Under a constraint between the potentials and eigenfunctions, the nonlinearization of the Lax pairs associated with the discrete hierarchy of a generalization of the Toda lattice equation is proposed, which leads to a new symplectic map and a class of finite-dimensional Hamiltonian systems. The generating function of the integrals of motion is presented, by which the symplectic map and these finite-dimensional Hamiltonian systems are further proved to be completely integrable in the Liouville sense. Finally, the representation of solutions for a lattice equation in the discrete hierarchy is obtained.
References
M. J. Ablowitz and J. F. Ladik, “Nonlinear differential-difference equations,” Journal of Mathematical Physics, vol. 16, pp. 598–603, 1975. MR0377223 0296.34062 10.1063/1.522558 M. J. Ablowitz and J. F. Ladik, “Nonlinear differential-difference equations,” Journal of Mathematical Physics, vol. 16, pp. 598–603, 1975. MR0377223 0296.34062 10.1063/1.522558
K. Daikoku, Y. Mizushima, and T. Tamama, “Computer experiments on new lattice solitons propagating in Volterra's system,” Japanese Journal of Applied Physics, vol. 14, no. 3, pp. 367–376, 1975. K. Daikoku, Y. Mizushima, and T. Tamama, “Computer experiments on new lattice solitons propagating in Volterra's system,” Japanese Journal of Applied Physics, vol. 14, no. 3, pp. 367–376, 1975.
M. Wadati, K. Konno, and Y. H. Ichikawa, “A generalization of inverse scattering method,” Journal of the Physical Society of Japan, vol. 46, no. 6, pp. 1965–1966, 1979. MR538119 10.1143/JPSJ.46.1965 M. Wadati, K. Konno, and Y. H. Ichikawa, “A generalization of inverse scattering method,” Journal of the Physical Society of Japan, vol. 46, no. 6, pp. 1965–1966, 1979. MR538119 10.1143/JPSJ.46.1965
X. Geng and C. Cao, “Quasi-periodic solutions of the $2+1$ dimensional modified Korteweg-de Vries equation,” Physics Letters A, vol. 261, no. 5-6, pp. 289–296, 1999. MR1724063 0937.35155 10.1016/S0375-9601(99)00553-8 X. Geng and C. Cao, “Quasi-periodic solutions of the $2+1$ dimensional modified Korteweg-de Vries equation,” Physics Letters A, vol. 261, no. 5-6, pp. 289–296, 1999. MR1724063 0937.35155 10.1016/S0375-9601(99)00553-8
X. Geng, H. H. Dai, and C. W. Cao, “Algebro-geometric constructions of the discrete Ablowitz-Ladik flows and applications,” Journal of Mathematical Physics, vol. 44, no. 10, pp. 4573–4588, 2003. MR2008934 1062.37092 10.1063/1.1605820 X. Geng, H. H. Dai, and C. W. Cao, “Algebro-geometric constructions of the discrete Ablowitz-Ladik flows and applications,” Journal of Mathematical Physics, vol. 44, no. 10, pp. 4573–4588, 2003. MR2008934 1062.37092 10.1063/1.1605820
X. Geng and T. Su, “Decomposition and straightening out of the discrete Kaup-Newell flows,” International Journal of Modern Physics B: Condensed Matter Physics, Statistical Physics, Applied Physics, vol. 25, no. 32, pp. 4513–4531, 2011. MR2875673 1247.37057 10.1142/S021797921105922X X. Geng and T. Su, “Decomposition and straightening out of the discrete Kaup-Newell flows,” International Journal of Modern Physics B: Condensed Matter Physics, Statistical Physics, Applied Physics, vol. 25, no. 32, pp. 4513–4531, 2011. MR2875673 1247.37057 10.1142/S021797921105922X
M. J. Ablowitz and J. F. Ladik, “A nonlinear difference scheme and inverse scattering,” vol. 55, no. 3, pp. 213–229, 1976. MR0471341 0338.35002 M. J. Ablowitz and J. F. Ladik, “A nonlinear difference scheme and inverse scattering,” vol. 55, no. 3, pp. 213–229, 1976. MR0471341 0338.35002
S. J. Orfanidis, “Sine-Gordon equation and nonlinear $\sigma $ model on a lattice,” Physical Review D: Particles and Fields, vol. 18, no. 10, pp. 3828–3832, 1978. MR515266 10.1103/PhysRevD.18.3828 S. J. Orfanidis, “Sine-Gordon equation and nonlinear $\sigma $ model on a lattice,” Physical Review D: Particles and Fields, vol. 18, no. 10, pp. 3828–3832, 1978. MR515266 10.1103/PhysRevD.18.3828
S. V. Manakov, “Complete integrability and stochastization of discrete dynamical systems,” Journal of Experimental and Theoretical Physics, vol. 40, no. 2, pp. 269–274, 1975. MR389107 S. V. Manakov, “Complete integrability and stochastization of discrete dynamical systems,” Journal of Experimental and Theoretical Physics, vol. 40, no. 2, pp. 269–274, 1975. MR389107
X. Geng, F. Li, and B. Xue, “A generalization of Toda lattices and their bi-Hamiltonian structures,” Modern Physics Letters B: Condensed Matter Physics, Statistical Physics, Applied Physics, vol. 26, no. 13, Article ID 1250078, p. 7, 2012. MR2916767 1263.82009 10.1142/S0217984912500789 X. Geng, F. Li, and B. Xue, “A generalization of Toda lattices and their bi-Hamiltonian structures,” Modern Physics Letters B: Condensed Matter Physics, Statistical Physics, Applied Physics, vol. 26, no. 13, Article ID 1250078, p. 7, 2012. MR2916767 1263.82009 10.1142/S0217984912500789
W.-X. Ma, “A soliton hierarchy associated with (3, $\mathbb{R}$),” Applied Mathematics and Computation, vol. 220, pp. 117–122, 2013. MR3091835 10.1016/j.amc.2013.04.062 W.-X. Ma, “A soliton hierarchy associated with (3, $\mathbb{R}$),” Applied Mathematics and Computation, vol. 220, pp. 117–122, 2013. MR3091835 10.1016/j.amc.2013.04.062
C. W. Cao, “Nonlinearization of the Lax system for AKNS hierarchy,” Science in China A, vol. 33, no. 5, pp. 528–536, 1990. MR1070539 0714.58026 C. W. Cao, “Nonlinearization of the Lax system for AKNS hierarchy,” Science in China A, vol. 33, no. 5, pp. 528–536, 1990. MR1070539 0714.58026
X. G. Geng, “Finite-dimensional discrete systems and integrable systems through nonlinearization of the discrete eigenvalue problem,” Journal of Mathematical Physics, vol. 34, no. 2, pp. 805–817, 1993. MR1201037 0784.58048 10.1063/1.530418 X. G. Geng, “Finite-dimensional discrete systems and integrable systems through nonlinearization of the discrete eigenvalue problem,” Journal of Mathematical Physics, vol. 34, no. 2, pp. 805–817, 1993. MR1201037 0784.58048 10.1063/1.530418
C. W. Cao and X. G. Geng, “C. Neumann and Bargmann systems associated with the coupled KdV soliton hierarchy,” Journal of Physics A: Mathematical and General, vol. 23, no. 18, pp. 4117–4125, 1990. MR1076724 0719.35082 10.1088/0305-4470/23/18/017 C. W. Cao and X. G. Geng, “C. Neumann and Bargmann systems associated with the coupled KdV soliton hierarchy,” Journal of Physics A: Mathematical and General, vol. 23, no. 18, pp. 4117–4125, 1990. MR1076724 0719.35082 10.1088/0305-4470/23/18/017
X. Geng and H. H. Dai, “Nonlinearization of the Lax pairs for discrete Ablowitz-Ladik hierarchy,” Journal of Mathematical Analysis and Applications, vol. 327, no. 2, pp. 829–853, 2007. MR2279968 1111.39011 10.1016/j.jmaa.2006.04.033 X. Geng and H. H. Dai, “Nonlinearization of the Lax pairs for discrete Ablowitz-Ladik hierarchy,” Journal of Mathematical Analysis and Applications, vol. 327, no. 2, pp. 829–853, 2007. MR2279968 1111.39011 10.1016/j.jmaa.2006.04.033
D. Du, “Complex form, reduction and Lie-Poisson structure for the nonlinearized spectral problem of the Heisenberg hierarchy,” Physica A: Statistical Mechanics and its Applications, vol. 303, no. 3-4, pp. 439–456, 2002. MR1917637 0979.82020 10.1016/S0378-4371(01)00562-3 D. Du, “Complex form, reduction and Lie-Poisson structure for the nonlinearized spectral problem of the Heisenberg hierarchy,” Physica A: Statistical Mechanics and its Applications, vol. 303, no. 3-4, pp. 439–456, 2002. MR1917637 0979.82020 10.1016/S0378-4371(01)00562-3
D. Du and C. Cao, “The Lie-Poisson representation of the nonlinearized eigenvalue problem of the Kac-van Moerbeke hierarchy,” Physics Letters A, vol. 278, no. 4, pp. 209–224, 2001. MR1827069 0972.37050 10.1016/S0375-9601(00)00776-3 D. Du and C. Cao, “The Lie-Poisson representation of the nonlinearized eigenvalue problem of the Kac-van Moerbeke hierarchy,” Physics Letters A, vol. 278, no. 4, pp. 209–224, 2001. MR1827069 0972.37050 10.1016/S0375-9601(00)00776-3
W. X. Ma and W. Strampp, “An explicit symmetry constraint for the Lax pairs and the adjoint Lax pairs of AKNS systems,” Physics Letters A, vol. 185, no. 3, pp. 277–286, 1994. MR1261401 0941.37530 10.1016/0375-9601(94)90616-5 W. X. Ma and W. Strampp, “An explicit symmetry constraint for the Lax pairs and the adjoint Lax pairs of AKNS systems,” Physics Letters A, vol. 185, no. 3, pp. 277–286, 1994. MR1261401 0941.37530 10.1016/0375-9601(94)90616-5
X. G. Geng, “The Hamiltonian structure and new finite-dimensional integrable system associated with Harry-Dym type equations,” Physics Letters A, vol. 194, no. 1-2, pp. 44–48, 1994. MR1299513 0961.35506 10.1016/0375-9601(94)00705-T X. G. Geng, “The Hamiltonian structure and new finite-dimensional integrable system associated with Harry-Dym type equations,” Physics Letters A, vol. 194, no. 1-2, pp. 44–48, 1994. MR1299513 0961.35506 10.1016/0375-9601(94)00705-T
Y. Wu and X. Geng, “A new integrable symplectic map associated with lattice soliton equations,” Journal of Mathematical Physics, vol. 37, no. 5, pp. 2338–2345, 1996. MR1385832 0864.58028 10.1063/1.531512 Y. Wu and X. Geng, “A new integrable symplectic map associated with lattice soliton equations,” Journal of Mathematical Physics, vol. 37, no. 5, pp. 2338–2345, 1996. MR1385832 0864.58028 10.1063/1.531512
B. Konopelchenko, J. Sidorenko, and W. Strampp, “$(1+1)$-dimensional integrable systems as symmetry constraints of $(2+1)$-dimensional systems,” Physics Letters A, vol. 157, no. 1, pp. 17–21, 1991. MR1117170 10.1016/0375-9601(91)90402-T B. Konopelchenko, J. Sidorenko, and W. Strampp, “$(1+1)$-dimensional integrable systems as symmetry constraints of $(2+1)$-dimensional systems,” Physics Letters A, vol. 157, no. 1, pp. 17–21, 1991. MR1117170 10.1016/0375-9601(91)90402-T
W.-X. Ma and Z. Zhou, “Binary symmetry constraints of $N$-wave interaction equations in $1+1$ and $2+1$ dimensions,” Journal of Mathematical Physics, vol. 42, no. 9, pp. 4345–4382, 2001. MR1852632 10.1063/1.1388898 W.-X. Ma and Z. Zhou, “Binary symmetry constraints of $N$-wave interaction equations in $1+1$ and $2+1$ dimensions,” Journal of Mathematical Physics, vol. 42, no. 9, pp. 4345–4382, 2001. MR1852632 10.1063/1.1388898
W.-X. Ma and R. Zhou, “Adjoint symmetry constraints leading to binary nonlinearization,” Journal of Nonlinear Mathematical Physics, vol. 9, supplement 1, pp. 106–126, 2002. MR1900189 10.2991/jnmp.2002.9.s1.10 W.-X. Ma and R. Zhou, “Adjoint symmetry constraints leading to binary nonlinearization,” Journal of Nonlinear Mathematical Physics, vol. 9, supplement 1, pp. 106–126, 2002. MR1900189 10.2991/jnmp.2002.9.s1.10
W.-X. Ma and X. Geng, “Bäcklund transformations of soliton systems from symmetry constraints,” in CRM Proceedings and Lecture Notes, vol. 29, pp. 313–323, 2001. \endinput MR1870929 1010.37048 W.-X. Ma and X. Geng, “Bäcklund transformations of soliton systems from symmetry constraints,” in CRM Proceedings and Lecture Notes, vol. 29, pp. 313–323, 2001. \endinput MR1870929 1010.37048
