Discrete Crum's theorems and lattice KdV-type equations

Cheng Zhang   ; Linyu Peng   ; Da-jun Zhang

Teoretičeskaâ i matematičeskaâ fizika, Tome 202 (2020) no. 2, pp. 187-206

Résumé

We develop Darboux transformations ($DTs$) and their associated Crum's formulas for two Schrödinger-type difference equations that are themselves discretized versions of the spectral problems of the KdV and modified KdV equations. With DTs viewed as a discretization process, classes of semidiscrete and fully discrete KdV-type systems, including the lattice versions of the potential KdV, potential modified KdV, and Schwarzian KdV equations, arise as the consistency condition for the differential/difference spectral problems and their DTs. The integrability of the underlying lattice models, such as Lax pairs, multidimensional consistency, $\tau$-functions, and soliton solutions, can be easily obtained by directly applying the discrete Crum's formulas.

Détail

Keywords: discrete Crum's theorem, exact discretization, discrete Schrödinger equation, lattice KdV equations.
Mots-clés : Darboux transformation

Bibliographie

[1] G. Darboux, “Sur une proposition relative aux équations linéaires”, C. R. Acad. Sci. Paris, 94 (1882), 1456–1459 | Zbl

[2] M. M. Crum, “Associated Sturm–Liouville systems”, Quart. J. Math., 6:1 (1955), 121–127 | DOI | MR

[3] V. B. Matveev, M. A. Salle, Darboux Transformations and Solitons, Springer, Berlin, 1991 | MR

[4] A. B. Shabat, “Dressing chains and lattices”, Nonlinearity, Integrability and All That: Twenty Years After NEEDS' 79 (Lecce, Italy, July 1–10, 1999), eds. M. Boiti, L. Martina, F. Pempinelly, B. Prinary, G. Soliani, World Sci., Singapore, 2000, 341–342 | MR

[5] D. Levi, R. Benguria, “Bäcklund transformations and nonlinear differential difference equations”, Proc. Nat. Acad. Sci. USA, 77:9 (1980), 5025–5027 | DOI | MR

[6] D. Levi, “Nonlinear differential difference equations as Bäcklund transformations”, J. Phys. A: Math. Gen., 14:5 (1981), 1083–1098 | DOI | MR

[7] V. B. Matveev, “Darboux transformation and the explicit solutions of differential-difference and difference-difference evolution equations I”, Lett. Math. Phys., 3:3 (1979), 217–222 | DOI | MR

[8] V. B. Matveev, “Darboux transformation and explicit solutions of the Kadomtsev–Petviashvili equation, depending on functional parameters”, Lett. Math. Phys., 3:3 (1979), 213–216 | DOI | MR

[9] V. Spiridonov, A. Zhedanov, “Discrete Darboux transformations, the discrete-time Toda lattice, and the Askey–Wilson polynomials”, Meth. Appl. Anal., 2:4 (1995), 369–398 | DOI | MR

[10] J. J. C. Nimmo, R. Willox, “Darboux transformations for the two-dimensional Toda system”, Proc. Roy. Soc. London Ser. A, 453:1967 (1997), 2497–2525 | MR

[11] J. J. C. Nimmo, “Darboux transformations and the discrete KP equation”, J. Phys. A: Math. Gen., 30:24 (1997), 8693–9704 | DOI | MR

[12] M. Sato, “Soliton equations as dynamical systems on infinite dimensional Grassmann manifolds”, Random Systems and Dynamical Systems (Kyoto, January 7–9, 1981), RIMS Kôkyûroku, 439, Kyoto Univ., Kyoto, 1981, 30–46 | MR | Zbl

[13] T. Ohta, J. Satsuma, D. Takahashi, T. Tokihiro, “An elementary introduction to Sato theory”, Prog. Theor. Phys. Suppl., 94 (1988), 210–241 | DOI | MR

[14] T. Miwa, M. Jimbo, E. Date, Solitons. Differential Equations, Symmetries and Infinite Dimensional Algebras, Cambridge Tracts in Mathematics, 135, Cambridge Univ. Press, Cambridge, 2000 | MR

[15] F. W. Nijhoff, “Lax pair for the Adler (lattice Krichever–Novikov) system”, Phys. Lett. A, 297:1–2 (2002), 49–58 | DOI | MR

[16] A. I. Bobenko, Yu. B. Suris, “Integrable systems on quad-graphs”, Int. Math. Res. Notices, 2002:11 (2002), 573–611 | DOI | MR

[17] J. Hietarinta, N. Joshi, F. W. Nijhoff, Discrete Systems and Integrability, Cambridge Univ. Press, Cambridge, 2016 | DOI | MR

[18] 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 | DOI | MR

[19] V. E. Adler, A. I. Bobenko, Yu. B. Suris, “Diskretnye nelineinye giperbolicheskie uravneniya. Klassifikatsiya integriruemykh sluchaev”, Funkts. analiz i ego pril., 43:1 (2009), 3–21 | DOI | DOI | MR | Zbl

[20] F. W. 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] J. Hietarinta, D. J. Zhang, “Soliton solutions for ABS lattice equations. II. Casoratians and bilinearization”, J. Phys. A: Math. Theor., 42:40 (2009), 404006, 30 pp. | DOI | MR

[22] E. Schrödinger, “A method of determining quantum-mechanical eigenvalues and eigenfunctions”, Proc. Roy. Irish Acad. Sect. A, 46 (1940), 9–16 | MR

[23] L. Infeld, T. E. Hull, “The factorization method”, Rev. Modern Phys., 23:1 (1951), 21–68 | DOI | MR

[24] F. W. Nijhoff, H. Capel, “The discrete Korteweg–de Vries equation”, Acta Appl. Math., 39:1–3 (1995), 133-158 | DOI | MR

[25] Q. P. Liu, Y. Q. Wang, “A note on a discrete Schrödinger spectral problem and associated evolution equations”, Acta Math. Sci. Ser. A, 26:5 (2006), 773–777 | MR

[26] Y. Shi, J. J. C. Nimmo, D.-J. Zhang, “Darboux and binary Darboux transformations for discrete integrable systems I. Discrete potential KdV equation”, J. Phys. A: Math. Theor., 47:2 (2013), 025205, 11 pp. | DOI | MR

[27] Y. Shi, J. J. C. Nimmo, J. X. Zhao, “Darboux and binary Darboux transformations for discrete integrable systems. II. Discrete potential mKdV equation”, SIGMA, 13 (2017), 036, 18 pp., arXiv: 1705.09896 | DOI | MR

[28] C.-W. Cao, G.-Y. Zhang, “Lax Pairs for discrete integrable equations via Darboux transformations”, Chinese Phys. Lett., 29:5 (2012), 050202, 2 pp. | DOI

[29] S. Konstantinou-Rizos, Darboux transformations, discrete integrable systems and related Yang–Baxter maps, PhD Thesis, University of Leeds, 2014, arXiv: 1410.5013

[30] S. V. Smirnov, “Faktorizatsiya preobrazovanii Darbu–Laplasa dlya diskretnykh giperbolicheskikh operatorov”, TMF, 199:2 (2019), 175–192 | DOI | DOI

[31] A. P. Fordy, J. Gibbons, “Factorization of operators. I. Miura transformations”, J. Math. Phys., 21:10 (1980), 2508–2510 | DOI | MR

[32] A. P. Fordy, J. Gibbons, “Factorization of operators. II”, J. Math. Phys., 22:6 (1981), 1170–1175 | DOI | MR

[33] M. Boiti, F. Pempinelli, B. Prinari, A. Spire, “An integrable discretization of KdV at large times”, Inverse Probl., 17:3 (2001), 515–526 | DOI | MR

[34] S. Butler, N. Joshi, “An inverse scattering transform for the lattice potential KdV equation”, Inverse Probl., 26:11 (2010), 115012, 28 pp. | DOI | MR

[35] A. N. Hone, “Exact discretization of the Ermakov–Pinney equation”, Phys. Lett. A, 263:4–6 (1999), 347–354 | DOI | MR

[36] A. P. Veselov, A. B. Shabat, “Odevayuschaya tsepochka i spektralnaya teoriya operatora Shredingera”, Funkts. analiz i ego pril., 27:2 (1993), 1–21 | DOI | MR | Zbl

[37] H. D. Wahlquist, F. B. Estabrook, “Bäcklund transformation for solutions of the Korteweg–de Vries equation”, Phys. Rev. Lett., 31:23 (1973), 1386–1390 | DOI | MR

[38] C. A. Evripidou, P. H. van der Kamp, C. Zhang, “Dressing the dressing chain”, SIGMA, 14 (2018), 059, 14 pp. | DOI

[39] R. Hirota, “Nonlinear partial difference equations. III. Discrete sine-Gordon equation”, J. Phys. Soc. Japan, 43:6 (1977), 2079–2086 | DOI | MR

[40] A. Dobrogowska, G. Jakimowicz, “Factorization method applied to the second order difference equations”, Appl. Math. Lett., 74 (2017), 161–166 | DOI | MR

[41] S. Odake, R. Sasaki, “Crum's theorem for ‘discrete’ quantum mechanics”, Progr. Theor. Phys., 122:5 (2009), 1067–1079 | DOI