## Inverse scattering and soliton solutions of nonlocal reverse-spacetime nonlinear Schrödinger equations

HTML articles powered by AMS MathViewer

- by Wen-Xiu Ma PDF
- Proc. Amer. Math. Soc.
**149**(2021), 251-263 Request permission

## Abstract:

The paper presents nonlocal reverse-spacetime PT-symmetric multicomponent nonlinear Schrödinger (NLS) equations under a specific nonlocal group reduction, and generates their inverse scattering transforms and soliton solutions by the Riemann-Hilbert technique. The Sokhotski-Plemelj formula is used to determine solutions to a class of associated Riemann-Hilbert problems and transform the systems that generalized Jost solutions need to satisfy. A formulation of solutions is developed for the Riemann-Hilbert problems associated with the reflectionless transforms, and the corresponding soliton solutions are constructed for the presented nonlocal reverse-spacetime PT-symmetric NLS equations.## References

- M. J. Ablowitz and Z. H. Musslimani,
*Integrable nonlocal nonlinear Schrödinger equation*, Phys. Rev. Lett.**110**(2013), 064105, DOI 10.1103/PhysRevLett.110.064105. - Mark J. Ablowitz and Ziad H. Musslimani,
*Integrable nonlocal nonlinear equations*, Stud. Appl. Math.**139**(2017), no. 1, 7–59. MR**3672137**, DOI 10.1111/sapm.12153 - Mark J. Ablowitz and Ziad H. Musslimani,
*Inverse scattering transform for the integrable nonlocal nonlinear Schrödinger equation*, Nonlinearity**29**(2016), no. 3, 915–946. MR**3465988**, DOI 10.1088/0951-7715/29/3/915 - Mark J. Ablowitz, Xu-Dan Luo, and Ziad H. Musslimani,
*Inverse scattering transform for the nonlocal nonlinear Schrödinger equation with nonzero boundary conditions*, J. Math. Phys.**59**(2018), no. 1, 011501, 42. MR**3743606**, DOI 10.1063/1.5018294 - Wen-Xiu Ma,
*Inverse scattering for nonlocal reverse-time nonlinear Schrödinger equations*, Appl. Math. Lett.**102**(2020), 106161, 7. MR**4040185**, DOI 10.1016/j.aml.2019.106161 - Jia-Liang Ji and Zuo-Nong Zhu,
*On a nonlocal modified Korteweg–de Vries equation: integrability, Darboux transformation and soliton solutions*, Commun. Nonlinear Sci. Numer. Simul.**42**(2017), 699–708. MR**3534967**, DOI 10.1016/j.cnsns.2016.06.015 - Li-Yuan Ma, Shou-Feng Shen, and Zuo-Nong Zhu,
*Soliton solution and gauge equivalence for an integrable nonlocal complex modified Korteweg–de Vries equation*, J. Math. Phys.**58**(2017), no. 10, 103501, 12. MR**3708198**, DOI 10.1063/1.5005611 - X. Huang and L. M. Ling,
*Soliton solutions for the nonlocal nonlinear Schrödinger equation*, Eur. Phys. J. Plus**131**(2016), 148, DOI 10.1140/epjp/i2016-16148-9. - Metin Gürses and AslıPekcan,
*Nonlocal nonlinear Schrödinger equations and their soliton solutions*, J. Math. Phys.**59**(2018), no. 5, 051501, 17. MR**3797911**, DOI 10.1063/1.4997835 - A. S. Fokas,
*Integrable multidimensional versions of the nonlocal nonlinear Schrödinger equation*, Nonlinearity**29**(2016), no. 2, 319–324. MR**3461601**, DOI 10.1088/0951-7715/29/2/319 - Cai-Qin Song, Dong-Mei Xiao, and Zuo-Nong Zhu,
*Solitons and dynamics for a general integrable nonlocal coupled nonlinear Schrödinger equation*, Commun. Nonlinear Sci. Numer. Simul.**45**(2017), 13–28. MR**3571096**, DOI 10.1016/j.cnsns.2016.09.013 - S. Novikov, S. V. Manakov, L. P. Pitaevskiĭ, and V. E. Zakharov,
*Theory of solitons*, Contemporary Soviet Mathematics, Consultants Bureau [Plenum], New York, 1984. The inverse scattering method; Translated from the Russian. MR**779467** - Deng-Shan Wang, Da-Jun Zhang, and Jianke Yang,
*Integrable properties of the general coupled nonlinear Schrödinger equations*, J. Math. Phys.**51**(2010), no. 2, 023510, 17. MR**2605060**, DOI 10.1063/1.3290736 - Yu Xiao and Engui Fan,
*A Riemann-Hilbert approach to the Harry-Dym equation on the line*, Chinese Ann. Math. Ser. B**37**(2016), no. 3, 373–384. MR**3490570**, DOI 10.1007/s11401-016-0966-4 - Xianguo Geng and Jianping Wu,
*Riemann-Hilbert approach and $N$-soliton solutions for a generalized Sasa-Satsuma equation*, Wave Motion**60**(2016), 62–72. MR**3427949**, DOI 10.1016/j.wavemoti.2015.09.003 - Wen-Xiu Ma,
*Riemann-Hilbert problems and $N$-soliton solutions for a coupled mKdV system*, J. Geom. Phys.**132**(2018), 45–54. MR**3836765**, DOI 10.1016/j.geomphys.2018.05.024 - Jianke Yang,
*General $N$-solitons and their dynamics in several nonlocal nonlinear Schrödinger equations*, Phys. Lett. A**383**(2019), no. 4, 328–337. MR**3907092**, DOI 10.1016/j.physleta.2018.10.051 - Wenxiu Ma and Ruguang Zhou,
*Adjoint symmetry constraints of multicomponent AKNS equations*, Chinese Ann. Math. Ser. B**23**(2002), no. 3, 373–384. MR**1930190**, DOI 10.1142/S0252959902000341 - Mark J. Ablowitz, David J. Kaup, Alan C. Newell, and Harvey Segur,
*The inverse scattering transform-Fourier analysis for nonlinear problems*, Studies in Appl. Math.**53**(1974), no. 4, 249–315. MR**450815**, DOI 10.1002/sapm1974534249 - V. S. Gerdjikov, Geometry, integrability and quantization, in:
*Proceedings of the 6th International Conference*(Varna, June 3-10, 2004), ed. I. M. Mladenov and A. C. Hirshfeld, 78–125, Softex, Sofia, 2005. - Evgeny V. Doktorov and Sergey B. Leble,
*A dressing method in mathematical physics*, Mathematical Physics Studies, vol. 28, Springer, Dordrecht, 2007. MR**2345237**, DOI 10.1007/1-4020-6140-4 - Wen-Xiu Ma, Xuelin Yong, Zhenyun Qin, Xiang Gu, and Yuan Zhou,
*A generalized Liouville’s formula*, preprint (2017). - Spyridon Kamvissis, Kenneth D. T.-R. McLaughlin, and Peter D. Miller,
*Semiclassical soliton ensembles for the focusing nonlinear Schrödinger equation*, Annals of Mathematics Studies, vol. 154, Princeton University Press, Princeton, NJ, 2003. MR**1999840**, DOI 10.1515/9781400837182 - F. D. Gakhov,
*Boundary value problems*, Pergamon Press, Oxford-New York-Paris; Addison-Wesley Publishing Co., Inc., Reading, Mass.-London, 1966. Translation edited by I. N. Sneddon. MR**0198152** - Tsutomu Kawata,
*Riemann spectral method for the nonlinear evolution equation*, Advances in nonlinear waves, Vol. I, Res. Notes in Math., vol. 95, Pitman, Boston, MA, 1984, pp. 210–225. MR**747833** - Ryogo Hirota,
*The direct method in soliton theory*, Cambridge Tracts in Mathematics, vol. 155, Cambridge University Press, Cambridge, 2004. Translated from the 1992 Japanese original and edited by Atsushi Nagai, Jon Nimmo and Claire Gilson; With a foreword by Jarmo Hietarinta and Nimmo. MR**2085332**, DOI 10.1017/CBO9780511543043 - N. C. Freeman and J. J. C. Nimmo,
*Soliton solutions of the Korteweg-de Vries and Kadomtsev-Petviashvili equations: the Wronskian technique*, Phys. Lett. A**95**(1983), no. 1, 1–3. MR**700477**, DOI 10.1016/0375-9601(83)90764-8 - Wen-Xiu Ma and Yuncheng You,
*Solving the Korteweg-de Vries equation by its bilinear form: Wronskian solutions*, Trans. Amer. Math. Soc.**357**(2005), no. 5, 1753–1778. MR**2115075**, DOI 10.1090/S0002-9947-04-03726-2 - V. B. Matveev and M. A. Salle,
*Darboux transformations and solitons*, Springer Series in Nonlinear Dynamics, Springer-Verlag, Berlin, 1991. MR**1146435**, DOI 10.1007/978-3-662-00922-2 - Wen-Xiu Ma and Yu-Juan Zhang,
*Darboux transformations of integrable couplings and applications*, Rev. Math. Phys.**30**(2018), no. 2, 1850003, 26. MR**3757744**, DOI 10.1142/S0129055X18500034 - Wen-Xiu Ma and Yuan Zhou,
*Lump solutions to nonlinear partial differential equations via Hirota bilinear forms*, J. Differential Equations**264**(2018), no. 4, 2633–2659. MR**3737849**, DOI 10.1016/j.jde.2017.10.033 - Wen-Xiu Ma,
*Lump and interaction solutions to linear PDEs in $2+1$ dimensions via symbolic computation*, Modern Phys. Lett. B**33**(2019), no. 36, 1950457, 10. MR**4044059**, DOI 10.1142/S0217984919504578 - Wen-Xiu Ma and Liqin Zhang,
*Lump solutions with higher-order rational dispersion relations*, Pramana - J. Phys.**94**(2020), 43, DOI 10.1007/s12043-020-1918-9. - Ruigang Zhang and Liangui Yang,
*Nonlinear Rossby waves in zonally varying flow under generalized beta approximation*, Dyn. Atmospheres Oceans**85**(2019), 16–27, DOI 10.1016/j.dynatmoce.2018.11.001. - Wen-Xiu Ma,
*Long-time asymptotics of a three-component coupled nonlinear Schrödinger system*, J. Geom. Phys.**153**(2020), 103669, 28. MR**4085269**, DOI 10.1016/j.geomphys.2020.103669 - Wen-Xiu Ma,
*Long-time asymptotics of a three-component coupled nonlinear Schrödinger system*, J. Geom. Phys.**153**(2020), 103669, 28. MR**4085269**, DOI 10.1016/j.geomphys.2020.103669 - Fritz Gesztesy and Helge Holden,
*Soliton equations and their algebro-geometric solutions. Vol. I*, Cambridge Studies in Advanced Mathematics, vol. 79, Cambridge University Press, Cambridge, 2003. $(1+1)$-dimensional continuous models. MR**1992536**, DOI 10.1017/CBO9780511546723 - Wen-Xiu Ma,
*Trigonal curves and algebro-geometric solutions to soliton hierarchies I*, Proc. A.**473**(2017), no. 2203, 20170232, 20. MR**3685475**, DOI 10.1098/rspa.2017.0232

## Additional Information

**Wen-Xiu Ma**- Affiliation: Department of Mathematics, Zhejiang Normal University, Jinhua 321004, Zhejiang, China; Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia; Department of Mathematics and Statistics, University of South Florida, Tampa, Florida 33620; School of Mathematics, South China University of Technology, Guangzhou 510640, China; and Department of Mathematical Sciences, North-West University, Mafikeng Campus, Mmabatho 2735, South Africa
- MR Author ID: 247034
- ORCID: 0000-0001-5309-1493
- Email: mawx@cas.usf.edu
- Received by editor(s): November 11, 2019
- Received by editor(s) in revised form: April 8, 2020, and May 7, 2020
- Published electronically: October 16, 2020
- Additional Notes: This work was supported in part by NSFC under the grants 11975145 and 11972291, NSF under the grant DMS-1664561, and the Natural Science Foundation for Colleges and Universities in Jiangsu Province (17 KJB 110020).
- Communicated by: Mourad E. H. Ismail
- © Copyright 2020 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**149**(2021), 251-263 - MSC (2010): Primary 37K15, 35Q55, 37K40
- DOI: https://doi.org/10.1090/proc/15174
- MathSciNet review: 4172602