American Institute of Mathematical Sciences

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December  2013, 5(4): 511-530. doi: 10.3934/jgm.2013.5.511

The Toda lattice, old and new

Carlos Tomei 1,

1. 

Departamento de Matemática, PUC-Rio, R. Mq. S. Vicente 225, Rio de Janeiro 22451-900, Brazil

Received  June 2013 Revised  October 2013 Published  December 2013

Originally a model for wave propagation on the line, the Toda lattice is a wonderful case study in mechanics and symplectic geometry. In Flaschka's variables, it becomes an evolution given by a Lax pair on the vector space of real, symmetric, tridiagonal matrices. Its very special asymptotic behavior was studied by Moser by introducing norming constants, which play the role of discrete inverse variables in analogy to the solution by inverse scattering of KdV. It is a completely integrable system on the coadjoint orbit of the upper triangular group. Recently, bidiagonal coordinates, which parameterize also non-Jacobi tridiagonal matrices, were used to reduce asymptotic questions to local theory. Larger phase spaces for the Toda lattice lead to the study of isospectral manifolds and different coadjoint orbits. Additionally, the time one map of the associated flow is computed by a familiar algorithm in numerical linear algebra.
    The text is mostly expositive and self contained, presenting alternative formulations of familiar results and applications to numerical analysis.
Keywords: QR algorithm, isospectral manifolds, Toda lattice, inverse scattering., completely integrable systems.
Mathematics Subject Classification: Primary: 65F15, 37S35; Secondary: 53D0.
Citation: Carlos Tomei. The Toda lattice, old and new. Journal of Geometric Mechanics, 2013, 5 (4) : 511-530. doi: 10.3934/jgm.2013.5.511
References:
[1]

M. Adler, On a trace functional for formal pseudo differential operators and the symplectic structure of the Korteweg-de-Vries type equations,, Invent. Math., 50 (): 219.  doi: 10.1007/BF01410079.  Google Scholar

[2]

M. Atiyah, Convexity and commuting Hamiltonians,, Bull. London Math. Soc., 14 (1982), 1.  doi: 10.1112/blms/14.1.1.  Google Scholar

[3]

M. J. Ablowitz, D. J. Kaup, A. C. Newell and H. Segur, The inverse scattering transform-Fourier analysis for nonlinear problems,, Stud. Appl. Math., 53 (1974), 249.   Google Scholar

[4]

R. Abraham and J. Marsden, Foundations of Mechanics,, Second edition, (1987).   Google Scholar

[5]

R. Beals, P. Deift and C. Tomei, Direct and Inverse Scattering on the Line,, Math. Surveys and Monographs, (1988).   Google Scholar

[6]

A. M. Bloch, H. Flaschka and T. Ratiu, A convexity theorem for isospectral manifolds of Jacobi matrices in a compact Lie algebra,, Duke Math. J., 61 (1990), 41.  doi: 10.1215/S0012-7094-90-06103-4.  Google Scholar

[7]

P. Deift, J. Demmel, L. C. Li and C. Tomei, The bidiagonal singular value decomposition and Hamiltonian mechanics,, SIAM J. Num. Anal., 28 (1991), 1463.  doi: 10.1137/0728076.  Google Scholar

[8]

P. Deift, L. C. Li, T. Nanda and C. Tomei, The Toda flow on a generic orbit is integrable,, Comm. Pure Appl. Math., 39 (1986), 183.  doi: 10.1002/cpa.3160390203.  Google Scholar

[9]

P. Deift, L. C. Li and C. Tomei, Matrix factorizations and integrable systems,, Comm. Pure Appl. Math., 42 (1989), 443.  doi: 10.1002/cpa.3160420405.  Google Scholar

[10]

P. Deift, L. C. Li and C. Tomei, Loop groups, discrete versions of some classical integrable systems, and rank 2 extensions,, Memoirs of the Amer. Math. Soc., 100 (1992).  doi: 10.1090/memo/0479.  Google Scholar

[11]

P. Deift, L. C. Li and C. Tomei, Toda flows with infinitely many variables,, J. Funct. Anal., 64 (1985), 358.  doi: 10.1016/0022-1236(85)90065-5.  Google Scholar

[12]

P. Deift, T. Nanda and C. Tomei, Ordinary differential equations for the symmetric eigenvalue problem,, SIAM J. Num. Anal., 20 (1983), 1.  doi: 10.1137/0720001.  Google Scholar

[13]

P. Deift, S. Rivera, C. Tomei and D. Watkins, A monotonicity property for Toda-type flows,, SIAM J. of Matrix Anal. and Appl., 12 (1991), 463.  doi: 10.1137/0612033.  Google Scholar

[14]

J. W. Demmel, Applied Numerical Linear Algebra,, SIAM, (1997).  doi: 10.1137/1.9781611971446.  Google Scholar

[15]

J. W. Demmel and W. Kahan, Accurate singular values of bidiagonal matrices,, SIAM J. Stat. Comput., 11 (1990), 873.  doi: 10.1137/0911052.  Google Scholar

[16]

H. Flaschka, The Toda lattice. I. Existence of integrals,, Phys. Rev. B (3), 9 (1974), 1924.  doi: 10.1103/PhysRevB.9.1924.  Google Scholar

[17]

D. Fried, The cohomology of an isospectral flow,, Proc. Amer. Math. Soc., 98 (1986), 363.  doi: 10.1090/S0002-9939-1986-0854048-6.  Google Scholar

[18]

L. Feher and I. Tsutsui, Regularization of Toda lattices by Hamiltonian reduction,, Jour. Geom. Phys., 21 (1997), 97.  doi: 10.1016/S0393-0440(96)00010-1.  Google Scholar

[19]

C. S. Gardner, J. M. Greene, M. D. Kruskal and R. M. Miura, Method for solving the Korteweg-de Vries equation,, Phys. Rev. Letter., 19 (1967), 1095.   Google Scholar

[20]

V. Guillemin and S. Sternberg, Symplectic Techniques in Physics,, Cambridge University Press, (1984).   Google Scholar

[21]

T. Kapeller and J. Pöschel, KdV & KAM,, Ergebnisse der Mathematik und ihrer Grenzgebiete, (2003).   Google Scholar

[22]

B. Kostant, Quantization and representation theory,, in Representation Theory of Lie Groups (ed. M. Atiyah), (1977), 287.   Google Scholar

[23]

Y. Kodama and B. Shipman, The finite non-periodic toda lattice: A geometric and topological viewpoint,, , (2008).   Google Scholar

[24]

I. M. Krichever, Methods of algebraic geometry in the theory of nonlinear equations,, Russ. Math. Surv., 32 (1977), 185.   Google Scholar

[25]

I. M. Krichever and S. P. Novikov, Holomorphich bundles over algebraic curves and nonlinear equations,, Russ. Math. Surv., 35 (1980), 53.   Google Scholar

[26]

P. Lax, Integrals of nonlinear equations of evolution and solitary waves,, Comm. Pure Appl. Math., 21 (1968), 467.  doi: 10.1002/cpa.3160210503.  Google Scholar

[27]

R. S. Leite, T. R. W. Richa and C. Tomei, Geometric proofs of some theorems of Schur-Horn type,, Lin. Alg. Appl., 286 (1999), 149.  doi: 10.1016/S0024-3795(98)10169-6.  Google Scholar

[28]

R. S. Leite, N. C. Saldanha and C. Tomei, An atlas for tridiagonal isospectral manifolds,, Lin. Alg. Appl., 429 (2008), 387.  doi: 10.1016/j.laa.2008.03.001.  Google Scholar

[29]

R. S. Leite, N. C. Saldanha and C. Tomei, The asymptotics of Wilkinson's shift: Loss of cubic convergence,, Found. Comp. Math., 10 (2010), 15.  doi: 10.1007/s10208-009-9047-3.  Google Scholar

[30]

R. S. Leite, N. C. Saldanha and C. Tomei, Dynamics of the symmetric eigenvalue problem with shift strategies,, Int. Math. Res. Notices, 2013 (2013), 4382.  doi: 10.1093/imrn/rns186.  Google Scholar

[31]

R. S. Leite and C. Tomei, Parametrization by polytopes of intersections of orbits by conjugation,, Lin. Alg. Appl., 361 (2003), 223.  doi: 10.1016/S0024-3795(02)00463-9.  Google Scholar

[32]

J. Moser, Finitely many points on the line under the influence of an exponential potential-an integrable system,, in Dynamical Systems, (1974), 467.   Google Scholar

[33]

J. Moser and A. Veselov, Discrete versions of some classical integrable systems and factorization of matrix polynomials,, Comm. Math. Phys., 139 (1991), 217.  doi: 10.1007/BF02352494.  Google Scholar

[34]

B. Parlett, The Symmetric Eigenvalue Problem,, Prentice-Hall Series in Computational Mathematics, (1980).   Google Scholar

[35]

A. M. Perelomov, Integrable Systems of Classical Mechanics and Lie Algebras. Vol. I,, Birkhäuser Verlag, (1990).  doi: 10.1007/978-3-0348-9257-5.  Google Scholar

[36]

J. Pöschel and E. Trubowitz, Inverse Spectral Theory,, Pure and Applied Mathematics, (1987).   Google Scholar

[37]

S. N. M. Ruijsenaars, Relativistic Toda systems,, Comm. Math. Phys., 133 (1990), 217.  doi: 10.1007/BF02097366.  Google Scholar

[38]

A. G. Reyman, M. A. Semenov-Tian-Shansky, Group-theoretical methods in the theory of finite-dimensional integrable-systems,, in Dynamical Systems VII (eds. V. I. Arnold and S. P. Novikov), (1994).   Google Scholar

[39]

W. Symes, Hamiltonian group actions and integrable systems,, Physica D, 1 (1980), 339.  doi: 10.1016/0167-2789(80)90017-2.  Google Scholar

[40]

W. Symes, The QR algorithm and scattering for the finite nonperiodic Toda lattice,, Physica D, 4 (): 275.  doi: 10.1016/0167-2789(82)90069-0.  Google Scholar

[41]

N. C. Saldanha and C. Tomei, Manifolds of normal or symmetric matrices of given spectrum and envelope,, in preparation., ().   Google Scholar

[42]

M. Toda, Wave propagation in anharmonic lattices,, J. Phys. Soc. Japan, 23 (1967), 501.   Google Scholar

[43]

C. Tomei, The topology of isospectral manifolds of tridiagonal matrices,, Duke Math. J., 51 (1984), 981.  doi: 10.1215/S0012-7094-84-05144-5.  Google Scholar

[44]

L. N.Trefethen and D. Bau, III, Numerical Linear Algebra,, SIAM, (1997).  doi: 10.1137/1.9780898719574.  Google Scholar

[45]

P. van Moerbeke, The spectrum of Jacobi matrices,, Invent. Math., 37 (1976), 45.  doi: 10.1007/BF01418827.  Google Scholar

[46]

D. S. Watkins and L. Elsner, On Rutishauser's approach to self-similar flows,, SIAM J. Matrix Anal. Appl., 11 (1990), 301.  doi: 10.1137/0611020.  Google Scholar

show all references

References:
[1]

M. Adler, On a trace functional for formal pseudo differential operators and the symplectic structure of the Korteweg-de-Vries type equations,, Invent. Math., 50 (): 219.  doi: 10.1007/BF01410079.  Google Scholar

[2]

M. Atiyah, Convexity and commuting Hamiltonians,, Bull. London Math. Soc., 14 (1982), 1.  doi: 10.1112/blms/14.1.1.  Google Scholar

[3]

M. J. Ablowitz, D. J. Kaup, A. C. Newell and H. Segur, The inverse scattering transform-Fourier analysis for nonlinear problems,, Stud. Appl. Math., 53 (1974), 249.   Google Scholar

[4]

R. Abraham and J. Marsden, Foundations of Mechanics,, Second edition, (1987).   Google Scholar

[5]

R. Beals, P. Deift and C. Tomei, Direct and Inverse Scattering on the Line,, Math. Surveys and Monographs, (1988).   Google Scholar

[6]

A. M. Bloch, H. Flaschka and T. Ratiu, A convexity theorem for isospectral manifolds of Jacobi matrices in a compact Lie algebra,, Duke Math. J., 61 (1990), 41.  doi: 10.1215/S0012-7094-90-06103-4.  Google Scholar

[7]

P. Deift, J. Demmel, L. C. Li and C. Tomei, The bidiagonal singular value decomposition and Hamiltonian mechanics,, SIAM J. Num. Anal., 28 (1991), 1463.  doi: 10.1137/0728076.  Google Scholar

[8]

P. Deift, L. C. Li, T. Nanda and C. Tomei, The Toda flow on a generic orbit is integrable,, Comm. Pure Appl. Math., 39 (1986), 183.  doi: 10.1002/cpa.3160390203.  Google Scholar

[9]

P. Deift, L. C. Li and C. Tomei, Matrix factorizations and integrable systems,, Comm. Pure Appl. Math., 42 (1989), 443.  doi: 10.1002/cpa.3160420405.  Google Scholar

[10]

P. Deift, L. C. Li and C. Tomei, Loop groups, discrete versions of some classical integrable systems, and rank 2 extensions,, Memoirs of the Amer. Math. Soc., 100 (1992).  doi: 10.1090/memo/0479.  Google Scholar

[11]

P. Deift, L. C. Li and C. Tomei, Toda flows with infinitely many variables,, J. Funct. Anal., 64 (1985), 358.  doi: 10.1016/0022-1236(85)90065-5.  Google Scholar

[12]

P. Deift, T. Nanda and C. Tomei, Ordinary differential equations for the symmetric eigenvalue problem,, SIAM J. Num. Anal., 20 (1983), 1.  doi: 10.1137/0720001.  Google Scholar

[13]

P. Deift, S. Rivera, C. Tomei and D. Watkins, A monotonicity property for Toda-type flows,, SIAM J. of Matrix Anal. and Appl., 12 (1991), 463.  doi: 10.1137/0612033.  Google Scholar

[14]

J. W. Demmel, Applied Numerical Linear Algebra,, SIAM, (1997).  doi: 10.1137/1.9781611971446.  Google Scholar

[15]

J. W. Demmel and W. Kahan, Accurate singular values of bidiagonal matrices,, SIAM J. Stat. Comput., 11 (1990), 873.  doi: 10.1137/0911052.  Google Scholar

[16]

H. Flaschka, The Toda lattice. I. Existence of integrals,, Phys. Rev. B (3), 9 (1974), 1924.  doi: 10.1103/PhysRevB.9.1924.  Google Scholar

[17]

D. Fried, The cohomology of an isospectral flow,, Proc. Amer. Math. Soc., 98 (1986), 363.  doi: 10.1090/S0002-9939-1986-0854048-6.  Google Scholar

[18]

L. Feher and I. Tsutsui, Regularization of Toda lattices by Hamiltonian reduction,, Jour. Geom. Phys., 21 (1997), 97.  doi: 10.1016/S0393-0440(96)00010-1.  Google Scholar

[19]

C. S. Gardner, J. M. Greene, M. D. Kruskal and R. M. Miura, Method for solving the Korteweg-de Vries equation,, Phys. Rev. Letter., 19 (1967), 1095.   Google Scholar

[20]

V. Guillemin and S. Sternberg, Symplectic Techniques in Physics,, Cambridge University Press, (1984).   Google Scholar

[21]

T. Kapeller and J. Pöschel, KdV & KAM,, Ergebnisse der Mathematik und ihrer Grenzgebiete, (2003).   Google Scholar

[22]

B. Kostant, Quantization and representation theory,, in Representation Theory of Lie Groups (ed. M. Atiyah), (1977), 287.   Google Scholar

[23]

Y. Kodama and B. Shipman, The finite non-periodic toda lattice: A geometric and topological viewpoint,, , (2008).   Google Scholar

[24]

I. M. Krichever, Methods of algebraic geometry in the theory of nonlinear equations,, Russ. Math. Surv., 32 (1977), 185.   Google Scholar

[25]

I. M. Krichever and S. P. Novikov, Holomorphich bundles over algebraic curves and nonlinear equations,, Russ. Math. Surv., 35 (1980), 53.   Google Scholar

[26]

P. Lax, Integrals of nonlinear equations of evolution and solitary waves,, Comm. Pure Appl. Math., 21 (1968), 467.  doi: 10.1002/cpa.3160210503.  Google Scholar

[27]

R. S. Leite, T. R. W. Richa and C. Tomei, Geometric proofs of some theorems of Schur-Horn type,, Lin. Alg. Appl., 286 (1999), 149.  doi: 10.1016/S0024-3795(98)10169-6.  Google Scholar

[28]

R. S. Leite, N. C. Saldanha and C. Tomei, An atlas for tridiagonal isospectral manifolds,, Lin. Alg. Appl., 429 (2008), 387.  doi: 10.1016/j.laa.2008.03.001.  Google Scholar

[29]

R. S. Leite, N. C. Saldanha and C. Tomei, The asymptotics of Wilkinson's shift: Loss of cubic convergence,, Found. Comp. Math., 10 (2010), 15.  doi: 10.1007/s10208-009-9047-3.  Google Scholar

[30]

R. S. Leite, N. C. Saldanha and C. Tomei, Dynamics of the symmetric eigenvalue problem with shift strategies,, Int. Math. Res. Notices, 2013 (2013), 4382.  doi: 10.1093/imrn/rns186.  Google Scholar

[31]

R. S. Leite and C. Tomei, Parametrization by polytopes of intersections of orbits by conjugation,, Lin. Alg. Appl., 361 (2003), 223.  doi: 10.1016/S0024-3795(02)00463-9.  Google Scholar

[32]

J. Moser, Finitely many points on the line under the influence of an exponential potential-an integrable system,, in Dynamical Systems, (1974), 467.   Google Scholar

[33]

J. Moser and A. Veselov, Discrete versions of some classical integrable systems and factorization of matrix polynomials,, Comm. Math. Phys., 139 (1991), 217.  doi: 10.1007/BF02352494.  Google Scholar

[34]

B. Parlett, The Symmetric Eigenvalue Problem,, Prentice-Hall Series in Computational Mathematics, (1980).   Google Scholar

[35]

A. M. Perelomov, Integrable Systems of Classical Mechanics and Lie Algebras. Vol. I,, Birkhäuser Verlag, (1990).  doi: 10.1007/978-3-0348-9257-5.  Google Scholar

[36]

J. Pöschel and E. Trubowitz, Inverse Spectral Theory,, Pure and Applied Mathematics, (1987).   Google Scholar

[37]

S. N. M. Ruijsenaars, Relativistic Toda systems,, Comm. Math. Phys., 133 (1990), 217.  doi: 10.1007/BF02097366.  Google Scholar

[38]

A. G. Reyman, M. A. Semenov-Tian-Shansky, Group-theoretical methods in the theory of finite-dimensional integrable-systems,, in Dynamical Systems VII (eds. V. I. Arnold and S. P. Novikov), (1994).   Google Scholar

[39]

W. Symes, Hamiltonian group actions and integrable systems,, Physica D, 1 (1980), 339.  doi: 10.1016/0167-2789(80)90017-2.  Google Scholar

[40]

W. Symes, The QR algorithm and scattering for the finite nonperiodic Toda lattice,, Physica D, 4 (): 275.  doi: 10.1016/0167-2789(82)90069-0.  Google Scholar

[41]

N. C. Saldanha and C. Tomei, Manifolds of normal or symmetric matrices of given spectrum and envelope,, in preparation., ().   Google Scholar

[42]

M. Toda, Wave propagation in anharmonic lattices,, J. Phys. Soc. Japan, 23 (1967), 501.   Google Scholar

[43]

C. Tomei, The topology of isospectral manifolds of tridiagonal matrices,, Duke Math. J., 51 (1984), 981.  doi: 10.1215/S0012-7094-84-05144-5.  Google Scholar

[44]

L. N.Trefethen and D. Bau, III, Numerical Linear Algebra,, SIAM, (1997).  doi: 10.1137/1.9780898719574.  Google Scholar

[45]

P. van Moerbeke, The spectrum of Jacobi matrices,, Invent. Math., 37 (1976), 45.  doi: 10.1007/BF01418827.  Google Scholar

[46]

D. S. Watkins and L. Elsner, On Rutishauser's approach to self-similar flows,, SIAM J. Matrix Anal. Appl., 11 (1990), 301.  doi: 10.1137/0611020.  Google Scholar

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