Binary Darboux transformation and new soliton solutions of the focusing nonlocal nonlinear Schrödinger equation. (English) Zbl 1504.35502
Summary: For the focusing nonlocal nonlinear Schrödinger (NNLS) equation, we establish the \(N\)-fold binary Darboux transformation (BDT) and give a complete proof on the form invariance of Lax pair. Then, by choosing the tanh-function solution as a seed, we implement the one-fold BDT and obtain two new types of soliton solutions including the exponential and exponential-and-rational types. Both two types of solutions are nonsingular with a wide range of parameter regimes, and they can describe the elastic soliton interactions over the nonzero background with the asymptotic phase difference \(\pi\) as \(x \to \pm \infty\). Also, we derive the expressions of all asymptotic solitons and discuss the parametric conditions for different soliton profiles. It turns out that each asymptotic soliton can display both the dark and antidark soliton profiles or exhibit no spatial localization, which confirms that the focusing NNLS equation admits a rich variety of soliton interactions like the defocusing NNLS case.
MSC:
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 35C08 | Soliton solutions |
| 37K35 | Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems |
| 35B40 | Asymptotic behavior of solutions to PDEs |
Keywords:
soliton solutions; nonlocal nonlinear Schrödinger equation; Darboux transformation; asymptotic analysis; soliton interactionsReferences:
| [1] | Ablowitz, M. J.; Clarkson, P. A., Solitons, Nonlinear Evolution Equations and Inverse Scattering (1992), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0762.35001 |
| [2] | Ablowitz, M. J.; Musslimani, Z. H., Integrable nonlocal nonlinear Schrödinger equation, Phys. Rev. Lett., 110, Article 064105 pp. (2013) |
| [3] | Ablowitz, M. J.; Musslimani, Z. H., Integrable discrete PT symmetric model, Phys. Rev. E, 90, Article 032912 pp. (2014) |
| [4] | Ablowitz, M. J.; Musslimani, Z. H., Inverse scattering transform for the integrable nonlocal nonlinear Schrödinger equation, Nonlinearity, 29, 915-946 (2016) · Zbl 1338.37099 |
| [5] | Ablowitz, M. J.; Musslimani, Z. H., Integrable nonlocal nonlinear equations, Stud. Appl. Math., 139, 7-59 (2016) · Zbl 1373.35281 |
| [6] | Ablowitz, M. J.; Musslimani, Z. H., Integrable nonlocal asymptotic reductions of physically significant nonlinear equations, J. Phys. A, 52, Article 15LT02 pp. (2019) · Zbl 1509.35271 |
| [7] | Ablowitz, M. J.; Musslimani, Z. H., Integrable space-time shifted nonlocal nonlinear equations, Phys. Lett. A, 409, Article 127516 pp. (2021) · Zbl 07411241 |
| [8] | Ablowitz, M. J.; Kaup, D. J.; Newell, A. C.; Segur, H., The inverse scattering transform-Fourier analysis for nonlinear problems, Stud. Appl. Math., 53, 249-315 (1973) · Zbl 0408.35068 |
| [9] | Ablowitz, M. J.; Feng, B. F.; Luo, X. D.; Musslimani, Z. H., Inverse scattering transform for the nonlocal reverse space-time nonlinear Schrödinger equation with nonzero boundary conditions, Theor. Math. Phys., 196, 1241-1267 (2018) · Zbl 1408.35169 |
| [10] | Ablowitz, M. J.; Feng, B. F.; Luo, X. D.; Musslimani, Z. H., Inverse scattering transform for the nonlocal reverse space-time Sine-Gordon, Sinh-Gordon and nonlinear Schrödinger equations with nonzero boundary conditions, Stud. Appl. Math., 141, 267-307 (2018) · Zbl 1420.35304 |
| [11] | Ablowitz, M. J.; Luo, X. D.; Musslimani, Z. H., Inverse scattering transform for the nonlocal nonlinear Schrödinger equation with nonzero boundary conditions, J. Math. Phys., 59, Article 011501 pp. (2018) · Zbl 1383.35204 |
| [12] | Cen, J.; Correa, F.; Fring, A., Integrable nonlocal Hirota equations, J. Math. Phys., 60, Article 081508 pp. (2019) · Zbl 1428.35491 |
| [13] | Chen, J. B.; Pelinovsky, D. E., Rogue periodic waves of the focusing nonlinear Schrödinger equation, Proc. R. Soc. A, 474, Article 20170814 pp. (2018) · Zbl 1402.35256 |
| [14] | Chen, K.; Deng, X.; Lou, S. Y.; Zhang, D. J., Solutions of nonlocal equations reduced from the AKNS hierarchy, Stud. Appl. Math., 141, 113-141 (2018) · Zbl 1398.35194 |
| [15] | Clarkson, P., Special polynomials associated with rational solutions of the defocusing nonlinear Schrödinger equation and the fourth Painlevé equation, Eur. J. Appl. Math., 17, 293-322 (2006) · Zbl 1126.35062 |
| [16] | Feng, B. F.; Luo, X. D.; Ablowitz, M. J.; Musslimani, Z. H., General soliton solution to a nonlocal nonlinear Schrödinger equation with zero and nonzero boundary conditions, Nonlinearity, 31, 5385-5409 (2018) · Zbl 1406.37049 |
| [17] | Fokas, A. S., Integrable multidimensional versions of the nonlocal nonlinear Schrödinger equation, Nonlinearity, 29, 319-324 (2016) · Zbl 1339.37066 |
| [18] | Gadzhimuradov, T. A.; Agalarov, A. M., Towards a gauge-equivalent magnetic structure of the nonlocal nonlinear Schrödinger equation, Phys. Rev. A, 93, Article 062124 pp. (2016) |
| [19] | Gerdjikov, V. S.; Saxena, A., Complete integrability of nonlocal nonlinear Schrödinger equation, J. Math. Phys., 58, Article 013502 pp. (2017) · Zbl 1364.37147 |
| [20] | Gerdjikov, V. S.; Grahovski, G. G.; Ivanov, R. I., The N-wave equations with PT symmetry, Theor. Math. Phys., 188, 1305-1321 (2016) · Zbl 1351.35090 |
| [21] | Gupta, S. K.; Sarma, A. K., Peregrine rogue wave dynamics in the continuous nonlinear Schrödinger system with parity-time symmetric Kerr nonlinearity, Commun. Nonlinear Sci. Numer. Simul., 36, 141-147 (2016) · Zbl 1470.35332 |
| [22] | Gürses, M.; Pekcan, A., Nonlocal nonlinear Schrödinger equations and their soliton solutions, J. Math. Phys., 59, Article 051501 pp. (2018) · Zbl 1392.35285 |
| [23] | Hirota, R., The Direct Method in Soliton Theory (2004), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1099.35111 |
| [24] | Huang, X.; Ling, L. M., Soliton solutions for the nonlocal nonlinear Schrödinger equation, Eur. Phys. J. Plus, 131, 148 (2016) |
| [25] | Inan, B.; Osman, M. S.; Turgut, A. K.; Baleanu, D., Analytical and numerical solutions of mathematical biology models: the Newell-Whitehead-Segel and Allen-Cahn equations, Math. Methods Appl. Sci., 43, 2588-2600 (2020) · Zbl 1454.35397 |
| [26] | Ji, J. L.; Zhu, Z. N., On a nonlocal modified Korteweg-de Vries equation: integrability, Darboux transformation and soliton solutions, Commun. Nonlinear Sci. Numer. Simul., 42, 699-708 (2017) · Zbl 1473.37081 |
| [27] | Khare, A.; Saxena, A., Periodic and hyperbolic soliton solutions of a number of nonlocal nonlinear equations, J. Math. Phys., 56, Article 032104 pp. (2015) · Zbl 1320.35146 |
| [28] | Konotop, V. V.; Yang, J.; Zezyulin, D. A., Nonlinear waves in PT-symmetric systems, Rev. Mod. Phys., 88, Article 035002 pp. (2016) |
| [29] | Li, M.; Xu, T., Dark and antidark soliton interactions in the nonlocal nonlinear Schrödinger equation with the self-induced parity-time-symmetric potential, Phys. Rev. E, 91, Article 033202 pp. (2015) |
| [30] | Li, M.; Xu, T.; Meng, D. X., Rational solitons in the parity-time-symmetric nonlocal nonlinear Schrödinger model, J. Phys. Soc. Jpn., 85, Article 124001 pp. (2016) |
| [31] | Li, M.; Zhang, X. F.; Xu, T.; Li, L. L., Asymptotic analysis and soliton interactions of the multi-pole solutions in the Hirota equation, J. Phys. Soc. Jpn., 89, Article 054004 pp. (2020) |
| [32] | Ling, L. M.; Zhao, L. C.; Guo, B. L., Darboux transformation and multi-dark soliton for N-component nonlinear Schrödinger equations, Nonlinearity, 28, 3243-3261 (2015) · Zbl 1326.37046 |
| [33] | Lou, S. Y., Alice-Bob systems, P-T-C symmetry invariant and symmetry breaking soliton solutions, J. Math. Phys., 59, Article 083507 pp. (2018) · Zbl 1395.35169 |
| [34] | Lou, S. Y., Prohibitions caused by nonlocality for nonlocal Boussinesq-KdV type systems, Stud. Appl. Math., 143, 123-138 (2019) · Zbl 1423.35313 |
| [35] | Mañas, M., Darboux transformations for nonlinear Schrödinger equations, J. Phys. A, 29, 7721-7737 (1996) · Zbl 0906.35097 |
| [36] | Matveev, V. B.; Salle, M. A., Darboux Transformations and Solitons (1991), Springer Press: Springer Press Berlin · Zbl 0744.35045 |
| [37] | Michor, J.; Sakhnovich, A. L., GBDT and algebro-geometric approaches to explicit solutions and wave functions for nonlocal NLS, J. Phys. A, 52, Article 025201 pp. (2019) · Zbl 1422.35151 |
| [38] | Osman, M. S., One-soliton shaping and inelastic collision between double solitons in the fifth-order variable-coefficient Sawada-Kotera equation, Nonlinear Dyn., 96, 1491-1496 (2019) · Zbl 1437.35603 |
| [39] | Osman, M. S.; Wazwaz, A. M., A general bilinear form to generate different wave structures of solitons for a (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation, Math. Methods Appl. Sci., 42, 6277-6283 (2019) · Zbl 1434.35144 |
| [40] | Rao, J. G.; C, Y.; Porsezian, K.; Mihalache, D.; He, J. S., PT-symmetric nonlocal Davey-Stewartson I equation: soliton solutions with nonzero background, Physica D, 401, Article 132180 pp. (2020) · Zbl 1453.37068 |
| [41] | Rybalko, Y.; Shepelsky, D., Long-time asymptotics for the integrable nonlocal nonlinear Schrödinger equation, J. Math. Phys., 60, Article 031504 pp. (2019) · Zbl 1436.35293 |
| [42] | Rybalko, Y.; Shepelsky, D., Long-time asymptotics for the integrable nonlocal focusing nonlinear Schrödinger equation for a family of sep-like initial data, Commun. Math. Phys., 382, 87-121 (2021) · Zbl 1462.35367 |
| [43] | Santini, P. M., The periodic Cauchy problem for PT-symmetric NLS, I: the first appearance of rogue waves, regular behavior or blow up at finite times, J. Phys. A, 51, Article 495207 pp. (2018) · Zbl 1411.35244 |
| [44] | Sarma, A. K.; Miri, M. A.; Musslimani, Z. H.; Christodoulides, D. N., Continuous and discrete Schrödinger systems with parity-time-symmetric nonlinearities, Phys. Rev. E, 89, Article 052918 pp. (2014) |
| [45] | Sinha, D.; Ghosh, P. K., Integrable nonlocal vector nonlinear Schrödinger equation with self-induced parity-time-symmetric potential, Phys. Lett. A, 381, 124-128 (2017) · Zbl 1377.81049 |
| [46] | Tang, X. Y.; Liang, Z. F., A general nonlocal nonlinear Schrödinger equation with shifted parity, charge-conjugate and delayed time reversal, Nonlinear Dyn., 92, 815-825 (2018) |
| [47] | Tian, S. F.; Zhang, T. T., Long-time asymptotic behavior for the Gerdjikov-Ivanov type of derivative nonlinear Schrödinger equation with time-periodic boundary condition, Proc. Am. Math. Soc., 146, 1713-1729 (2018) · Zbl 1427.35259 |
| [48] | Wang, D. S.; Zhang, D. J.; Yang, J. K., Integrable properties of the general coupled nonlinear Schrödinger equations, J. Math. Phys., 51, Article 023510 pp. (2010) · Zbl 1309.35145 |
| [49] | Wen, X. Y.; Yan, Z. Y.; Yang, Y. Q., Dynamics of higher-order rational solitons for the nonlocal nonlinear Schrödinger equation with the self-induced parity-time-symmetric potential, Chaos, 26, Article 063123 pp. (2016) |
| [50] | Xu, T.; Chen, Y.; Li, M.; Meng, D. X., General stationary solutions of the nonlocal nonlinear Schrödinger equation and their relevance to the \(\mathcal{P} \mathcal{T} \)-symmetric system, Chaos, 29, Article 123124 pp. (2019) · Zbl 1429.35182 |
| [51] | Xu, T.; Lan, S.; Li, M.; Li, L. L.; Zhang, G. W., Mixed soliton solutions of the defocusing nonlocal nonlinear Schrödinger equation, Physica D, 390, 47-61 (2019) · Zbl 1448.37089 |
| [52] | Xu, T.; Li, L. L.; Li, M.; Li, C. X.; Zhang, X. F., Rational solutions of the defocusing nonlocal nonlinear Schrödinger equation: asymptotic analysis and soliton interactions, Proc. R. Soc. A, 477, Article 20210512 pp. (2021) |
| [53] | Yan, Z. Y., Integrable PT-symmetric local and nonlocal vector nonlinear Schrödinger equations: a unified two-parameter model, Appl. Math. Lett., 47, 61-68 (2015) · Zbl 1322.35132 |
| [54] | Yang, B.; Chen, Y., Dynamics of high-order solitons in the nonlocal nonlinear Schrödinger equations, Nonlinear Dyn., 94, 489-502 (2018) · Zbl 1412.35315 |
| [55] | Yang, B.; Yang, J. K., Transformations between nonlocal and local integrable equations, Stud. Appl. Math., 140, 178-201 (2018) · Zbl 1392.35297 |
| [56] | Yang, B.; Yang, J. K., Rogue waves in the nonlocal PT-symmetric nonlinear Schrödinger equation, Lett. Math. Phys., 109, 945-973 (2019) · Zbl 1412.35316 |
| [57] | Yang, J. K., Physically significant nonlocal nonlinear Schrödinger equation and its soliton solutions, Phys. Rev. E, 98, Article 042202 pp. (2018) |
| [58] | Yang, J. K., General N-solitons and their dynamics in several nonlocal nonlinear Schrödinger equations, Phys. Lett. A, 383, 328-337 (2019) · Zbl 1473.35519 |
| [59] | Yu, F. J., A novel non-isospectral hierarchy and soliton wave dynamics for a parity-time-symmetric nonlocal vector nonlinear Gross-Pitaevskii equations, Commun. Nonlinear Sci. Numer. Simul., 78, Article 104852 pp. (2019) · Zbl 1480.35364 |
| [60] | Zhang, G. Q.; Yan, Z. Y.; Chen, Y., Novel higher-order rational solitons and dynamics of the defocusing integrable nonlocal nonlinear Schrödinger equation via the determinants, Appl. Math. Lett., 69, 113-120 (2017) · Zbl 1375.35511 |
| [61] | Zhang, Y. S.; Qiu, D. Q.; Cheng, Y.; He, J. S., Rational solution of the nonlocal nonlinear Schrödinger equation and its application in optics, Rom. J. Phys., 62, 108 (2017) |
| [62] | Zhong, W. P.; Yang, Z. P.; Belic, M.; Zhong, W. Y., Breather solutions of the nonlocal nonlinear self-focusing Schrödinger equation, Phys. Lett. A, 395, Article 127228 pp. (2021) · Zbl 07409508 |
| [63] | Zhou, Z. X., Darboux transformations and global solutions for a nonlocal derivative nonlinear Schrödinger equation, Commun. Nonlinear Sci. Numer. Simul., 62, 480-488 (2018) · Zbl 1470.35344 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
