Algebro-geometric solutions to the lattice potential modified Kadomtsev-Petviashvili equation. (English) Zbl 1511.37090
Summary: Algebro-geometric solutions of the lattice potential modified Kadomtsev-Petviashvili (lpmKP) equation are constructed. A Darboux transformation of the Kaup-Newell spectral problem is employed to generate a Lax triad for the lpmKP equation, as well as to define commutative integrable symplectic maps which generate discrete flows of eigenfunctions. These maps share the same integrals with the finite-dimensional Hamiltonian system associated to the Kaup-Newell spectral problem. We investigate asymptotic behaviors of the Baker-Akhiezer functions and obtain their expression in terms of Riemann theta function. Finally, algebro-geometric solutions for the lpmKP equation are reconstructed from these Baker-Akhiezer functions.
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.) |
| 37K20 | Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions |
| 37K35 | Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems |
| 39A36 | Integrable difference and lattice equations; integrability tests |
Keywords:
lattice potential modified Kadomtsev-Petviashvili equation; algebro-geometric solution; Kaup-Newell spectral problem; Baker-Akhiezer functionReferences:
| [1] | Adler, V. E.; Bobenko, A. I.; Suris, Y. 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 |
| [2] | Adler, V. E.; Bobenko, A. I.; Suris, Y. B., Classification of integrable discrete equations of octahedron type, Int. Math. Res. Not., 2012, 1822-1889 (2012) · Zbl 1277.39006 · doi:10.1093/imrn/rnr083 |
| [3] | Babelon, O.; Bernard, D.; Talon, M., Introduction to Classical Integrable Systems (2003), Cambridge: Cambridge University Press, Cambridge · Zbl 1045.37033 |
| [4] | Belokolos, E. D.; Bobenko, A. I.; Enolskij, V. Z.; Its, A. R.; Matveev, V. B., Algebro-Geometric Approach to Nonlinear Integrable Equations (1994), Berlin: Spinger, Berlin · Zbl 0809.35001 |
| [5] | Bobenko, A.; Suris, Y., Integrable systems on quad-graphs, Int. Math. Res. Not., 2002, 573-611 (2002) · Zbl 1004.37053 · doi:10.1155/s1073792802110075 |
| [6] | Bruschi, M.; Ragnisco, O.; Santini, P. M.; Tu, G-Z, Integrable symplectic maps, Physica D, 49, 273-294 (1991) · Zbl 0734.58023 · doi:10.1016/0167-2789(91)90149-4 |
| [7] | Butler, S., Multidimensional inverse scattering of integrable lattice equations, Nonlinearity, 25, 1613-1634 (2012) · Zbl 1251.39003 · doi:10.1088/0951-7715/25/6/1613 |
| [8] | Cao, C., Nonlinearization of the Lax system for AKNS hierarchy, Sci. China A, 33, 528-536 (1990) · Zbl 0714.58026 |
| [9] | Cao, C.; Xu, X., A finite genus solution of the H1 model, J. Phys. A: Math. Theor., 45 (2012) · Zbl 1234.35216 · doi:10.1088/1751-8113/45/5/055213 |
| [10] | Cao, C.; Xu, X.; Zhang, D-j; Nijhoff, F. W.; Shi, Y.; Zhang, D. J., On the lattice potential KP equation, Asymptotic, Algebraic and Geometric Aspects of Integrable Systems (2020), Berlin: Springer, Berlin |
| [11] | Cao, C.; Yang, X., A (2 + 1)-dimensional derivative Toda equation in the context of the Kaup-Newell spectral problem, J. Phys. A: Math. Theor., 41 (2008) · Zbl 1151.35092 · doi:10.1088/1751-8113/41/2/025203 |
| [12] | Cao, C.; Zhang, G., A finite genus solution of the Hirota equation via integrable symplectic maps, J. Phys. A: Math. Theor., 45 (2012) · Zbl 1248.37062 · doi:10.1088/1751-8113/45/9/095203 |
| [13] | Cao, C.; Zhang, G., Integrable symplectic maps associated with the ZS-AKNS spectral problem, J. Phys. A: Math. Theor., 45 (2012) · Zbl 1387.37069 · doi:10.1088/1751-8113/45/26/265201 |
| [14] | Cao, C-W; Zhang, G-Y, Lax pairs for discrete integrable equations via Darboux transformations, Chin. Phys. Lett., 29 (2012) · doi:10.1088/0256-307x/29/5/050202 |
| [15] | Chen, D-y, k-constraint for the modified Kadomtsev-Petviashvili system, J. Math. Phys., 43, 1956-1965 (2002) · Zbl 1059.37050 · doi:10.1063/1.1446665 |
| [16] | Chen, K.; Deng, X.; Zhang, D-j, Symmetry constraint of the differential-difference KP hierarchy and a second discretization of the ZS-AKNS system, J. Nonlinear Math. Phys., 24, 18-35 (2017) · Zbl 1420.35288 · doi:10.1080/14029251.2017.1418051 |
| [17] | Chen, K.; Zhang, C.; Zhang, D-j, Squared eigenfunction symmetry of the D_ΔmKP hierarchy and its constraint, Stud. Appl. Math., 147, 752-791 (2021) · Zbl 1487.35340 · doi:10.1111/sapm.12399 |
| [18] | Cheng, Y.; Li, Y-s, The constraint of the Kadomtsev-Petviashvili equation and its special solutions, Phys. Lett. A, 157, 22-26 (1991) · doi:10.1016/0375-9601(91)90403-u |
| [19] | Dickson, R.; Gesztesy, F.; Unterkofler, K., Algebro-geometric solutions of the Boussinesq hierarchy, Rev. Math. Phys., 11, 823-879 (1999) · Zbl 0971.35065 · doi:10.1142/s0129055x9900026x |
| [20] | Dubrovin, B. A., Periodic problems for the Korteweg-de Vries equation in the class of finite band potentials, Funct. Anal. Appl., 9, 215-223 (1975) · Zbl 0358.35022 · doi:10.1007/bf01078183 |
| [21] | Dubrovin, B. A.; Novikov, S. P., Periodic and conditionally periodic analogs of the many soliton solutions of the Korteweg-de Vries equation, Sov. Phys. - JETP, 40, 1058-1063 (1975) |
| [22] | Faddeev, L. D.; Takhtajan, L. A., Hamiltonian Methods in the Theory of Solitons (1987), Berlin: Springer, Berlin · Zbl 0632.58004 |
| [23] | Farkas, H. M.; Kra, I., Riemann Surfaces (1992), New York: Springer, New York · Zbl 0764.30001 |
| [24] | Geng, X.; Wu, L.; He, G., Algebro-geometric constructions of the modified Boussinesq flows and quasi-periodic solutions, Physica D, 240, 1262-1288 (2011) · Zbl 1223.37093 · doi:10.1016/j.physd.2011.04.020 |
| [25] | Geng, X.; Wu, L.; He, G., Quasi-periodic solutions of the Kaup-Kupershmidt hierarchy, J. Nonlinear Sci., 23, 527-555 (2013) · Zbl 1309.37070 · doi:10.1007/s00332-012-9160-3 |
| [26] | Geng, X.; Zhai, Y.; Dai, H. H., Algebro-geometric solutions of the coupled modified Korteweg-de Vries hierarchy, Adv. Math., 263, 123-153 (2014) · Zbl 1304.37046 · doi:10.1016/j.aim.2014.06.013 |
| [27] | Gerdjikov, V. S.; Vilasi, G.; Yanovski, A. B., Integrable Hamiltonian Hierarchies (2008), Berlin: Springer, Berlin · Zbl 1167.37001 |
| [28] | Gesztesy, F.; Holden, H., Soliton Equations and Their Algebro-Geometric Solutions: (1 + 1)-Dimensional Continuous Models (2003), Cambridge: Cambridge University Press, Cambridge · Zbl 1061.37056 |
| [29] | Gesztesy, F.; Holden, H.; Michor, J.; Teschl, G., Soliton Equations and Their Algebro-Geometric Solutions: (1 + 1)-Dimensional Discrete Models, vol 2 (2009), Cambridge: Cambridge University Press, Cambridge |
| [30] | Griffiths, J. P.; Harris, J., Principles of Algebraic Geometry (1978), New York: Wiley, New York · Zbl 0408.14001 |
| [31] | Hietarinta, J.; Joshi, N.; Nijhoff, F. W., Discrete Systems and Integrability (2016), Cambridge: Cambridge University Press, Cambridge · Zbl 1362.37130 |
| [32] | Hietarinta, J.; Zhang, D-j, Soliton solutions for ABS lattice equations: II. Casoratians and bilinearization, J. Phys. A: Math. Theor., 42 (2009) · Zbl 1184.35276 · doi:10.1088/1751-8113/42/40/404006 |
| [33] | Hietarinta, J.; Zhang, D-j; Euler, N.; Zhang, D. J., Discrete Boussinesq-type equations, Nonlinear Systems and Their Remarkable Mathematical Structures, vol 3 (2021), Boca Raton, FL: CRC Press, Boca Raton, FL |
| [34] | Its, A. R.; Matveev, V. B., Hill operators with a finite number of lacunae, Funct. Anal. Appl., 9, 65-66 (1975) · Zbl 0318.34038 · doi:10.1007/bf01078185 |
| [35] | Its, A. R.; Matveev, V. B., Schrödinger operators with the finite-band spectrum and the N-soliton solutions of the Korteweg-de Vries equation, Theor. Math. Phys., 23, 343-355 (1975) · doi:10.1007/bf01038218 |
| [36] | Kaup, D. J.; Newell, A. C., On the Coleman correspondence and the solution of the massive Thirring model, Lett. Nuovo Cimento, 20, 325-331 (1977) · doi:10.1007/bf02783605 |
| [37] | Kaup, D. J.; Newell, A. C., An exact solution for a derivative nonlinear Schrödinger equation, J. Math. Phys., 19, 798-801 (1978) · Zbl 0383.35015 · doi:10.1063/1.523737 |
| [38] | Khanizadeh, F.; Mikhailov, A. V.; Wang, J. P., Darboux transformations and recursion operators for differential-difference equations, Theor. Math. Phys, 177, 1606-1654 (2013) · Zbl 1301.37041 · doi:10.1007/s11232-013-0124-z |
| [39] | Konopelchenko, B.; Strampp, W., New reductions of the Kadomtsev-Petviashvili and two-dimensional Toda lattice hierarchies via symmetry constraints, J. Math. Phys., 33, 3676-3686 (1992) · Zbl 0762.35103 · doi:10.1063/1.529862 |
| [40] | Konopelchenko, B.; Sidorenko, J.; Strampp, W., (1 + 1)-dimensional integrable systems as symmetry constraints of (2 + 1)-dimensional systems, Phys. Lett. A, 157, 17-21 (1991) · doi:10.1016/0375-9601(91)90402-t |
| [41] | Konstantinou-Rizos, S.; Mikhailov, A. V.; Xenitidis, P., Reduction groups and related integrable difference systems of nonlinear Schrödinger type, J. Math. Phys., 56 (2015) · Zbl 1328.35213 · doi:10.1063/1.4928048 |
| [42] | Krichever, I. M., Methods of algebraic geometry in the theory of non-linear equations, Russ. Math. Surv., 32, 185-213 (1977) · Zbl 0386.35002 · doi:10.1070/rm1977v032n06abeh003862 |
| [43] | Levi, D., Nonlinear differential-difference equations as Bäcklund transformations, J. Phys. A: Math. Gen., 14, 1083-1098 (1981) · Zbl 0465.35081 · doi:10.1088/0305-4470/14/5/028 |
| [44] | Levi, D.; Benguria, R., Bäcklund transformations and nonlinear differential-difference equations, Proc. Natl Acad. Sci. USA, 77, 5025-5027 (1980) · Zbl 0453.35072 · doi:10.1073/pnas.77.9.5025 |
| [45] | Matveev, V. B., 30 years of finite-gap integration theory, Phil. Trans. R. Soc. A, 366, 837-875 (2008) · Zbl 1153.37419 · doi:10.1098/rsta.2007.2055 |
| [46] | Miwa, T., On Hirota’s difference equations, Proc. Japan Acad. A, 58, 9-12 (1982) · Zbl 0508.39009 · doi:10.3792/pjaa.58.9 |
| [47] | Mumford, D., Tata Lectures on Theta I (1983), Boston, MA: Birkhäuser, Boston, MA · Zbl 0509.14049 |
| [48] | Nijhoff, F. W., Lax pair for the Adler (lattice Krichever-Novikov) system, Phys. Lett. A, 297, 49-58 (2002) · Zbl 0994.35105 · doi:10.1016/s0375-9601(02)00287-6 |
| [49] | Nijhoff, F. W.; 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 |
| [50] | Nijhoff, F. W.; Capel, H.; Wiersma, G.; Martini, R., Integrable lattice systems in two and three dimensions, Geometric Aspects of the Einstein Equations and Integrable Systems (1985), Berlin: Springer, Berlin |
| [51] | Nijhoff, F. W.; Capel, H. W.; Wiersma, G. L.; Quispel, G. R W., Bäcklund transformations and three-dimensional lattice equations, Phys. Lett. A, 105, 267-272 (1984) · doi:10.1016/0375-9601(84)90994-0 |
| [52] | Nijhoff, F. W.; Walker, A. J., The discrete and continuous Painlevé VI hierarchy and the Garnier systems, Glasg. Math. J., 43, 109-123 (2001) · Zbl 0990.39015 · doi:10.1017/s0017089501000106 |
| [53] | Nimmo, J. J C., Darboux transformations and the discrete KP equation, J. Phys. A: Math. Gen., 30, 8693-8704 (1997) · Zbl 0926.39008 · doi:10.1088/0305-4470/30/24/028 |
| [54] | Novikov, S. P., The periodic problem for the Korteweg-de vries equation, Funct. Anal. Appl., 8, 236-246 (1974) · Zbl 0299.35017 · doi:10.1007/BF01075697 |
| [55] | Tamizhmani, K. M.; Kanaga Vel, S., Gauge equivalence and l-reductions of the differential-difference KP equation, Chaos Solitons Fractals, 11, 137-143 (2000) · Zbl 1160.37400 · doi:10.1016/s0960-0779(98)00277-x |
| [56] | Toda, M., Theory of Nonlinear Lattices (1981), Berlin: Springer, Berlin · Zbl 0465.70014 |
| [57] | Veselov, A. P., Integrable maps, Russ. Math. Surv., 46, 3-45 (1991) · Zbl 0746.58033 · doi:10.1070/rm1991v046n05abeh002856 |
| [58] | Veselov, A. P.; Zakharov, V. E., What is an integrable mapping?, What Is Integrability? (1991), Berlin: Springer, Berlin · Zbl 0724.00014 |
| [59] | Wu, Y.; Geng, X., A new hierarchy of integrable differential-difference equations and Darboux transformation, J. Phys. A: Math. Gen., 31, L677-L684 (1998) · Zbl 0931.35190 · doi:10.1088/0305-4470/31/38/004 |
| [60] | Xu, X.; Cao, C.; Nijhoff, F. W., Algebro-geometric integration of the Q1 lattice equation via nonlinear integrable symplectic maps, Nonlinearity, 34, 2897-2918 (2021) · Zbl 1475.37083 · doi:10.1088/1361-6544/abddca |
| [61] | Xu, X.; Cao, C.; Zhang, G., Finite genus solutions to the lattice Schwarzian Korteweg-de Vries equation, J. Nonlinear Math. Phys., 27, 633-646 (2020) · Zbl 1441.37076 · doi:10.1080/14029251.2020.1819608 |
| [62] | Xu, X.; Jiang, M.; Nijhoff, F. W., Integrabe symplectic maps associated with discrete Korteweg-de Vries-type equations, Stud. Appl. Math., 146, 233-278 (2021) · Zbl 1476.37080 · doi:10.1111/sapm.12346 |
| [63] | 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 |
| [64] | Ru-Guang, Z.; Jie, C., Two hierarchies of new differential-difference equations related to the Darboux transformations of the Kaup-Newell hierarchy, Commun. Theor. Phys., 63, 1-6 (2015) · Zbl 1305.37039 · doi:10.1088/0253-6102/63/1/01 |
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