• [1] M. J. Ablowitz and Z. H. Musslimani, Integrable nonlocal nonlinear Schrödinger equation, Phys. Rev. Lett. 110 (2013), 064105, DOI 10.1103/PhysRevLett.110.064105.
• [2] Mark J. Ablowitz and Ziad H. Musslimani, Integrable nonlocal nonlinear equations, Stud. Appl. Math. 139 (2017), no. 1, 7–59. MR 3672137, https://doi.org/10.1111/sapm.12153
• [3] 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, https://doi.org/10.1088/0951-7715/29/3/915
• [4] 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, https://doi.org/10.1063/1.5018294
• [5] Wen-Xiu Ma, Inverse scattering for nonlocal reverse-time nonlinear Schrödinger equations, Appl. Math. Lett. 102 (2020), 106161, 7. MR 4040185, https://doi.org/10.1016/j.aml.2019.106161
• [6] 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, https://doi.org/10.1016/j.cnsns.2016.06.015
• [7] 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, https://doi.org/10.1063/1.5005611
• [8] 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.
• [9] Metin Gürses and Aslı Pekcan, Nonlocal nonlinear Schrödinger equations and their soliton solutions, J. Math. Phys. 59 (2018), no. 5, 051501, 17. MR 3797911, https://doi.org/10.1063/1.4997835
• [10] A. S. Fokas, Integrable multidimensional versions of the nonlocal nonlinear Schrödinger equation, Nonlinearity 29 (2016), no. 2, 319–324. MR 3461601, https://doi.org/10.1088/0951-7715/29/2/319
• [11] 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, https://doi.org/10.1016/j.cnsns.2016.09.013
• [12] 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
• [13] 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, https://doi.org/10.1063/1.3290736
• [14] Yu Xiao and Engui Fan, A Riemann-Hilbert approach to the Harry-Dym equation on the line, Chin. Ann. Math. Ser. B 37 (2016), no. 3, 373–384. MR 3490570, https://doi.org/10.1007/s11401-016-0966-4
• [15] 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, https://doi.org/10.1016/j.wavemoti.2015.09.003
• [16] Wen-Xiu Ma, Riemann-Hilbert problems and 𝑁-soliton solutions for a coupled mKdV system, J. Geom. Phys. 132 (2018), 45–54. MR 3836765, https://doi.org/10.1016/j.geomphys.2018.05.024
• [17] 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, https://doi.org/10.1016/j.physleta.2018.10.051
• [18] 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, https://doi.org/10.1142/S0252959902000341
• [19] 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, https://doi.org/10.1137/1015113
• [20] 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.
• [21] Evgeny V. Doktorov and Sergey B. Leble, A dressing method in mathematical physics, Mathematical Physics Studies, vol. 28, Springer, Dordrecht, 2007. MR 2345237
• [22] Wen-Xiu Ma, Xuelin Yong, Zhenyun Qin, Xiang Gu, and Yuan Zhou, A generalized Liouville's formula, preprint (2017).
• [23] 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
• [24] F. D. Gakhov, Boundary value problems, Translation edited by I. N. Sneddon, Pergamon Press, Oxford-New York-Paris; Addison-Wesley Publishing Co., Inc., Reading, Mass.-London, 1966. MR 0198152
• [25] 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
• [26] 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
• [27] 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, https://doi.org/10.1016/0375-9601(83)90764-8
• [28] 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, https://doi.org/10.1090/S0002-9947-04-03726-2
• [29] V. B. Matveev and M. A. Salle, Darboux transformations and solitons, Springer Series in Nonlinear Dynamics, Springer-Verlag, Berlin, 1991. MR 1146435
• [30] 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, https://doi.org/10.1142/S0129055X18500034
• [31] 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, https://doi.org/10.1016/j.jde.2017.10.033
• [32] 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, https://doi.org/10.1142/S0217984919504578
• [33] 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.
• [34] 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.
• [35] Wen-Xiu Ma, Long-time asymptotics of a three-component coupled nonlinear Schrödinger system, J. Geom. Phys. 153 (2020), 103669, 28. MR 4085269, https://doi.org/10.1016/j.geomphys.2020.103669
• [36] Wen-Xiu Ma, Long-time asymptotics of a three-component coupled nonlinear Schrödinger system, J. Geom. Phys. 153 (2020), 103669, 28. MR 4085269, https://doi.org/10.1016/j.geomphys.2020.103669
• [37] 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
• [38] Wen-Xiu Ma, Trigonal curves and algebro-geometric solutions to soliton hierarchies I, Proc. A. 473 (2017), no. 2203, 20170232, 20. MR 3685475, https://doi.org/10.1098/rspa.2017.0232
  Wen-Xiu Ma, Trigonal curves and algebro-geometric solutions to soliton hierarchies II, Proc. A. 473 (2017), no. 2203, 20170233, 20. MR 3685476, https://doi.org/10.1098/rspa.2017.0233

References

[1]

M. J. Ablowitz and Z. H. Musslimani, Integrable nonlocal nonlinear Schrödinger equation, Phys. Rev. Lett. 110 (2013), 064105, DOI 10.1103/PhysRevLett.110.064105.

[2]

Mark J. Ablowitz and Ziad H. Musslimani, Integrable nonlocal nonlinear equations, Stud. Appl. Math. 139 (2017), no. 1, 7-59. MR 3672137, https://doi.org/10.1111/sapm.12153

[3]

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, https://doi.org/10.1088/0951-7715/29/3/915

[4]

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, https://doi.org/10.1063/1.5018294

[5]

Wen-Xiu Ma, Inverse scattering for nonlocal reverse-time nonlinear Schrödinger equations, Appl. Math. Lett. 102 (2020), 106161, 7. MR 4040185, https://doi.org/10.1016/j.aml.2019.106161

[6]

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, https://doi.org/10.1016/j.cnsns.2016.06.015

[7]

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, https://doi.org/10.1063/1.5005611

[8]

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.

[9]

Metin Gürses and Aslı Pekcan, Nonlocal nonlinear Schrödinger equations and their soliton solutions, J. Math. Phys. 59 (2018), no. 5, 051501, 17. MR 3797911, https://doi.org/10.1063/1.4997835

[10]

A. S. Fokas, Integrable multidimensional versions of the nonlocal nonlinear Schrödinger equation, Nonlinearity 29 (2016), no. 2, 319-324. MR 3461601, https://doi.org/10.1088/0951-7715/29/2/319

[11]

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, https://doi.org/10.1016/j.cnsns.2016.09.013

[12]

S. Novikov, S. V. Manakov, L. P. Pitaevskiĭ, and V. E. Zakharov, Theory of solitons: The inverse scattering method; Translated from the Russian, Contemporary Soviet Mathematics, Consultants Bureau [Plenum], New York, 1984. MR 779467

[13]

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, https://doi.org/10.1063/1.3290736

[14]

Yu Xiao and Engui Fan, A Riemann-Hilbert approach to the Harry-Dym equation on the line, Chin. Ann. Math. Ser. B 37 (2016), no. 3, 373-384. MR 3490570, https://doi.org/10.1007/s11401-016-0966-4

[15]

Xianguo Geng and Jianping Wu, Riemann-Hilbert approach and -soliton solutions for a generalized Sasa-Satsuma equation, Wave Motion 60 (2016), 62-72. MR 3427949, https://doi.org/10.1016/j.wavemoti.2015.09.003

[16]

Wen-Xiu Ma, Riemann-Hilbert problems and -soliton solutions for a coupled mKdV system, J. Geom. Phys. 132 (2018), 45-54. MR 3836765, https://doi.org/10.1016/j.geomphys.2018.05.024

[17]

Jianke Yang, General -solitons and their dynamics in several nonlocal nonlinear Schrödinger equations, Phys. Lett. A 383 (2019), no. 4, 328-337. MR 3907092, https://doi.org/10.1016/j.physleta.2018.10.051

[18]

Wen-Xiu Ma and Ruguang Zhou, Adjoint symmetry constraints of multicomponent AKNS equations, Chinese Ann. Math. Ser. B 23 (2002), no. 3, 373-384. MR 1930190, https://doi.org/10.1142/S0252959902000341

[19]

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, https://doi.org/10.1137/1015113

[20]

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.

[21]

Evgeny V. Doktorov and Sergey B. Leble, A dressing method in mathematical physics, Mathematical Physics Studies, vol. 28, Springer, Dordrecht, 2007. MR 2345237

[22]

Wen-Xiu Ma, Xuelin Yong, Zhenyun Qin, Xiang Gu, and Yuan Zhou, A generalized Liouville's formula, preprint (2017).

[23]

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

[24]

F. D. Gakhov, Boundary value problems, Elsevier Science, London, 2014. MR 0198152

[25]

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

[26]

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

[27]

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, https://doi.org/10.1016/0375-9601(83)90764-8

[28]

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, https://doi.org/10.1090/S0002-9947-04-03726-2

[29]

V. B. Matveev and M. A. Salle, Darboux transformations and solitons, Springer Series in Nonlinear Dynamics, Springer-Verlag, Berlin, 1991. MR 1146435

[30]

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, https://doi.org/10.1142/S0129055X18500034

[31]

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, https://doi.org/10.1016/j.jde.2017.10.033

[32]

Wen-Xiu Ma, Lump and interaction solutions to linear PDEs in dimensions via symbolic computation, Modern Phys. Lett. B 33 (2019), no. 36, 1950457, 10. MR 4044059, https://doi.org/10.1142/S0217984919504578

[33]

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.

[34]

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.

[35]

Wen-Xiu Ma, Long-time asymptotics of a three-component coupled mKdV system, Mathematics 7 (2019), 573, DOI 10.3390/math7070573. MR 4085269

[36]

Wen-Xiu Ma, Long-time asymptotics of a three-component coupled nonlinear Schródinger system, J. Geom. Phys. 153 (2020), 103669. MR 4085269, https://doi.org/10.1016/j.geomphys.2020.103669

[37]

Fritz Gesztesy and Helge Holden, Soliton equations and their algebro-geometric solutions. Vol. I: -dimensional continuous models, Cambridge Studies in Advanced Mathematics, vol. 79, Cambridge University Press, Cambridge, 2003. MR 1992536

[38]

Wen-Xiu Ma, Trigonal curves and algebro-geometric solutions to soliton hierarchies I, II, Proc. A. 473 (2017), no. 2203, 20170232, 20170233. MR 3685475, MR 3685476, https://doi.org/10.1098/rspa.2017.0232, DOI 10.1098/rspa.2017.0233