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A numerical study of the 3-periodic wave solutions to Toda-type equations. (English) Zbl 1490.65179

Summary: In this paper, we present an efficient numerical scheme to calculate \(N\)-periodic wave solutions to the Toda-type equations. The starting point is the algebraic condition for having \(N\)-periodic wave solutions proposed by Akira Nakamura. The basic idea is to formulate the condition as a nonlinear least square problem and then use the Gauss-Newton method to solve it. By use of this numerical scheme, we calculate the 3-periodic wave solutions to some discrete integrable equations such as the Toda lattice equation, the Lotka-Volterra equation, the differential-difference KP equation and so on.

MSC:

65M22 Numerical solution of discretized equations for initial value and initial-boundary value problems involving PDEs
65H10 Numerical computation of solutions to systems of equations
65Q10 Numerical methods for difference equations
39A23 Periodic solutions of difference equations
39A36 Integrable difference and lattice equations; integrability tests
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[1] B.A. Dubrovin, Theta functions and nonlinear equations, Russ. Math. Surv. 36 (1981) 11-92. · Zbl 0549.58038
[2] A. Nakamura, A direct method of calculating periodic wave solutions to nonlinear evolution equations. I. Exact two-periodic wave solution, J. Phys. Soc. Japan, 47(5) (1979) 1701-1705. · Zbl 1334.35006
[3] S.P. Novikov, The periodic problem for the Korteweg-de Vries equation, Funct. Anal. Appl. 8 (1974) 236-246. · Zbl 0299.35017
[4] B.A. Dubrovin, Periodic problems for the Korteweg-deVries equation in the class of finite band potentials, Funct. Anal. Appl. 9 (1975) 215-223. · Zbl 0358.35022
[5] B.A. Dubrovin, S.P. Novikov, Periodic and conditionally periodic analogues of the many-soliton solutions of the Kortweg-deVries equation, Sov. Phys. JETP. 40 (1975) 1058-1063.
[6] P.D. Lax, Periodic solutions of the KdV equation, Comm. Pure. Appl. Math. 28 (1975) 141-188. · Zbl 0295.35004
[7] A.R. Its, V.B. Matveev, Hill’s operator with finitely many gaps, Funct. Anal. Appl. 9 (1975) 65-66. · Zbl 0318.34038
[8] H.P. McKean and P. van Moerbeke, The spectrum of Hill’s equation, Inv. Math. 30(3) (1975) 217-274. · Zbl 0319.34024
[9] Y.C. Ma and M.J. Ablowitz, The periodic cubic Schrödinger equation, Stud Appl. Math. 65 (1981) 113-158. · Zbl 0493.35032
[10] M.G. Forest and D. W. McLaughlin, Spectral theory for the periodic sine-Gordon equation: A concrete viewpoint, J. Math. Phys. 23 (1982) 1248-1277. · Zbl 0498.35072
[11] E. Date and S. Tanaka, Periodic multi-soliton solutions of Korteweg-de Vries equation and Toda lattice, Suppl. Prog. Theor. Phys. 59 (1976) 107-126.
[12] I.M. Krichever, Algebraic-geometric construction of the Zaharov-Sabat equations and their periodic solutions, Dokl. Akad. Nauk SSSR, 227 (1976) 394-397. · Zbl 0361.35007
[13] E.D. Belokolos, A.I. Bobenko, V.Z. Enol’skii, A.R. Its, V.B. Matveev, Algebro-Geometric Ap-proach to Nonlinear Integrable Equations, Springer-Verlag, Berlin, 1994. · Zbl 0809.35001
[14] C.W. Cao, Y.T. Wu and X.G. Geng, On quasi-periodic solutions of the 2+1 dimensional Caudrey-Dodd-Gibbon-Kotera-Sawada equation, Phys. Lett. A, 256(1) (1999) 59-65.
[15] X.G. Geng, L.H. Wu and G.L. He, Quasi-periodic solutions of the Kaup-Kupershmidt hier-archy, J. Nonlinear Sci. 23(4) (2013) 527-555. · Zbl 1309.37070
[16] X.G. Geng, L.H. Wu and G.L. He, Algebro-geometric constructions of the modified Boussi-nesq flows and quasi-periodic solutions, Phys. D, 240(16) (2011) 1262-1288. · Zbl 1223.37093
[17] X.G. Geng, L.H. Wu and G.L. He, Quasi-Periodic Solutions of Nonlinear Evolution Equa-tions Associated with a 3×3 Matrix Spectral Problem, Stud. Appl. Math. 127(2) (2011) 107-140. · Zbl 1243.37058
[18] A. Nakamura, A Direct Method of Calculating Periodic Wave Solutions to Nonlinear Evo-lution Equations. II. Exact One-and Two-Periodic Wave Solution of the Coupled Bilinear Equations, J. Phys. Soc. Japan, 48(4) (1980) 1365-1370. · Zbl 1334.35250
[19] R. Hirota and M. Ito, A Direct Approach to Multi-Periodic Wave Solutions to Nonlinear Evolution Equations, J. Phys. Soc. Japan, 50(1) (1981) 338-342.
[20] W.X. Ma and E.G. Fan, Linear superposition principle applying to Hirota bilinear equations, Comput. Math. Appl. 61(4) (2011) 950-959. · Zbl 1217.35164
[21] L. Luo and E.G. Fan, Bilinear approach to the quasi-periodic wave solutions of Modified Nizhnik-Novikov-Vesselov equation in (2+1) dimensions, Phys. Lett. A, 374(30) (2010) 3001-3006. · Zbl 1237.35140
[22] E.G. Fan, K.W. Chow and J.H. Li, On doubly periodic standing wave solutions of the coupled Higgs field equation. Stud. Appl. Math. 128(1) (2012) 86-105. · Zbl 1241.35165
[23] T. Trogdon and B. Deconinck, Numerical computation of the finite-genus solutions of the Korteweg-de Vries equation via Riemann-Hilbert problems, Appl. Math. Lett. 26 (2013) 5-9. · Zbl 1255.65177
[24] T. Trogdon and B. Deconinck, A numerical dressing method for the nonlinear superposition of solutions of the KdV equation, Nonlinearity, 27 (2014) 67-86. · Zbl 1302.65234
[25] J. Frauendiener and C. Klein, Hyperelliptic theta-functions and spectral methods, J. Comput. Appl. Math. 167 (2004) 193-218. · Zbl 1052.65107
[26] J. Frauendiener and C. Klein, Hyperelliptic theta-Functions and spectral methods: KdV and KP Solutions, Lett. Math. Phys. 76 (2006) 249-267. · Zbl 1127.14032
[27] C. Kalla and C. Klein, On the numerical evaluation of algebro-geometric solutions to inte-grable equations, Nonlinearity, 25 (2012) 569-596. · Zbl 1251.37067
[28] R. Hirota, The direct method in soliton theory, Cambridge University Press, 2004. · Zbl 1099.35111
[29] Y.N. Zhang, X.B. Hu and J.Q. Sun, A numerical study of the 3-periodic wave solutions to KdV-type equations, J. Comput. Phys. 355 (2018) 566-581. · Zbl 1380.65093
[30] M. Ito, An extension of nonlinear evolution equations of the K-dV (mK-dV) type to higher orders, J. Phys. Soc. Japan, 49 (1980) 771-778. · Zbl 1334.35282
[31] J. Hietarinta, A search for bilinear equations passing Hirota’s three-soliton condition. I. KdV-type bilinear equations, J. Math. Phys. 28 (1987) 1732-1742. · Zbl 0641.35073
[32] M. Toda, Vibration of a chain with nonlinear interaction, J. Phys. Soc. Japan, 22 (1967) 431-436.
[33] M. Toda, Wave Propagation in anharmonic lattices, J. Phys. Soc. Japan, 23 (1967) 501-506.
[34] M. Toda, Waves in nonlinear lattice, Prog. Theor. Phys. Suppl. 45 (1970) 174-200.
[35] M. Toda, Theory of Nonlinear Lattices (Springer Series in Solid-State Sciences), Springer, 1989. · Zbl 0694.70001
[36] Y. Kodama, S. Matsutani and E. Previato, Quasi-periodic and periodic solutions of the Toda lattice via the hyperelliptic sigma function, Ann. Inst. Fourier (Grenoble) 63(2) (2013) 655-688. · Zbl 1279.14044
[37] J. Wei, X.G. Geng, Zeng X. Quasi-periodic solutions to the hierarchy of four-component Toda lattices, J. Geom. Phys. 106 (2016) 26-41. · Zbl 1339.37061
[38] C.W. Cao, G.Y. Zhang, A finite genus solution of the Hirota equation via integrable symplec-tic maps, J. Phys. A, 45(9) (2012) 095203. · Zbl 1248.37062
[39] C.W. Cao, X.X. Xu, A finite genus solution of the H1 model, J. Phys. A, 45(5) (2012) 055213. · Zbl 1234.35216
[40] J.Y. Zhu, X.G. Geng, Algebro-geometric constructions of the (2+1)-dimensional differential-difference equation, Phys. Lett. A, 368(6) (2007) 464-469. · Zbl 1209.37085
[41] X.G. Geng, D. Gong, Quasi-periodic solutions of the discrete mKdV hierarchy, Int. J. Geom. Methods Mod. Phys. 10(03) (2013) 1250094[37pp]. · Zbl 1282.35331
[42] P. Zhao, E.G. Fan, A unified construction for the algebro-geometric quasiperiodic solutions of the Lotka-Volterra and relativistic Lotka-Volterra hierarchy, J. Math. Phys. 56(4) (2015) 043501. · Zbl 1323.37041
[43] E.G. Fan, K.W. Chow, On the periodic solutions for both nonlinear differential and difference equations: a unified approach, Phys. Lett. A, 374(35) (2010) 3629-3634. · Zbl 1238.35060
[44] A. Björck, Numerical methods for least squares problems, SIAM, 1996. · Zbl 0847.65023
[45] R. Hirota, Nonlinear partial difference equations. II. Discrete time Toda equation, J. Phys. Soc. Japan, 43(6) (1977) 2074-2078. · Zbl 1334.39014
[46] A.J. Lotka, Contribution to the theory of periodic reactions, J. Phys. Chem. 14(3) (1910) 271-274.
[47] R. Hirota, Nonlinear partial difference equations. I. A difference analogue of the Korteweg-de Vries equation, J. Phys. Soc. Japan, 43(4) (1977) 1424-1433. · Zbl 1334.39013
[48] R. Hirota, S. Tsujimoto, Conserved Quantities of a Class of Nonlinear Difference-Difference Equations, J. Phys. Soc. Japan, 64(9) (1995) 3125-3127. · Zbl 0972.39501
[49] X.B. Hu, P.A. Clarkson, R. Bullough, New integrable differential-difference systems, J. Phys. A: Math. Gen. 30(20) (1997) L669-L676. · Zbl 0924.35157
[50] K. Narita, Soliton solution to extended Volterra equation, J. Phys. Soc. Japan, 51(5) (1982) 1682-1685.
[51] E. Date, M. Jimbo, T. Miwa, Method for generating discrete soliton equations. V, J. Phys. Soc. Japan, 52(3) (1983) 766-771. · Zbl 0571.35105
[52] E. Date, M. Jimbo, T. Miwa, Method for generating discrete soliton equations. II, J. Phys. Soc. Japan, 51(12) (1982) 4125-4131.
[53] T. Tamizhmani, S. Kanaga Vel, K.M. Tamizhmani, Wronskian and rational solutions of the differential-difference KP equation, J. Phys. A: Math. Gen. 31(37) (1998) 7627-7633. · Zbl 0931.35154
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