@article{TMF_2022_213_3_a4,
author = {Jian Li and Tiecheng Xia and Handong Guo},
title = {Long-time asymptotics for the~nonlocal {Kundu{\textendash}nonlinear-Schr\"odinger} equation by the~nonlinear steepest descent method},
journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
pages = {459--481},
year = {2022},
volume = {213},
number = {3},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/TMF_2022_213_3_a4/}
}
TY - JOUR AU - Jian Li AU - Tiecheng Xia AU - Handong Guo TI - Long-time asymptotics for the nonlocal Kundu–nonlinear-Schrödinger equation by the nonlinear steepest descent method JO - Teoretičeskaâ i matematičeskaâ fizika PY - 2022 SP - 459 EP - 481 VL - 213 IS - 3 UR - http://geodesic.mathdoc.fr/item/TMF_2022_213_3_a4/ LA - ru ID - TMF_2022_213_3_a4 ER -
%0 Journal Article %A Jian Li %A Tiecheng Xia %A Handong Guo %T Long-time asymptotics for the nonlocal Kundu–nonlinear-Schrödinger equation by the nonlinear steepest descent method %J Teoretičeskaâ i matematičeskaâ fizika %D 2022 %P 459-481 %V 213 %N 3 %U http://geodesic.mathdoc.fr/item/TMF_2022_213_3_a4/ %G ru %F TMF_2022_213_3_a4
Jian Li; Tiecheng Xia; Handong Guo. Long-time asymptotics for the nonlocal Kundu–nonlinear-Schrödinger equation by the nonlinear steepest descent method. Teoretičeskaâ i matematičeskaâ fizika, Tome 213 (2022) no. 3, pp. 459-481. http://geodesic.mathdoc.fr/item/TMF_2022_213_3_a4/
[1] C. M. Bender, S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having $\mathscr{P\!T}$ symmetry”, Phys. Rev. Lett., 80:24 (1998), 5243–5246, arXiv: physics/9712001 | DOI | MR
[2] R. El-Ganainy, K. G. Makris, D. N. Christodoulides, Z. H. Musslimani, “Theory of coupled optical $PT$-symmetric structures”, Opt. Lett., 32:17 (2007), 2632–2634 | DOI
[3] K. G. Makris, R. El-Ganainy, D. N. Christodoulides, Z. H. Musslimani, “Beam dynamics in $\mathscr{P\!T}$ symmetric optical lattices”, Phys. Rev. Lett., 100:10 (2008), 103904, 4 pp. | DOI
[4] A. Guo, G. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. Siviloglou, D. N. Christodoulides, “Observation of $\mathscr{P\!T}$-symmetry breaking in complex optical potentials”, Phys. Rev. Lett., 103:9 (2009), 093902, 4 pp. | DOI
[5] H. Cartarius, G. Wunner, “Model of a $\mathscr{P\!T}$-symmetric Bose–Einstein condensate in a $\delta$-function double-well potential”, Phys. Rev. A, 86:1 (2012), 013612, 5 pp., arXiv: 1203.1885 | DOI | MR
[6] J. Schindler, A. Li, M. C. Zheng, F. M. Ellis, T. Kottos, “Experimental study of active $LRC$ circuits with $\mathscr{P\!T}$ symmetries”, Phys. Rev. A, 84:4 (2011), 040101, 5 pp. | DOI
[7] T. A. Gadzhimuradov, A. M. Agalarov, “Towards a gauge-equivalent magnetic structure of the nonlocal nonlinear Schrödinger equation”, Phys. Rev. A, 93:6 (2011), 062124, 6 pp. | DOI
[8] D. R. Nelson, N. M. Shnerb, “Non-Hermitian localization and population biology”, Phys. Rev. E., 58:2 (1998), 1383–1403, arXiv: cond-mat/9708071 | DOI | MR
[9] M. J. Ablowitz, Z. H. Musslimani, “Integrable nonlocal nonlinear Schrödinger equation”, Phys. Rev. Lett., 110:6 (2013), 064105, 5 pp. | DOI
[10] J.-L. Ji, Z.-N. Zhu, “On a nonlocal modified Korteweg–de Vries equation: Integrability, Darboux transformation and soliton solutions”, Commun. Nonlinear Sci. Numer. Simul., 42 (2017), 699–708 | DOI | MR
[11] A. S. Fokas, “Integrable multidimensional versions of the nonlocal nonlinear Schrödinger equation”, Nonlinearity, 29:2 (2016), 319–324 | DOI | MR | Zbl
[12] M. J. Ablowitz, Z. H. Musslimani, “Integrable nonlocal nonlinear equations”, Stud. Appl. Math., 139:1 (2016), 7–59, arXiv: 1610.02594 | DOI | MR
[13] D.-F. Bian, B.-L. Guo, L.-M. Ling, “High-order soliton solution of Landau–Lifshitz equation”, Stud. Appl. Math., 134:2 (2015), 181–214 | DOI | MR
[14] A.-Y. Chen, W.-J. Zhu, Z.-J. Qiao, W.-T. Huang, “Algebraic traveling wave solutions of a non-local hydrodynamic-type model”, Math. Phys. Anal. Geom., 17:3–4 (2014), 465–482 | DOI | MR
[15] X. Shi, J. Li, C. Wu, “Dynamics of soliton solutions of the nonlocal Kundu-nonlinear Schrödinger equation”, Chaos, 29:2 (2019), 023120, 12 pp. | DOI | MR
[16] Ya. Rybalko, D. Shepelsky, “Long-time asymptotics for the integrable nonlocal nonlinear Schrödinger equation with step-like initial data”, J. Differ. Equ., 270:1 (2021), 694–724 | DOI | MR
[17] Ya. Rybalko, D. Shepelsky, “Long-time asymptotics for the integrable nonlocal nonlinear Schrödinger equation”, J. Math. Phys., 60:3 (2019), 031504, 16 pp., arXiv: 1710.07961 | DOI | MR
[18] S. V. Manakov, “Nelineinaya difraktsiya Fraungofera”, ZhETF, 65:10 (1973), 1392–1398
[19] M. J. Ablowitz, A. C. Newell, “The decay of the continuous spectrum for solutions of the Korteweg–de Vries equation”, J. Math. Phys., 14:9 (1973), 1277–1284 | DOI | MR
[20] V. E. Zakharov, S. V. Manakov, “Asimptoticheskoe povedenie nelineinykh, volnovykh sistem, integriruemykh metodom obratnoi zadachi”, ZhETF, 71:1 (1976), 203–215 | MR
[21] A. R. Its, “Asimptotika reshenii nelineinogo uravneniya Shredingera i izomonodromnye deformatsii sistem lineinykh differentsialnykh uravnenii”, Dokl. AN SSSR, 261:1 (1981), 14–18 | MR | Zbl
[22] R. Beals, R. R. Coifman, “Scattering and inverse scattering for first order systems”, Commun. Pure Appl. Math., 37:1 (1981), 39–90 | DOI | MR
[23] R. Buckingham, S. Venakides, “Long-time asymptotics of the nonlinear Schrödinger equation shock problem”, Comm. Pure Appl. Math., 60:9 (2007), 1349–1414 | DOI | MR
[24] A. Boutet de Monvel, A. Its, V. Kotlyarov, “Long-time asymptotics for the focusing NLS equation with time-periodic boundary condition on the half-line”, Commun. Math. Phys., 290:2 (2009), 479–522 | DOI | MR
[25] P. Deift, X. Zhou, “A steepest descent method for oscillatory Riemann–Hilbert problems”, Ann. Math., 137:2 (1993), 295–368 | DOI | MR
[26] P. Deift, S. Venakides, X. Zhou, “New results in small dispersion KdV by an extension of the steepest descent method for Riemann–Hilbert problems”, Int. Math. Res. Notices, 1997:6 (1997), 285–299 | DOI | MR
[27] P. Deift, J. Park, “Long-time asymptotics for solutions of the NLS equation with a delta potential and even initial data”, Int. Math. Res. Notices, 2011:24 (2011), 5505–5624 | DOI | MR
[28] A. H. Vartanian, “Long-time asymptotics of solutions to the Cauchy problem for the defocusing nonlinear Schrödinger equation with finite-density initial data. II. Dark solitons on continua”, Math. Phys. Anal. Geom., 5:4 (2002), 319–413 | DOI | MR
[29] A. Boutet de Monvel, A. Kostenko, D. Shepelsky, G. Teschl, “Long-time asymptotics for the Camassa–Holm equation”, SIAM J. Math. Anal., 41:4 (2009), 1559–1588 | DOI | MR
[30] D.-S. Wang, X. Wang, “Long-time asymptotics and the bright $N$-soliton solutions of the Kundu–Eckhaus equation via the Riemann–Hilbert approach”, Nonlinear Anal. Real World Appl., 41 (2018), 334–361 | DOI | MR
[31] W.-X. Ma, “Long-time asymptotics of a three-component coupled nonlinear Schrödinger system”, J. Geom. Phys., 153 (2020), 103669, 28 pp. | DOI | MR
[32] J. Xu, E. Fan, “Long-time asymptotics for the Fokas–Lenells equation with decaying initial value problem: without solitons”, J. Differ. Equ., 259:3 (2015), 1098–1148 | DOI | MR
[33] J. Xu, E. G. Fan, “A Riemann–Hilbert approach to the initial-boundary problem for derivative nonlinear Schrödinger equation”, Acta Math. Sci., 34:4 (2014), 973–994 | DOI | MR
[34] J. Lenells, “The nonlinear steepest descent method for Riemann–Hilbert problems of low regularity”, Indiana Univ. Math. J., 66:4 (2017), 1287–1332 | DOI | MR
[35] J. Lenells, “Nonlinear Fourier transforms and the mKdV equation in the quarter plane”, Stud. Appl. Math., 136:1 (2016), 3–63 | DOI | MR
[36] X.-G. Geng, M.-M. Chen, K.-D. Wang, “Long-time asymptotics of the coupled modified Korteweg–de Vries equation”, J. Geom. Phys., 142 (2019), 151–167 | DOI | MR
[37] M. J. Ablowitz, Z. H. Musslimani, “Inverse scattering transform for the integrable nonlocal nonlinear Schrödinger equation”, Nonlinearity, 29:3 (2016), 915–946 | DOI | MR
[38] X.-G. Geng, K.-D. Wang, M.-M. Chen, “Long-time asymptotics for the spin-1 Gross–Pitaevskii equation”, Commun. Math. Phys., 382:1 (2021), 585–611 | DOI | MR