-Hilbert problems of a six-component fourth-order AKNS system and its soliton solutions. Comp. Appl. Math., 2018, 376359-6375

[32] Ma W X. Riemann-Hilbert problems and soliton solutions of a multicomponent mKdV system and its reduction. Math. Method. App., 2019, 42: 1099-1113

[33] Ma W X, Huang Y H, Wang F D. Inverse scattering transforms and soliton solutions of nonlocal reverse-space nonlinear Schrödinger hierarchies. Stud. Appl. Math., 2020, 145: 563-585

[34] Ma W X. Riemann-Hilbert problems and soliton solutions of nonlocal reverse-time NLS hierarchies. Acta. Math. Sci., 2022, 42: 127-140

[35] Ma W X. Riemann-Hilbert problems and soliton solutions for a coupled mKdV system. J. Geom. Phys., 2018, 132: 45-54

[36] Ma W X. Riemann-Hilbert problems of a six-component mKdV system and its soliton solutions. Acta. Math. Sci., 2019, 39: 509-523

[37] Zhang Y S, Tao X X, Xu S W. The bound-state soliton solutions of the complex modified KdV equation. Inverse Prob., 2020, 36(6): 065003

[38] Geng X G, Wu J P. Riemann-Hilbert approach and N-soliton solutions for a generalized Sasa-Satsuma equation. Wave Motion, 2016, 6062-72

[39] Guo B L, Ling L M. Riemann-Hilbert approach and N-soliton formula for coupled derivative Schrödinger equation. J. Math. Phys., 2012, 53(7): 073506

[40] Fokas A S. A Unified Approach to Boundary Value Problems. 2008, Philadelphia, SIAM

[41] Wu J P, Geng X G. Inverse scattering transform and soliton classification of the coupled modified Korteweg-de Vries equation. Commun. Nonlinear. Sci. Num. Simul., 2017, 5383-93

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