Abstract
In this article, we focus on the inverse scattering transform for the Gerdjikov–Ivanov equation with nonzero boundary at infinity. An appropriate two-sheeted Riemann surface is introduced to map the original spectral parameter k into a single-valued parameter z. Based on the Lax pair of the Gerdjikov–Ivanov equation, we derive its Jost solutions with nonzero boundary. Further asymptotic behaviors, analyticity and the symmetries of the Jost solutions and the spectral matrix are in detail derived. The formula of N-soliton solutions is obtained via transforming the problem of nonzero boundary into the corresponding matrix Riemann–Hilbert problem. As examples of N-soliton formula, for \(N=1\) and \(N=2\), respectively, different kinds of soliton solutions and breather solutions are explicitly presented according to different distributions of the spectrum. The dynamical features of those solutions are characterized in the particular case with a quartet of discrete eigenvalues. It is shown that distribution of the spectrum and non-vanishing boundary also affect feature of soliton solutions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Zakharov, V.E., Shabat, A.B.: Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media. Sov. Phys. JETP 34, 62–69 (1972)
Demontis, F., Prinari, B., van der Mee, C., Vitale, F.: The inverse scattering transform for the defocusing nonlinear schrödinger equations with nonzero boundary conditions. Stud. Appl. Math. 131, 1–40 (2013)
Kakei, S., Sasa, N., Satsuma, J.: Bilinearization of a generialized derivative nonlinear Schrödinger equation. J. Phys. Soc. Jpn. 64, 1519–1523 (1995)
Kundu, A.: Exact solutions to higher-order nonlinear equations through gauge transformation. Physica D 25, 399–406 (1987)
Fan, E.G.: A family of completely integrable multi-Hamiltonian systems explicitly related to some celebrated equations. J. Math. Phys. 42, 4327–4344 (2001)
Kaup, D.J., Newell, A.C.: An exact solution for a derivative nonlinear Schrödinger equation. J. Math. Phys. 19, 798–801 (1978)
Chen, H.H., Lee, Y.C., Liu, C.S.: Integrability of nonlinear Hamiltonian systems by inverse scattering method. Phys. Scr. 20, 490–492 (1979)
Gerdjikov, V.S., Ivanov, I.: A quadratic pencil of general type and nonlinear evolution equations. II. Hierarchies of Hamiltonian structures. Bulg. J. Phys. 10, 130–143 (1983)
Tzoar, N., Jain, M.: Self-phase modulation in long-geometry optical waveguides. Phys. Rev. A 23, 1266–1270 (1981)
Kodama, Y.: Optical solitons in a monomode fiber. J. Stat. Phys. 39, 597–614 (1985)
Rogister, A.: Parallel propagation of nonlinear low-frequency waves in high-\(\beta \) plasma. Phys. Fluids 14, 2733–2739 (1971)
Nakatsuka, H., Grischkowsky, D., Balant, A.C.: Nonlinear picosecond-pulse propagation through optical fibers with positive group velocity dispersion. Phys. Rev. Lett. 47, 910–913 (1981)
Einar, M.: Nonlinear alfv\(\acute{e}\)n waves and the dnls equation: oblique aspects. Phys. Scr. 40, 227–237 (1989)
Agrawal, G.P.: Nonlinear Fiber Optics. Academic Press, Boston (2007)
Fan, E.G.: Darboux transformation and solion-like solutions for the Gerdjikov–Ivanov equation. J. Phys. A 33, 6925–6933 (2000)
Dai, H.H., Fan, E.G.: Variable separation and algebro-geometric solutions of the Gerdjikov–Ivanov equation. Chaos Solitons Fractals 22, 93–101 (2004)
Hou, Y., Fan, E.G., Zhao, P.: Algebro-geometric solutions for the Gerdjikov–Ivanov hierarchy. J. Math. Phys. 54, 1–30 (2013)
He, B., Meng, Q.: Bifurcations and new exact travelling wave solutions for the Gerdjikov–Ivanov equation. Commun. Nonlinear Sci. Numer. Simul. 15, 1783–1790 (2010)
Kakei, S., Kikuchi, T.: Solutions of a derivative nonlinear Schrödinger hierarchy and its similarity reduction. Glasgow Math. J. 47, 99–107 (2005)
Nie, H., Zhu, J.Y., Geng, X.G.: Trace formula and new form of N-soliton to the Gerdjikov–Ivanov equation. Anal. Math. Phys. 8, 415–426 (2018)
Gardner, C.S., Greene, J.M., Kruskal, M.D., Miura, R.M.: Method for solving the Korteweg–deVries equation. Phys. Rev. Lett. 19, 1095–1097 (1967)
Ablowitz, M.J., Clarkson, P.A.: Soliton, Nonlinear Evolution Equations and Inverse Scattering. Cambridge University Press, Cambridge (1991)
Yang, J.K.: Nonlinear Waves in Integrable and Nonintegrable Systems. Soc Indus Appl Math, Philadelphia (2010)
Deift, P., Zhou, X.: A steepest descent method for oscillatory Riemann–Hilbert problems. Ann. Math. 137, 295–368 (1993)
Biondini, G., Kova\(\breve{\rm {c}}\)i\(\breve{\rm {c}}\), G.: Inverse scattering transform for the focusing nonlinear Schrödinger equation with nonzero boundary conditions. J. Math. Phys. 55, 1–22 (2014)
Pichler, M., Biondini, G.: On the focusing non-linear Schrödinger equation with non-zero boundary conditions and double poles. IMA J. Appl. Math. 82, 131–151 (2017)
Biondini, G., Kraus, D.: Inverse scattering transform for the defocusing Manakov system with nonzero boundary conditions. SIAM J. Math. Anal. 47, 706–757 (2015)
Prinari, B., Ablowitz, M.J., Biondini, G.: Inverse scattering transform for the vector nonlinear Schrödinger equation with nonvanishing boundary conditions. J. Math. Phys. 47, 1–32 (2006)
Biondini, G., Kraus, D.K., Prinari, B.: The three-component defocusing nonlinear Schrödinger equation with nonzero boundary conditions. Commun. Math. Phys. 348, 475–533 (2016)
Acknowledgements
This work is supported by the National Science Foundation of China (Grant Nos. 11671095, 51879045).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Zhang, Z., Fan, E. Inverse scattering transform for the Gerdjikov–Ivanov equation with nonzero boundary conditions. Z. Angew. Math. Phys. 71, 149 (2020). https://doi.org/10.1007/s00033-020-01371-z
Received:
Revised:
Published:
DOI: https://doi.org/10.1007/s00033-020-01371-z