On the Riemann-Hilbert problem of the matrix Lakshmanan-Porsezian-Daniel system with a \(4\times4\) AKNS-type matrix Lax pair. (English. Russian original) Zbl 1515.37069
Theor. Math. Phys. 210, No. 3, 337-352 (2022); translation from Teor. Mat. Fiz. 210, No. 3, 387-404 (2022).
Summary: The initial-boundary value problems for the matrix Lakshmanan-Porsezian-Daniel system are studied by utilizing the Fokas unified transform approach. First, the spectral analysis of the \(4\times4\) Ablowitz-Kaup-Newell-Segur-type matrix Lax pair is performed. Second, solutions of the matrix Lakshmanan-Porsezian-Daniel system are reconstructed from a \(4\times4\) matrix Riemann-Hilbert problem. It is proved in addition that the spectral functions are not independent but are related by the so-called global relation.
MSC:
| 37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |
| 37K15 | Inverse spectral and scattering methods for infinite-dimensional Hamiltonian and Lagrangian systems |
| 45D05 | Volterra integral equations |
| 35Q15 | Riemann-Hilbert problems in context of PDEs |
| 35Q51 | Soliton equations |
Keywords:
Volterra integral equations; Riemann-Hilbert problem; matrix Lakshmanan-Porsezian-Daniel system; initial-boundary value problem; Fokas unified transform approachReferences:
| [1] | Lakshmanan, M.; Porsezian, K.; Daniel, M., Effect of discreteness on the continuum limit of the Heisenberg spin chain, Phys. Lett. A, 133, 483-488 (1988) |
| [2] | Porsezian, K.; Daniel, M.; Lakshmanan, M., On the integrability aspects of the one-dimensional classical continuum isotropic Heisenberg spin chain, J. Math. Phys., 33, 1807-1816 (1992) · Zbl 1112.82309 |
| [3] | Zhang, H.-Q.; Tian, B.; Meng, X.-H.; Lu, X.; Liu, W.-J., Conservation laws, soliton solutions and modulational instability for the higher-order dispersive nonlinear Schrödinger equation, Eur. Phys. J. B, 72, 233-239 (2009) · Zbl 1188.35184 |
| [4] | Wang, L. H.; Porsezian, K.; He, J. S., Breather and rogue wave solutions of a generalized nonlinear Schrödinger equation, Phys. Rev. E, 87 (2013) |
| [5] | Wang, L.; Zhang, J.-H.; Wang, Z.-Q.; Liu, C.; Li, M.; Qi, F.-H.; Guo, R., Breather-to-soliton transitions, nonlinear wave interactions, and modulational instability in a higher-order generalized nonlinear Schrödinger equation, Phys. Rev. E, 93 (2016) |
| [6] | Manafian, J.; Foroutan, M.; Guzali, A., Applications of the ETEM for obtaining optical soliton solutions for the Lakshmanan-Porsezian-Daniel model, Eur. Phys. J. Plus, 132 (2017) |
| [7] | Alqahtani, R. T.; Babatin, M. M.; Biswas, A., Bright optical solitons for Lakshmanan-Porsezian-Daniel model by semi-inverse variational principle, Optik, 154, 109-114 (2018) |
| [8] | Biswas, A.; Yildirim, Y.; Yasar, E.; Zhou, Q.; Moshokoa, S. P.; Belic, M., Optical solitons for Lakshmanan-Porsezian-Daniel equation by modified simple equation method, Optik, 160, 24-32 (2018) |
| [9] | Javid, A.; Raza, N., Singular and dark optical solitons to the well posed Lakshmanan- Porsezian-Daniel model, Optik, 171, 120-129 (2018) |
| [10] | Ekici, M., Optical solitons in birefringent fibers for Lakshmanan-Porsezian-Daniel model by extended Jacobi’s elliptic function expansion scheme, Optik, 172, 651-656 (2018) |
| [11] | Rezazadeh, H.; Kumar, D.; Neirameh, A.; Eslami, M.; Mirzazadeh, M., Applications of three methods for obtaining optical soliton solutions for the Lakshmanan-Porsezian-Daniel model with Kerr law nonlinearity, Pramana J. Phys., 94 (2020) |
| [12] | Xin, H., Optical envelope patterns in nonlinear media modeled by the Lakshmanan- Porsezian- Daniel equation, Optik, 227 (2021) |
| [13] | Liu, W.; Qiu, D.-Q.; Wu, Z.-W.; He, J.-S., Dynamical behavior of solution in integrable nonlocal Lakshmanan-Porsezian-Daniel equation, Commun. Theor. Phys., 65, 671-676 (2016) · Zbl 1345.35104 |
| [14] | Yang, Y.; Suzuki, T.; Cheng, X., Darboux transformations and exact solutions for the integrable nonlocal Lakshmanan-Porsezian-Daniel equation, Appl. Math. Lett., 99 (2020) · Zbl 1428.35542 |
| [15] | Yépez-Martínez, H.; Gómez-Aguilar, J. F., M-derivative applied to the soliton solutions for the Lakshmanan-Porsezian-Daniel equation with dual-dispersion for optical fibers, Opt. Quant. Electron., 51 (2019) |
| [16] | Veeresha, P.; Prakasha, D. G.; Baskonus, H. M.; Yel, G., An efficient analytical approach for fractional Lakshmanan-Porsezian-Daniel model, Math. Methods Appl. Sci., 43, 4136-4155 (2020) · Zbl 1454.35352 |
| [17] | Liu, D.-Y.; Tian, B.; Xie, X.-Y., Bound-state solutions, Lax pair and conservation laws for the coupled higher-order nonlinear Schrödinger equations in the birefringent or two-mode fiber, Modern Phys. Lett. B, 31 (2017) |
| [18] | Sun, W.-R.; Liu, D.-Y.; Xie, X.-Y., Vector semirational rogue waves and modulation instability for the coupled higherorder nonlinear Schrödinger equations in the birefringent optical fibers, Chaos, 27 (2017) · Zbl 1390.35340 |
| [19] | Xu, T.; He, G., Higher-order interactional solutions and rogue wave pairs for the coupled Lakshmanan-Porsezian-Daniel equations, Nonlinear Dyn., 98, 1731-1744 (2019) · Zbl 1430.37082 |
| [20] | Ye, Y.; Hou, C.; Cheng, D.; Chen, S., Rogue wave solutions of the vector Lakshmanan- Porsezian-Daniel equation, Phys. Lett. A., 384 (2020) · Zbl 1482.37072 |
| [21] | Dong, M.-J.; Tian, L.-X., Characteristics of rogue waves on a soliton background of the vector Lakshmanan-Porsezian-Daniel equation, Math. Methods Appl. Sci., 44, 5225-5237 (2021) · Zbl 1473.35476 |
| [22] | Zhang, Z.; Tian, B.; Liu, L.; Sun, Y.; Du, Z., Lax pair, breather-to-soliton conversions, localized and periodic waves for a coupled higher-order nonlinear Schrödinger system in a birefringent optical fiber, Eur. Phys. J. Plus, 134 (2019) |
| [23] | Veli, S. Saravana; Latha, M. M., A generalized Davydov model with interspine coupling and its integrable discretization, Phys. Scr., 86 (2012) · Zbl 1271.81209 |
| [24] | Sun, W.-R.; Tian, B.; Wang, Y.-F.; Zhen, H.-L., Soliton excitations and interactions for the three-coupled fourth-order nonlinear Schrödinger equations in the alpha helical proteins, Eur. Phys. J. D, 69 (2015) |
| [25] | Du, Z.; Tian, B.; Qu, Q.-X.; Chai, H.-P.; Wu, X.-Y., Semirational rogue waves for the three-coupled fourth-order nonlinear Schrödinger equations in an alpha helical protein, Superlattice Microstruct., 112, 362-373 (2017) |
| [26] | Sun, W.-R., Vector solitons and rogue waves of the matrix Lakshmanan-Porsezian-Daniel equation, Nonlinear Dyn., 102, 1743-1751 (2020) |
| [27] | Zhang, H.-Q.; Li, J.; Xu, T.; Zhang, Y.-X.; Hu, W.; Tian, B., Optical soliton solutions for two coupled nonlinear Schrödinger systems via Darboux transformation, Phys. Scr., 76, 452-460 (2007) · Zbl 1131.35390 |
| [28] | Lü, X.; Tian, B., Vector bright soliton behaviors associated with negative coherent coupling, Phys. Rev. E, 85 (2012) |
| [29] | Sun, W.-R.; Wang, L., Matter rogue waves for the three-component Gross-Pitaevskii equations in the spinor Bose-Einstein condensates, Proc. R. Soc. A, 474 (2018) · Zbl 1402.76165 |
| [30] | Sun, W.-R.; Wang, L., Vector rogue waves, rogue wave-to-soliton conversions and modulation instability of the higher-order matrix nonlinear Schrödinger equation, Eur. Phys. J. Plus, 133 (2018) |
| [31] | Fokas, A. S., A unified transform method for solving linear and certain nonlinear PDEs, Proc. Roy. Soc. London Ser. A, 453, 1411-1443 (1997) · Zbl 0876.35102 |
| [32] | Lenells, J., Initial-boundary value problems for integrable evolution equations with \(3\times 3\) Lax pairs, Phys. D, 241, 857-875 (2012) · Zbl 1251.35006 |
| [33] | Lenells, J., The Degasperis-Procesi equation on the half-line, Nonlinear Anal., 76, 122-139 (2013) · Zbl 1253.35138 |
| [34] | Xu, J.; Fan, E., The unified transform method for the Sasa-Satsuma equation on the half- line, Proc. Roy. Soc. London Ser. A, 469 (2013) · Zbl 1348.35249 |
| [35] | Xu, J.; Fan, E., The three-wave equation on the half-line, Phys. Lett. A, 378, 26-33 (2014) · Zbl 1396.31001 |
| [36] | Monvel, A. Boutet de; Shepelsky, D., The Ostrovsky-Vakhnenko equation by a Riemann- Hilbert approach, J. Phys. A: Math. Theor., 48 (2015) · Zbl 1311.35175 |
| [37] | Monvel, A. Boutet de; Shepelsky, D.; Zielinski, L., The short pulse equation by a Riemann- Hilbert approach, Lett. Math. Phys., 107, 1345-1373 (2017) · Zbl 1370.35238 |
| [38] | Geng, X. G.; Liu, H.; Zhu, J. Y., Initial-boundary value problems for the coupled nonlinear Schrödinger equation on the half-line, Stud. Appl. Math., 135, 310-346 (2015) · Zbl 1338.35408 |
| [39] | Liu, H.; Geng, X., Initial-boundary problems for the vector modified Korteweg-de Vries equation via Fokas unified transform method, J. Math. Anal. Appl., 440, 578-596 (2016) · Zbl 1342.35306 |
| [40] | Tian, S.-F., Initial-boundary value problems of the coupled modified Korteweg-de Vries equation on the half-line via the Fokas method, J. Phys. A: Math. Theor., 50 (2017) · Zbl 1377.37100 |
| [41] | Zhu, Q.-Z.; Fan, E.-G.; Xu, J., Initial-boundary value problem for two-component Gerdjikov-Ivanov equation with \(3\times 3\) Lax pair on half-line, Commun. Theor. Phys., 68, 425-438 (2017) · Zbl 1377.35235 |
| [42] | Yan, Z., An initial-boundary value problem for the integrable spin-1 Gross-Pitaevskii equations with a \(4\times4\) Lax pair on the half-line, Chaos, 27 (2017) · Zbl 1394.35486 |
| [43] | Yan, Z., Initial-boundary value problem for the spin-1 Gross-Pitaevskii system with a \(4\times4\) Lax pair on a finite interval, J. Math. Phys., 60 (2019) · Zbl 1421.37029 |
| [44] | Hu, B.; Zhang, L.; Xia, T.; Zhang, N., On the Riemann-Hilbert problem of the Kundu equation, Appl. Math. Comput., 381 (2020) · Zbl 1474.35228 |
| [45] | Hu, B.-B.; Xia, T.-C.; Zhang, N.; Wang, J.-B., Initial-boundary value problems for the coupled higher-order nonlinear Schrödinger equations on the half-line, Internat. J. Nonlinear Numer. Simul., 19, 83-92 (2018) · Zbl 1401.35025 |
| [46] | Huang, L., The initial-boundary-value problems for the Hirota equation on the half-line, Chin. Ann. Math. Ser. B, 41, 117-132 (2020) · Zbl 1433.35210 |
| [47] | Hu, B.-B.; Zhang, L.; Xia, T.-C., On the Riemann-Hilbert problem of a generalized derivative nonlinear Schrödinger equation, Commun. Theor. Phys., 73 (2021) · Zbl 1521.35130 |
| [48] | Hu, B.; Zhang, L.; Zhang, N., On the Riemann-Hilbert problem for the mixed Chen-Lee-Liu derivative nonlinear Schrödinger equation, J. Comput. Appl. Math., 390 (2021) · Zbl 1465.35329 |
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.
