May  2017, 37(5): 2259-2264. doi: 10.3934/dcds.2017098

On uniqueness properties of solutions of the Toda and Kac-van Moerbeke hierarchies

1. 

Department of Mathematical Sciences, Norwegian University of Science and Technology, NO–7491 Trondheim, Norway

2. 

Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1,1090 Wien, Austria

3. 

International Erwin Schrödinger Institute for Mathematical Physics, Boltzmanngasse 9,1090 Wien, Austria

Received  October 2016 Revised  November 2016 Published  February 2017

Fund Project: Research supported by the Norwegian Research Council project DIMMA 213638.

We prove that a solution of the Toda lattice cannot decay too fast at two different times unless it is trivial. In fact, we establish this result for the entire Toda and Kac-van Moerbeke hierarchies.

Citation: Isaac Alvarez-Romero, Gerald Teschl. On uniqueness properties of solutions of the Toda and Kac-van Moerbeke hierarchies. Discrete and Continuous Dynamical Systems, 2017, 37 (5) : 2259-2264. doi: 10.3934/dcds.2017098
References:
[1]

I. Alvarez-Romero and G. Teschl, A dynamic uncertainty principle for Jacobi operators, Journal of Mathematical Analysis and Applications, 449 (2017), 580-588.  doi: 10.1016/j.jmaa.2016.12.028.

[2]

W. BullaF. GesztesyH. Holden and G. Teschl, Algebro-Geometric Quasi-Periodic Finite-Gap Solutions of the Toda and Kac-van Moerbeke Hierarchies, Mem. Amer. Math. Soc., 135 (1998).  doi: 10.1090/memo/0641.

[3]

L. EscauriazaC. E. KenigG. Ponce and L. Vega, On uniqueness properties of solutions of Schrödinger equations, Comm. Partial Differential Equations, 31 (2006), 1811-1823. 

[4]

L. Faddeev and L. Takhtajan, Hamiltonian Methods in the Theory of Solitons Springer, Berlin, 1987. doi: 10.1007/978-3-540-69969-9.

[5]

H. Flaschka, The Toda lattice. I. Existence of integrals, Phys. Rev. B, 9 (1974), 1924-1925.  doi: 10.1103/PhysRevB.9.1924.

[6]

F. Gesztesy, H. Holden, J. Michor and G. Teschl, Soliton Equations and Their Algebro-Geometric Solutions. Volume Ⅱ: $(1+1)$-Dimensional Discrete Models Cambridge Studies in Advanced Mathematics 114, Cambridge University Press, Cambridge, 2008.

[7]

A. Ionescu and C. E. Kenig, Lp Carleman inequalities and uniqueness of solutions of nonlinear Schrödinger equations, Acta Math., 193 (2004), 193-239.  doi: 10.1007/BF02392564.

[8]

Ph. Jaming, Yu. Lyubarskii, E. Malinnikova and K. -M. Perfekt, Uniqueness for discrete Schrödinger evolutions, Rev. Mat. Iberoamericana (to appear). arXiv: 1505.05398.

[9]

C. E. KenigG. Ponce and L. Vega, On unique continuation of solutions to the generalized KdV equation, Math. Res. Lett., 10 (2003), 833-846.  doi: 10.4310/MRL.2003.v10.n6.a10.

[10]

H. Krüger and G. Teschl, Long-time asymptotics for the Toda lattice for decaying initial data revisited, Rev. Math. Phys., 21 (2009), 61-109.  doi: 10.1007/s00209-008-0391-9.

[11]

H. Krüger and G. Teschl, Unique continuation for discrete nonlinear wave equations, Proc. Amer. Math. Soc., 140 (2012), 1321-1330.  doi: 10.1090/S0002-9939-2011-10980-8.

[12]

B. Ya. Levin, Lectures on Entire Functions Translations of Mathematical Monographs, Amer. Math. Soc. , Providence RI, 1996.

[13]

J. Michor and G. Teschl, On the equivalence of different Lax pairs for the Kac-van Moerbeke hierarchy, in Modern Analysis and Applications, V. Adamyan (ed.) et al. , Oper. Theory Adv. Appl. , Birkhäuser, Basel, 191 (2009), 445–454. doi: 10.1007/978-3-7643-9921-4_27.

[14]

G. Teschl, Jacobi operators and completely integrable nonlinear lattices Math. Surv. and Mon. Amer. Math. Soc. , Rhode Island, 72 2000. doi: 10.1090/surv/072.

[15]

G. Teschl, Almost everything you always wanted to know about the Toda equation, Jahresber. Deutsch. Math.-Verein., 103 (2001), 149-162. 

[16]

G. Teschl, On the spatial asymptotics of solutions of the Toda lattice, Discrete Contin. Dyn. Syst., 27 (2010), 1233-1239.  doi: 10.3934/dcds.2010.27.1233.

[17]

M. Toda, Theory of Nonlinear Lattices, 2nd enl. ed. , Springer, Berlin, 1989.

show all references

References:
[1]

I. Alvarez-Romero and G. Teschl, A dynamic uncertainty principle for Jacobi operators, Journal of Mathematical Analysis and Applications, 449 (2017), 580-588.  doi: 10.1016/j.jmaa.2016.12.028.

[2]

W. BullaF. GesztesyH. Holden and G. Teschl, Algebro-Geometric Quasi-Periodic Finite-Gap Solutions of the Toda and Kac-van Moerbeke Hierarchies, Mem. Amer. Math. Soc., 135 (1998).  doi: 10.1090/memo/0641.

[3]

L. EscauriazaC. E. KenigG. Ponce and L. Vega, On uniqueness properties of solutions of Schrödinger equations, Comm. Partial Differential Equations, 31 (2006), 1811-1823. 

[4]

L. Faddeev and L. Takhtajan, Hamiltonian Methods in the Theory of Solitons Springer, Berlin, 1987. doi: 10.1007/978-3-540-69969-9.

[5]

H. Flaschka, The Toda lattice. I. Existence of integrals, Phys. Rev. B, 9 (1974), 1924-1925.  doi: 10.1103/PhysRevB.9.1924.

[6]

F. Gesztesy, H. Holden, J. Michor and G. Teschl, Soliton Equations and Their Algebro-Geometric Solutions. Volume Ⅱ: $(1+1)$-Dimensional Discrete Models Cambridge Studies in Advanced Mathematics 114, Cambridge University Press, Cambridge, 2008.

[7]

A. Ionescu and C. E. Kenig, Lp Carleman inequalities and uniqueness of solutions of nonlinear Schrödinger equations, Acta Math., 193 (2004), 193-239.  doi: 10.1007/BF02392564.

[8]

Ph. Jaming, Yu. Lyubarskii, E. Malinnikova and K. -M. Perfekt, Uniqueness for discrete Schrödinger evolutions, Rev. Mat. Iberoamericana (to appear). arXiv: 1505.05398.

[9]

C. E. KenigG. Ponce and L. Vega, On unique continuation of solutions to the generalized KdV equation, Math. Res. Lett., 10 (2003), 833-846.  doi: 10.4310/MRL.2003.v10.n6.a10.

[10]

H. Krüger and G. Teschl, Long-time asymptotics for the Toda lattice for decaying initial data revisited, Rev. Math. Phys., 21 (2009), 61-109.  doi: 10.1007/s00209-008-0391-9.

[11]

H. Krüger and G. Teschl, Unique continuation for discrete nonlinear wave equations, Proc. Amer. Math. Soc., 140 (2012), 1321-1330.  doi: 10.1090/S0002-9939-2011-10980-8.

[12]

B. Ya. Levin, Lectures on Entire Functions Translations of Mathematical Monographs, Amer. Math. Soc. , Providence RI, 1996.

[13]

J. Michor and G. Teschl, On the equivalence of different Lax pairs for the Kac-van Moerbeke hierarchy, in Modern Analysis and Applications, V. Adamyan (ed.) et al. , Oper. Theory Adv. Appl. , Birkhäuser, Basel, 191 (2009), 445–454. doi: 10.1007/978-3-7643-9921-4_27.

[14]

G. Teschl, Jacobi operators and completely integrable nonlinear lattices Math. Surv. and Mon. Amer. Math. Soc. , Rhode Island, 72 2000. doi: 10.1090/surv/072.

[15]

G. Teschl, Almost everything you always wanted to know about the Toda equation, Jahresber. Deutsch. Math.-Verein., 103 (2001), 149-162. 

[16]

G. Teschl, On the spatial asymptotics of solutions of the Toda lattice, Discrete Contin. Dyn. Syst., 27 (2010), 1233-1239.  doi: 10.3934/dcds.2010.27.1233.

[17]

M. Toda, Theory of Nonlinear Lattices, 2nd enl. ed. , Springer, Berlin, 1989.

[1]

Carlos Tomei. The Toda lattice, old and new. Journal of Geometric Mechanics, 2013, 5 (4) : 511-530. doi: 10.3934/jgm.2013.5.511

[2]

Gerald Teschl. On the spatial asymptotics of solutions of the Toda lattice. Discrete and Continuous Dynamical Systems, 2010, 27 (3) : 1233-1239. doi: 10.3934/dcds.2010.27.1233

[3]

José G. Llorente. Mean value properties and unique continuation. Communications on Pure and Applied Analysis, 2015, 14 (1) : 185-199. doi: 10.3934/cpaa.2015.14.185

[4]

Muriel Boulakia. Quantification of the unique continuation property for the nonstationary Stokes problem. Mathematical Control and Related Fields, 2016, 6 (1) : 27-52. doi: 10.3934/mcrf.2016.6.27

[5]

Laurent Bourgeois. Quantification of the unique continuation property for the heat equation. Mathematical Control and Related Fields, 2017, 7 (3) : 347-367. doi: 10.3934/mcrf.2017012

[6]

Zhongqi Yin. A quantitative internal unique continuation for stochastic parabolic equations. Mathematical Control and Related Fields, 2015, 5 (1) : 165-176. doi: 10.3934/mcrf.2015.5.165

[7]

A. Alexandrou Himonas, Gerard Misiołek, Feride Tiǧlay. On unique continuation for the modified Euler-Poisson equations. Discrete and Continuous Dynamical Systems, 2007, 19 (3) : 515-529. doi: 10.3934/dcds.2007.19.515

[8]

Gunther Uhlmann, Jenn-Nan Wang. Unique continuation property for the elasticity with general residual stress. Inverse Problems and Imaging, 2009, 3 (2) : 309-317. doi: 10.3934/ipi.2009.3.309

[9]

Can Zhang. Quantitative unique continuation for the heat equation with Coulomb potentials. Mathematical Control and Related Fields, 2018, 8 (3&4) : 1097-1116. doi: 10.3934/mcrf.2018047

[10]

Andreas Henrici. Symmetries of the periodic Toda lattice, with an application to normal forms and perturbations of the lattice with Dirichlet boundary conditions. Discrete and Continuous Dynamical Systems, 2015, 35 (7) : 2949-2977. doi: 10.3934/dcds.2015.35.2949

[11]

Ihyeok Seo. Carleman estimates for the Schrödinger operator and applications to unique continuation. Communications on Pure and Applied Analysis, 2012, 11 (3) : 1013-1036. doi: 10.3934/cpaa.2012.11.1013

[12]

Jan Boman. Unique continuation of microlocally analytic distributions and injectivity theorems for the ray transform. Inverse Problems and Imaging, 2010, 4 (4) : 619-630. doi: 10.3934/ipi.2010.4.619

[13]

Mouhamed Moustapha Fall, Veronica Felli. Unique continuation properties for relativistic Schrödinger operators with a singular potential. Discrete and Continuous Dynamical Systems, 2015, 35 (12) : 5827-5867. doi: 10.3934/dcds.2015.35.5827

[14]

Roberto Triggiani. Unique continuation of boundary over-determined Stokes and Oseen eigenproblems. Discrete and Continuous Dynamical Systems - S, 2009, 2 (3) : 645-677. doi: 10.3934/dcdss.2009.2.645

[15]

Guojie Zheng, Dihong Xu, Taige Wang. A unique continuation property for a class of parabolic differential inequalities in a bounded domain. Communications on Pure and Applied Analysis, 2021, 20 (2) : 547-558. doi: 10.3934/cpaa.2020280

[16]

Agnid Banerjee. A note on the unique continuation property for fully nonlinear elliptic equations. Communications on Pure and Applied Analysis, 2015, 14 (2) : 623-626. doi: 10.3934/cpaa.2015.14.623

[17]

Taige Wang, Dihong Xu. A quantitative strong unique continuation property of a diffusive SIS model. Discrete and Continuous Dynamical Systems - S, 2022, 15 (6) : 1599-1614. doi: 10.3934/dcdss.2022024

[18]

Matthias Täufer, Martin Tautenhahn. Scale-free and quantitative unique continuation for infinite dimensional spectral subspaces of Schrödinger operators. Communications on Pure and Applied Analysis, 2017, 16 (5) : 1719-1730. doi: 10.3934/cpaa.2017083

[19]

Peng Gao. Unique continuation property for stochastic nonclassical diffusion equations and stochastic linearized Benjamin-Bona-Mahony equations. Discrete and Continuous Dynamical Systems - B, 2019, 24 (6) : 2493-2510. doi: 10.3934/dcdsb.2018262

[20]

Qiaoyi Hu, Zhijun Qiao. Analyticity, Gevrey regularity and unique continuation for an integrable multi-component peakon system with an arbitrary polynomial function. Discrete and Continuous Dynamical Systems, 2016, 36 (12) : 6975-7000. doi: 10.3934/dcds.2016103

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (142)
  • HTML views (63)
  • Cited by (1)

Other articles
by authors

[Back to Top]