Geodesic
Local and nonlocal complex discrete sine-Gordon equation. Solutions and continuum limits
Teoretičeskaâ i matematičeskaâ fizika, Tome 211 (2022) no. 3, pp. 375-393 Cet article a éte moissonné depuis la source Math-Net.Ru

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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.
Keywords: local and nonlocal complex discrete sine-Gordon equation, dynamics, continuum limit.
Mots-clés : Cauchy matrix solution 
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Xiao-bo Xiang; Wei Feng; Song-lin Zhao. Local and nonlocal complex discrete sine-Gordon equation. Solutions and continuum limits. Teoretičeskaâ i matematičeskaâ fizika, Tome 211 (2022) no. 3, pp. 375-393. http://geodesic.mathdoc.fr/item/TMF_2022_211_3_a1/

[1] M. J. Ablowitz, D. J. Kaup, A. C. Newell, H. Segur, “The inverse scattering transform-Fourier analysis for nonlinear problems”, Stud. Appl. Math., 53:4 (1974), 249–315 | DOI | MR

[2] M. J. Ablowitz, Z. H. Musslimani, “Integrable nonlocal nonlinear Schrödinger equation”, Phys. Rev. Lett., 110:8 (2013), 064105, 5 pp. | DOI

[3] M. J. Ablowitz, Z. H. Musslimani, “Inverse scattering transform for the integrable nonlocal nonlinear Schrödinger equation”, Nonlinearity, 29:3 (2016), 915–946 | DOI | MR

[4] S. Y. Lou, “Alice–Bob systems, $\hat{P}-\hat{T}-\hat{C}$ symmetry invariant and symmetry breaking soliton solutions”, J. Math. Phys., 59:8 (2018), 083507, 20 pp. | DOI | MR

[5] S. Y. Lou, “Multi-place physics and multi-place nonlocal systems”, Commun. Theor. Phys., 72:5 (2020), 057001, 13 pp. | DOI | MR

[6] S. Y. Lou, F. Huang, “Alice–Bob physics: Coherent solutions of nonlocal KdV systems”, Sci. Rep., 7 (2017), 869, 11 pp., arXiv: 1606.03154 | DOI

[7] M. J. Ablowitz, Z. H. Musslimani, “Integrable nonlocal nonlinear equations”, Stud. Appl. Math., 139:1 (2016), 7–59 | DOI | MR

[8] B. Yang, J. Yang, “Transformations between nonlocal and local integrable equations”, Stud. Appl. Math., 40:2 (2017), 178–201 | DOI | MR

[9] M. Li, T. Xu, “Dark and antidark soliton interactions in the nonlocal nonlinear Schrödinger equation with the self-induced parity-time-symmetric potential”, Phys. Rev. E, 91:3 (2015), 033202, 8 pp. | DOI | MR

[10] Z. Yan, “Integrable $\mathscr{P\!T}$-symmetric local and nonlocal vector nonlinear Schrödinger equations: A unified two-parameter model”, Appl. Math. Lett., 47 (2015), 61–68 | DOI | MR

[11] Z.-X. Zhou, “Darboux transformations and global solutions for a nonlocal derivative nonlinear Schrödinger equation”, Commun. Nonlinear Sci. Numer. Simul., 62 (2018), 480–488, arXiv: 1612.04892 | DOI | MR

[12] L.-Y. Ma, S.-F. Shen, Z.-N. Zhu, “Soliton solution and gauge equivalence for an integrable nonlocal complex modified Korteweg–de Vries equation”, J. Math. Phys., 58:10 (2017), 103501, 12 pp. | DOI | MR

[13] M. J. Ablowitz, X.-D. Luo, Z. H. Musslimani, “Inverse scattering transform for the nonlocal nonlinear Schrödinger equation with nonzero boundary conditions”, J. Math. Phys., 59:1 (2018), 011501, 42 pp. | DOI | MR

[14] B. Yang, Y. Chen, “Several reverse-time integrable nonlocal nonlinear equations: Rogue-wave solutions”, Chaos, 28:5 (2018), 053104, 6 pp. | DOI | MR

[15] K. Chen, X. Deng, S. Lou, D.-J. Zhang, “Solutions of nonlocal equations reduced from the AKNS hierarchy”, Stud. Appl. Math., 141:1 (2018), 113–141 | DOI | MR

[16] W.-X. Ma, Y. Huang, F. Wang, “Inverse scattering transforms and soliton solutions of nonlocal reverse-space nonlinear Schrödinger hierarchies”, Stud. Appl. Math., 145:3 (2020), 563–585 | DOI | MR

[17] W. Feng, S.-L. Zhao, Y.-Y. Sun, “Double Casoratian solutions to the nonlocal semi-discrete modified Korteweg–de Vries equation”, Internat. J. Modern Phys. B, 34:5 (2020), 2050021, 14 pp. | DOI | MR

[18] S.-Z. Liu, J. Wang, D.-J. Zhang, “The Fokas–Lenells equations: Bilinear approach”, Stud. Appl. Math., 148:2 (2022), 651–688 | DOI

[19] W. Feng, S.-L. Zhao, “Cauchy matrix type solutions for the nonlocal nonlinear Schrödinger equation”, Rep. Math. Phys., 84:1 (2019), 75–83 | DOI | MR

[20] F. Nijhoff, J. Atkinson, J. Hietarinta, “Soliton solutions for ABS lattice equations: I. Cauchy matrix approach”, J. Phys. A: Math. Theor., 42:40 (2009), 404005, 34 pp. | DOI | MR

[21] V. E. Adler, A. I. Bobenko, Yu. B. Suris, “Classification of integrable equations on quad- graphs. The consistency approach”, Commun. Math. Phys., 233:3 (2003), 513–543, arXiv: nlin/0202024 | DOI | MR | Zbl

[22] F. Nijhoff, J. Atkinson, “Elliptic $N$-soliton solutions of ABS lattice equations”, Int. Math. Res. Not., 2010:20 (2010), 3837–3895 | DOI | MR

[23] D.-J. Zhang, S.-L. Zhao, “Solutions to ABS lattice equations via generalized Cauchy matrix approach”, Stud. Appl. Math., 131:1 (2013), 72–103 | DOI | MR

[24] J. Sylvester, “Sur l'équation en matrices $px=xq$”, C. R. Acad. Sci. Paris, 99:2 (1884), 67–71 | Zbl

[25] A. Sakhnovich, “Exact solutions of nonlinear equations and the method of operator identities”, Linear Algebra Appl., 182 (1993), 109–126 | DOI | MR

[26] B. Carl, C. Schiebold, “Ein direkter Ansatz zur Untersuchung von Solitonengleichungen”, Jahresber. Deutsch. Math.-Verein., 102:3 (2000), 102–148 | MR

[27] S. Carillo, C. Schiebold, “Noncommutative Korteweg–de Vries and modified Korteweg–de Vries hierarchies via recursion methods”, J. Math. Phys., 50:7 (2009), 073510, 14 pp. | DOI | MR | Zbl

[28] S. Carillo, C. Schiebold, “Matrix Korteweg–de Vries and modified Korteweg–de Vries hierarchies: Noncommutative soliton solutions”, J. Math. Phys., 52:5 (2011), 053507, 21 pp. | DOI | MR

[29] A. L. Sakhnovich, “Bäcklund–Darboux transformation for non-isospectral canonical system and Riemann–Hilbert problem”, SIGMA, 3 (2007), 054, 11 pp. | DOI | MR | Zbl

[30] A. L. Sakhnovich, L. A. Sakhnovich, I. Ya. Roitberg, Inverse problems and nonlinear evolution equations. Solutions, Darboux matrices and Weyl–Titchmarsh functions, De Gruyter Studies in Mathematics, 47, De Gruyter, Berlin, 2013 | DOI | MR

[31] A. Dimakis, F. Müller-Hoissen, “Bidifferential graded algebras and integrable systems”, Discrete Contin. Dyn. Syst., 2009, suppl. (2009), 208–219 | DOI | MR | Zbl

[32] A. Dimakis, F. Müller-Hoissen, “Solutions of matrix NLS systems and their discretizations: a unified treatment”, Inverse Problems, 26:9 (2010), 095007, 55 pp. | DOI | MR

[33] A. Dimakis, F. Müller-Hoissen, “Bidifferential calculus approach to AKNS hierarchies and their solutions”, SIGMA, 6 (2010), 055, 27 pp., arXiv: 1004.1627 | DOI | MR

[34] D.-D. Zhang, P. H. van der Kamp, D.-J. Zhang, “Multi-component extension of CAC systems”, SIGMA, 16 (2020), 060, 30 pp., arXiv: 1912.00713 | DOI | MR

[35] T. Bridgman, W. Hereman, G. R. W. Quispel, P. H. van der Kamp, “Symbolic computation of Lax Pairs of partial difference equations using consistency around the cube”, Found. Comput. Math., 13:4 (2013), 517–544 | DOI | MR

[36] G. R. W. Quispel, F. W. Nijhoff, H. W. Capel, J. van der Linden, “Linear integral equations and nonlinear difference-difference equations”, Phys. A, 125:2–3 (1984), 344–380 | DOI | MR

[37] S.-L. Zhao, “A discrete negative AKNS equation: generalized Cauchy matrix approach”, J. Nonlinear Math. Phys., 23:4 (2016), 544–562 | DOI | MR

[38] S.-L. Zhao, Y. Shi, “Discrete and semidiscrete models for AKNS equation”, Z. Naturforsch. A, 72:3 (2017), 281–290 | DOI

[39] S.-L. Zhao, W. Feng, Y.-Y. Jin, “Discrete analogues for two nonlinear Schrödinger type equations”, Commun. Nonlinear Sci. Numer. Simul., 72 (2019), 329–341 | DOI | MR

[40] S.-L. Zhao, “Discrete potential Ablowitz–Kaup–Newell–Segur equation”, J. Differ. Equ. Appl., 25:8 (2019), 1134–1148 | DOI | MR

[41] W. Huang, L. Xue, Q. P. Liu, “Integrable discretizations for classical Boussinesq system”, J. Phys. A: Math. Theor., 54:4 (2021), 045201, 29 pp. | DOI | MR

[42] Shuai Chzhan, Sun-Lin Chzhao, In Shi, “Diskretnoe uravnenie Ablovitsa–Kaupa–Nyuella–Sigura vtorogo poryadka i ego modifikatsiya”, TMF, 210:3 (2022), 350–374 | DOI | DOI

[43] D.-J. Zhang, J. Ji, S.-L. Zhao, “Soliton scattering with amplitude changes of a negative order AKNS equation”, Phys. D, 238:23–24 (2009), 2361–2367 | DOI | MR