Local and nonlocal complex discrete sine-Gordon equation. Solutions and continuum limits. (English. Russian original) Zbl 1515.37082
Theor. Math. Phys. 211, No. 3, 758-774 (2022); translation from Teor. Mat. Fiz. 211, No. 3, 375-393 (2022).
Summary: We study local and nonlocal complex reductions of a discrete negative-order Ablowitz-Kaup-Newell-Segur equation. For the resulting local and nonlocal complex discrete sine-Gordon equations, we construct solutions of the Cauchy matrix type, including soliton solutions and Jordan-block solutions. The dynamics of 1-soliton solutions are analyzed and illustrated. Continuum limits of the resulting local and nonlocal complex discrete sine-Gordon equations are discussed.
MSC:
| 37K60 | Lattice dynamics; integrable lattice equations |
| 37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |
| 39A36 | Integrable difference and lattice equations; integrability tests |
| 39A14 | Partial difference equations |
References:
| [1] | 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 (1974) · Zbl 0408.35068 · doi:10.1002/sapm1974534249 |
| [2] | Ablowitz, M. J.; Musslimani, Z. H., Integrable nonlocal nonlinear Schrödinger equation, Phys. Rev. Lett., 110 (2013) · doi:10.1103/PhysRevLett.110.064105 |
| [3] | Ablowitz, M. J.; Musslimani, Z. H., Inverse scattering transform for the integrable nonlocal nonlinear Schrödinger equation, Nonlinearity, 29, 915-946 (2016) · Zbl 1338.37099 · doi:10.1088/0951-7715/29/3/915 |
| [4] | Lou, S. Y., Alice-Bob systems, \( \hat{P}-\hat{T}-\hat{C}\) symmetry invariant and symmetry breaking soliton solutions, J. Math. Phys., 59 (2018) · Zbl 1395.35169 · doi:10.1063/1.5051989 |
| [5] | Lou, S. Y., Multi-place physics and multi-place nonlocal systems, Commun. Theor. Phys., 72 (2020) · Zbl 1451.81011 · doi:10.1088/1572-9494/ab770b |
| [6] | Lou, S. Y.; Huang, F., Alice-Bob physics: Coherent solutions of nonlocal KdV systems, Sci. Rep., 7 (2017) · doi:10.1038/s41598-017-00844-y |
| [7] | Ablowitz, M. J.; Musslimani, Z. H., Integrable nonlocal nonlinear equations, Stud. Appl. Math., 139, 7-59 (2016) · Zbl 1373.35281 · doi:10.1111/sapm.12153 |
| [8] | Yang, B.; Yang, J., Transformations between nonlocal and local integrable equations, Stud. Appl. Math., 40, 178-201 (2017) · Zbl 1392.35297 |
| [9] | 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 (2015) · doi:10.1103/PhysRevE.91.033202 |
| [10] | Yan, Z., Integrable \(\mathscr{P\!T} \)-symmetric local and nonlocal vector nonlinear Schrödinger equations: A unified two-parameter model, Appl. Math. Lett., 47, 61-68 (2015) · Zbl 1322.35132 · doi:10.1016/j.aml.2015.02.025 |
| [11] | 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 · doi:10.1016/j.cnsns.2018.01.008 |
| [12] | Ma, L.-Y.; Shen, S.-F.; Zhu, Z.-N., Soliton solution and gauge equivalence for an integrable nonlocal complex modified Korteweg-de Vries equation, J. Math. Phys., 58 (2017) · Zbl 1380.37132 · doi:10.1063/1.5005611 |
| [13] | 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 (2018) · Zbl 1383.35204 · doi:10.1063/1.5018294 |
| [14] | Yang, B.; Chen, Y., Several reverse-time integrable nonlocal nonlinear equations: Rogue- wave solutions, Chaos, 28 (2018) · Zbl 1391.35362 · doi:10.1063/1.5019754 |
| [15] | Chen, K.; Deng, X.; Lou, S.; Zhang, D.-J., Solutions of nonlocal equations reduced from the AKNS hierarchy, Stud. Appl. Math., 141, 113-141 (2018) · Zbl 1398.35194 · doi:10.1111/sapm.12215 |
| [16] | Ma, W.-X.; Huang, Y.; Wang, F., Inverse scattering transforms and soliton solutions of nonlocal reverse-space nonlinear Schrödinger hierarchies, Stud. Appl. Math., 145, 563-585 (2020) · Zbl 1454.35348 · doi:10.1111/sapm.12329 |
| [17] | Feng, W.; Zhao, S.-L.; Sun, Y.-Y., Double Casoratian solutions to the nonlocal semi-discrete modified Korteweg-de Vries equation, Internat. J. Modern Phys. B, 34 (2020) · doi:10.1142/S0217979220500216 |
| [18] | Liu, S.-Z.; Wang, J.; Zhang, D.-J., The Fokas-Lenells equations: Bilinear approach, Stud. Appl. Math., 148, 651-688 (2022) · Zbl 1529.35474 · doi:10.1111/sapm.12454 |
| [19] | Feng, W.; Zhao, S.-L., Cauchy matrix type solutions for the nonlocal nonlinear Schrödinger equation, Rep. Math. Phys., 84, 75-83 (2019) · Zbl 1441.35218 · doi:10.1016/S0034-4877(19)30070-9 |
| [20] | Nijhoff, F.; Atkinson, J.; Hietarinta, J., Soliton solutions for ABS lattice equations: I. Cauchy matrix approach, J. Phys. A: Math. Theor., 42 (2009) · Zbl 1184.35281 · doi:10.1088/1751-8113/42/40/404005 |
| [21] | Adler, V. E.; Bobenko, A. I.; Suris, Yu. B., Classification of integrable equations on quad- graphs. The consistency approach, Commun. Math. Phys., 233, 513-543 (2003) · Zbl 1075.37022 · doi:10.1007/s00220-002-0762-8 |
| [22] | Nijhoff, F.; Atkinson, J., Elliptic \(N\)-soliton solutions of ABS lattice equations, Int. Math. Res. Not., 2010, 3837-3895 (2010) · Zbl 1217.37068 |
| [23] | Zhang, D.-J.; Zhao, S.-L., Solutions to ABS lattice equations via generalized Cauchy matrix approach, Stud. Appl. Math., 131, 72-103 (2013) · Zbl 1338.37113 · doi:10.1111/sapm.12007 |
| [24] | Sylvester, J., Sur l’équation en matrices \(px=xq\), C. R. Acad. Sci. Paris, 99, 67-71 (1884) · JFM 16.0108.03 |
| [25] | Sakhnovich, A., Exact solutions of nonlinear equations and the method of operator identities, Linear Algebra Appl., 182, 109-126 (1993) · Zbl 0770.35071 · doi:10.1016/0024-3795(93)90495-A |
| [26] | Carl, B.; Schiebold, C., Ein direkter Ansatz zur Untersuchung von Solitonengleichungen, Jahresber. Deutsch. Math.-Verein., 102, 102-148 (2000) · Zbl 0970.35121 |
| [27] | Carillo, S.; Schiebold, C., Noncommutative Korteweg-de Vries and modified Korteweg– de Vries hierarchies via recursion methods, J. Math. Phys., 50 (2009) · Zbl 1256.37036 · doi:10.1063/1.3155080 |
| [28] | Carillo, S.; Schiebold, C., Matrix Korteweg-de Vries and modified Korteweg-de Vries hierarchies: Noncommutative soliton solutions, J. Math. Phys., 52 (2011) · Zbl 1317.35220 · doi:10.1063/1.3576185 |
| [29] | Sakhnovich, A. L., Bäcklund-Darboux transformation for non-isospectral canonical system and Riemann-Hilbert problem, SIGMA, 3 (2007) · Zbl 1136.37040 |
| [30] | Sakhnovich, A. L.; Sakhnovich, L. A.; Roitberg, I. Ya., Inverse problems and nonlinear evolution equations. Solutions, Darboux matrices and Weyl-Titchmarsh functions (2013), Berlin: De Gruyter, Berlin · Zbl 1283.47003 |
| [31] | Dimakis, A.; Müller-Hoissen, F., Bidifferential graded algebras and integrable systems, Discrete Contin. Dyn. Syst., 2009, suppl., 208-219 (2009) · Zbl 1184.37052 |
| [32] | Dimakis, A.; Müller-Hoissen, F., Solutions of matrix NLS systems and their discretizations: a unified treatment, Inverse Problems, 26 (2010) · Zbl 1205.35291 · doi:10.1088/0266-5611/26/9/095007 |
| [33] | Dimakis, A.; Müller-Hoissen, F., Bidifferential calculus approach to AKNS hierarchies and their solutions, SIGMA, 6 (2010) · Zbl 1218.37076 |
| [34] | Zhang, D.-D.; Kamp, P. H. van der; Zhang, D.-J., Multi-component extension of CAC systems, SIGMA, 16 (2020) · Zbl 1448.37092 |
| [35] | Bridgman, T.; Hereman, W.; Quispel, G. R. W.; Kamp, P. H. van der, Symbolic computation of Lax Pairs of partial difference equations using consistency around the cube, Found. Comput. Math., 13, 517-544 (2013) · Zbl 1306.37067 · doi:10.1007/s10208-012-9133-9 |
| [36] | Quispel, G. R. W.; Nijhoff, F. W.; Capel, H. W.; Linden, J. van der, Linear integral equations and nonlinear difference-difference equations, Phys. A, 125, 344-380 (1984) · Zbl 0598.45009 · doi:10.1016/0378-4371(84)90059-1 |
| [37] | Zhao, S.-L., A discrete negative AKNS equation: generalized Cauchy matrix approach, J. Nonlinear Math. Phys., 23, 544-562 (2016) · Zbl 1420.39008 · doi:10.1080/14029251.2016.1237201 |
| [38] | Zhao, S.-L.; Shi, Y., Discrete and semidiscrete models for AKNS equation, Z. Naturforsch. A, 72, 281-290 (2017) · doi:10.1515/zna-2016-0443 |
| [39] | Zhao, S.-L.; Feng, W.; Jin, Y.-Y., Discrete analogues for two nonlinear Schrödinger type equations, Commun. Nonlinear Sci. Numer. Simul., 72, 329-341 (2019) · Zbl 1464.39017 · doi:10.1016/j.cnsns.2019.01.003 |
| [40] | Zhao, S.-L., Discrete potential Ablowitz-Kaup-Newell-Segur equation, J. Differ. Equ. Appl., 25, 1134-1148 (2019) · Zbl 1423.39014 · doi:10.1080/10236198.2019.1662410 |
| [41] | Huang, W.; Xue, L.; Liu, Q. P., Integrable discretizations for classical Boussinesq system, J. Phys. A: Math. Theor., 54 (2021) · Zbl 1519.37100 · doi:10.1088/1751-8121/abd2fb |
| [42] | Zhang, Shuai; Zhao, Song-Lin; Shi, Ying, Discrete second-order Ablowitz-Kaup-Newell-Segur equation and its modified form, Theoret. and Math. Phys., 210, 304-326 (2022) · Zbl 1515.37083 · doi:10.1134/S0040577922030023 |
| [43] | Zhang, D.-J.; Ji, J.; Zhao, S.-L., Soliton scattering with amplitude changes of a negative order AKNS equation, Phys. D, 238, 2361-2367 (2009) · Zbl 1183.37121 · doi:10.1016/j.physd.2009.09.018 |
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.
