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 & Continuous Dynamical Systems - A, 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[4]

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

[5]

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

[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.Google Scholar

[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. Google Scholar

[8]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[12]

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

[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. Google Scholar

[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. Google Scholar

[15]

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

[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. Google Scholar

[17]

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

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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[4]

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

[5]

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

[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.Google Scholar

[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. Google Scholar

[8]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[12]

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

[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. Google Scholar

[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. Google Scholar

[15]

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

[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. Google Scholar

[17]

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

[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 & Continuous Dynamical Systems - A, 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 & 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 & 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 & 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 & 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 & Continuous Dynamical Systems - A, 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 & 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 & 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 & Continuous Dynamical Systems - A, 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 & 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 & 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 & Continuous Dynamical Systems - A, 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 & Continuous Dynamical Systems - S, 2009, 2 (3) : 645-677. doi: 10.3934/dcdss.2009.2.645

[15]

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

[16]

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

[17]

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

[18]

Qiaoyi Hu, Zhijun Qiao. Persistence properties and unique continuation for a dispersionless two-component Camassa-Holm system with peakon and weak kink solutions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2613-2625. doi: 10.3934/dcds.2016.36.2613

[19]

Shaojie Yang, Tianzhou Xu. Symmetry analysis, persistence properties and unique continuation for the cross-coupled Camassa-Holm system. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 329-341. doi: 10.3934/dcds.2018016

[20]

Peng Gao. Carleman estimates and Unique Continuation Property for 1-D viscous Camassa-Holm equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 169-188. doi: 10.3934/dcds.2017007

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (11)
  • HTML views (6)
  • Cited by (0)

Other articles
by authors

[Back to Top]