References
[1] M.J. Ablowitz, B.-F. Feng, X.-D. Luo, and Z.H. Musslimani, General soliton solution to a nonlocal nonlinear Schrödinger equation with zero and nonzero boundaryconditions, Nonlinearity 31 5385 (2018). CrossRef

[2] M.J. Ablowitz, D.J. Kaup, A.C. Newell, and H. Segur, The Inverse ScatteringTransform-Fourier Analysis for Nonlinear Problems, Stud. Appl. Math. 53 (1974),249–315. CrossRef

[3] M.J. Ablowitz, X.-D. Luo, and J. Cole, Solitons, the Korteweg–de Vries equationwith step boundary values, and pseudo-embedded eigenvalues, J. Math. Phys. 59091406 (2018). CrossRef

[4] M.J. Ablowitz and Z.H. Musslimani, Integrable nonlocal nonlinear Schrödingerequation, Phys. Rev. Lett. 110 064105 (2013). CrossRef

[5] M.J. Ablowitz and Z.H. Musslimani, Inverse scattering transform for the integrablenonlocal nonlinear Schrödinger equation, Nonlinearity 29 (2016), 915–946. CrossRef

[6] K. Andreiev and I. Egorova, On the long-time asymptotics for the Korteweg–deVries equation with steplike initial data associated with rarefaction waves, Zh. Mat.Fiz. Anal. Geom. 13 (2017), 325–343. CrossRef

[7] K. Andreiev, I. Egorova, T.L. Lange, and G. Teschl, Rarefaction waves of theKorteweg–de Vries equation via nonlinear steepest descent, J. Differential Equations, 261 (2016) 5371–5410. CrossRef

[8] C.M. Bender and S. Boettcher, Real spectra in non-Hermitian Hamiltonians havingP-T symmetry, Phys. Rev. Lett. 80 (1998), 5243. CrossRef

[9] G. Biondini, Riemann problems and dispersive shocks in self-focusing media, Phys.Rev. E, 98 (2018), 052220-7. CrossRef

[10] G. Biondini, E. Fagerstrom, and B. Prinari, Inverse scattering transform for thedefocusing nonlinear Schrödinger equation with fully asymmetric non-zero boundaryconditions, Phys. D: Nonlinear Phenomena, 333 (2016), 117–136. CrossRef

[11] G. Biondini and B. Prinari, On the spectrum of the Dirac operator and the existence of discrete eigenvalues for the defocusing nonlinear Schrödinger equation,Stud. Appl. Math. 132 2 (2014), 138–159. CrossRef

[12] Yu. Bludov, V. Konotop, and B. Malomed, Stable dark solitons in PT-symmetricdual-core waveguides, Phys. Rev. A 87 013816 (2013). CrossRef

[13] A. Boutet de Monvel, V.P. Kotlyarov, and D. Shepelsky, Focusing NLS Equation:Long-Time Dynamics of Step-Like Initial Data, Int. Math. Res. Not. 7 (2011), 1613–1653 CrossRef

[14] D.C. Brody, PT-symmetry, indefinite metric, and nonlinear quantum mechanics, J.Phys. A: Math. Theor. 50 485202 (2017). CrossRef

[15] R. Buckingham and S. Venakides, Long-time asymptotics of the nonlinear Schrödinger equation shock problem, Comm. Pure Appl. Math. 60 (2007), 1349–1414. CrossRef

[16] K. Chen and D.J. Zhang, Solutions of the nonlocal nonlinear Schrödinger hierarchyvia reduction, Appl. Math. Lett., 75 (2018), 82–88. CrossRef

[17] P.A. Deift, A.R. Its, and X. Zhou, Long-time asymptotics for integrable nonlinear wave equations. In Important developments in Soliton Theory 1980–1990 (Eds.A.S. Fokas and V.E. Zakharov), Springer, New York, 1993, 181–204. CrossRef

[18] P. Deift, S. Kamvissis, T. Kriecherbauer, and X. Zhou, The Toda rarefaction problem, Comm. Pure Appl. Math., XLIX (1996), 35–83. CrossRef

[19] P.A. Deift, S. Venakides, and X. Zhou, The collisionless shock region for the longtime behavior of solutions of the KdV equation, Comm. Pure Appl. Math. 47 (1994),No. 2, 199–206. CrossRef

[20] P.A. Deift, S. Venakides, and X. Zhou, New results in small dispersion KdV by anextension of the steepest descent method for Riemann–Hilbert problems, Int. Math.Res. Not. 6 (1997), 286–299. CrossRef

[21] P.A. Deift and X. Zhou, A steepest descend method for oscillatory Riemann–Hilbertproblems. Asymptotics for the MKdV equation, Ann. Math. 137 (1993), No. 2, 295–368. CrossRef

[22] I. Egorova, J. Michor and G. Teschl, Long-time asymptotics for the Toda shockproblem: non-overlapping spectra, Zh. Mat. Fiz. Anal. Geom. 14 (2018), 406–451. CrossRef

[23] G.A. El and M.A. Hoefer, Dispersive shock waves and modulation theory, Phys. D:Nonlinear Phenomena 333 (2016), 11–65. CrossRef

[24] L.D. Faddeev and L.A. Takhtajan, Hamiltonian Methods in the Theory of Solitons,Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1987. CrossRef

[25] A.S. Fokas, A.R. Its, A.A. Kapaev, and V. Yu. Novokshenov, Painleve Transcendents. The Riemann–Hilbert Approach, Amer. Math. Soc, Providence, RI, 2006. CrossRef

[26] T. Gadzhimuradov and A. Agalarov, Towards a gauge-equivalent magnetic structureof the nonlocal nonlinear Schrödinger equation, Phys. Rev. A 93 (2016), 062124. CrossRef

[27] V.S. Gerdjikov and A. Saxena, Complete integrability of nonlocal nonlinearSchrödinger equation, J. Math. Phys. 58 (2017), 013502. CrossRef

[28] A.V. Gurevich and L.P. Pitaevskii, Nonstationary structure of a collisionless shockwave, Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki 65 (1973), 590–604.

[29] M. Gürses and A. Pekcan, Nonlocal nonlinear Schrödinger equations and their soliton solutions, J. Math. Phys. 59 (2018), 051501. CrossRef

[30] R. Jenkins, Regularization of a sharp shock by the defocusing nonlinear Schrödingerequation, Nonlinearity 28 (2015), 2131–2180. CrossRef

[31] E.Ya. Hruslov, Asymptotics of the solution of the cauchy problem for the Kortewegde Vries equation with initial data of step type, Math. USSR-Sb. 28 (1976), 229–248. CrossRef

[32] A.R. Its, Asymptotic behavior of the solutions to the nonlinear Schrödinger equation, and isomonodromic deformations of systems of linear differential equations,Doklady Akad. Nauk SSSR 261 (1981), No. 1, 14–18.

[33] V.V. Konotop, J. Yang, and D.A. Zezyulin, Nonlinear waves in PT-symmetric systems, Rev. Mod. Phys. 88 (2016), 035002. CrossRef

[34] V.P. Kotlyarov and E.Ya. Khruslov, Solitons of the nonlinear Schrödinger equation,which are generated by the continuous spectrum, Teoreticheskaya i Matematicheskaya Fizika 68 (1986), No. 2, 172–186. CrossRef

[35] V.P. Kotlyarov and, A.M. Minakov, Riemann–Hilbert problem to the modifiedKorteveg–de Vries equation: Long-time dynamics of the step-like initial data, J.Math. Phys. 51 (2010), 093506. CrossRef

[36] V.P. Kotlyarov and A. Minakov. Dispersive shock wave, generalized Laguerre polynomials, and asymptotic solitons of the focusing nonlinear Schrödinger equation, J.Math. Phys. 60 (2019), 123501. CrossRef

[37] J. Lenells, The nonlinear steepest descent method for Riemann–Hilbert problems oflow regularity, Indiana Univ. Math. 66 (2017), 1287–1332. CrossRef

[38] S.Y. Lou, Alice–Bob systems, P̂ − T̂ − Ĉ symmetry invariant and symmetry breakingsoliton solutions, J. Math. Phys.59 (2018), 083507. CrossRef

[39] S. Lou and F. Huang, Alice-Bob Physics: Coherent Solutions of Nonlocal KdVSystems, Scientific Reports 7 (2017), 869. CrossRef

[40] K.T.-R. McLaughlin and P.D. Miller, The ∂¯ steepest descent method and the asymptotic behavior of polynomials orthogonal on the unit circle with fixed and exponentially varying nonanalytic weights, Int. Math. Res. Pap. Art. 177 (2006), 48673.

[41] J. Michor and A. L. Sakhnovich, GBDT and algebro-geometric approaches to explicit solutions and wave functions for nonlocal NLS, J. Phys. A: Math. Theor. 52(2018), 025201. CrossRef

[42] A. Minakov, Asymptotics of step-like solutions for the Camassa–Holm equation, J.Differential Equations 261, No. 11 (2016). CrossRef

[43] M. Onorato, A.R. Osborne, and M. Serio, Modulational instability in crossing seastates: A possible mechanism for the formation of freak waves, Phys. Rev. Lett. 96(2006), 014503. CrossRef

[44] Ya. Rybalko and D. Shepelsky, Long-time asymptotics for the integrable nonlocalnonlinear Schrödinger equation, J. Math. Phys. 60 (2019), 031504. CrossRef

[45] Ya. Rybalko and, D. Shepelsky, Long-time asymptotics for the integrable nonlocalnonlinear Schrödinger equation with step-like initial data, J. Differential Equations270 (2021), 694–724. CrossRef

[46] Ya. Rybalko and D. Shepelsky, Long-time asymptotics for the integrable nonlocalfocusing nonlinear Schrödinger equation for a family of step-like initial data, Comm.Math. Phys. (accepted), preprint, https://arxiv.org/abs/1908.06415.

[47] Ya. Rybalko, D. Shepelsky, Curved wedges in the long-time asymptotics for theintegrable nonlocal nonlinear Schrödinger equation, preprint, https://arxiv.org/abs/2004.05987.

[48] S. Venakides, P. Deift, and R. Oba, The Toda shock problem, Comm. Pure Appl.Math. 44 (1991), 1171–1242. CrossRef

[49] A. Sarma, M. Miri, Z. Musslimani, and D. Christodoulides, Continuous and discreteSchrödinger systems with parity-time-symmetric nonlinearities, Phys. Rev. E 89(2014). CrossRef

[50] J. Yang, General N-solitons and their dynamics in several nonlocal nonlinearSchrödinger equations, Phys. Lett. A 383 (2019), No. 4, 328–337. CrossRef

[51] B. Yang and J. Yang, General rogue waves in the nonlocal PT-symmetric nonlinearSchrödinger equation, Lett. Math. Phys. 109 (2019), 945–973. CrossRef

[52] V.E. Zakharov and L.A. Ostrovsky, Modulation instability: The beginning, Phys.D 238 (2009), 540–548. CrossRef

[53] Y. Zhang, D. Qiu, Y. Cheng, and J. He, Rational Solution of the Nonlocal NonlinearSchroedinger Equation and Its Application in Optics, Romanian Journal of Physics62 (2017), 108.

[54] M. Znojil and D.I. Borisov, Two patterns of PT-symmetry break- down in a nonnumerical six-state simulation, Ann. Phys., NY 394 (2018), 40–49. CrossRef