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| The '''Toda lattice''', introduced by {{harvs|txt|first=Morikazu |last=Toda|authorlink=Morikazu Toda|year=1967}}, is a simple model for a one-dimensional crystal | | The title of the author is Figures. Years in the past he moved to North Dakota and his family members enjoys it. What I love doing is taking part in baseball but I haven't made a dime with it. Managing individuals is his profession.<br><br>Here is my web page :: [http://vip.akwacity.com/oxwall/blogs/post/10278 home std test kit] |
| in [[solid state physics]]. It is given by a chain of particles with nearest neighbor interaction
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| described by the equations of motion
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| :<math> \begin{align}
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| \frac{d}{dt} p(n,t) &= e^{-(q(n,t) - q(n-1,t))} - e^{-(q(n+1,t) - q(n,t))}, \\
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| \frac{d}{dt} q(n,t) &= p(n,t),
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| \end{align} </math>
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| where <math>q(n,t)</math> is the displacement of the <math>n</math>-th particle from its equilibrium position,
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| and <math>p(n,t)</math> is its momentum (mass <math>m=1</math>). | |
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| The Toda lattice is a prototypical example of a [[completely integrable system]] with [[soliton]] solutions. To see this one uses [[Hermann Flaschka|Flaschka]]'s variables
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| :<math> a(n,t) = \frac{1}{2} {\rm e}^{-(q(n+1,t) - q(n,t))/2}, \qquad b(n,t) = -\frac{1}{2} p(n,t) </math>
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| such that the Toda lattice reads
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| :<math> \begin{align}
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| \dot{a}(n,t) &= a(n,t) \Big(b(n+1,t)-b(n,t)\Big), \\
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| \dot{b}(n,t) &= 2 \Big(a(n,t)^2-a(n-1,t)^2\Big).
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| \end{align}</math>
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| Then one can verify that the Toda lattice is equivalent to the Lax equation
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| :<math>\frac{d}{dt} L(t) = [P(t), L(t)]</math>
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| where [''L'', ''P''] = ''LP'' - ''PL'' is the [[commutator]] of two [[operator (mathematics)|operator]]s. The operators ''L'' and ''P'', the [[Lax pair]], are [[linear operators]] in the [[Hilbert space]] of square summable sequences <math>\ell^2(\mathbb{Z})</math> given by
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| :<math> \begin{align}
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| L(t) f(n) &= a(n,t) f(n+1) + a(n-1,t) f(n-1) + b(n,t) f(n), \\
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| P(t) f(n) &= a(n,t) f(n+1) - a(n-1,t) f(n-1).
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| \end{align}</math>
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| The matrix <math>L(t)</math> has the property that its eigenvalues are invariant in time. These eigenvalues constitute independent integrals of motion, therefore the Toda lattice is completely integrable.
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| In particular, the Toda lattice can be solved by virtue of the [[inverse scattering transform]] for the [[Jacobi operator]] ''L''. The main result implies that arbitrary (sufficiently fast) decaying initial conditions asymptotically for large ''t'' split into a sum of solitons and a decaying [[Dispersion (water waves)|dispersive]] part.
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| ==References==
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| *{{citation|id={{MR|2493113}}|last=Krüger|first=Helge|last2=Teschl|first2= Gerald
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| |title=Long-time asymptotics of the Toda lattice for decaying initial data revisited
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| |journal=Rev. Math. Phys.|volume= 21 |year=2009|issue= 1|pages= 61-109|doi=10.1142/S0129055X0900358X}}
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| *{{citation|id={{MR|1711536}}|title=Jacobi Operators and Completely Integrable Nonlinear Lattices
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| |first=Gerald|last=Teschl|authorlink=Gerald Teschl|publisher=Amer. Math. Soc.|location=Providence|year=2000|url=http://www.mat.univie.ac.at/~gerald/ftp/book-jac/|isbn=0-8218-1940-2}}
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| *{{citation|id={{MR|1879178}}|last=Teschl|first= Gerald
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| |title=Almost everything you always wanted to know about the Toda equation
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| |journal=Jahresbericht der Deutschen Mathematiker-Vereinigung |volume= 103 |year=2001|issue= 4|pages= 149–162|url=http://www.mat.univie.ac.at/~gerald/ftp/articles/Toda.html}}
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| *Integrable Hamiltonians with Exponential Potential, Eugene Gutkin, Physica 16D (1985) 398-404.
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| *{{citation|first=Morikazu|last= Toda|title=Vibration of a chain with a non-linear interaction|journal= J. Phys. Soc. Japan |volume= 22 |year=1967|pages= 431–436|doi=10.1143/JPSJ.22.431}}
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| *{{citation|id={{MR|0971987}}|title=Theory of Nonlinear Lattices
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| |first=Morikazu|last= Toda|edition=2|publisher=Springer|location=Berlin|year=1989|isbn=978-0-387-10224-5|doi=10.1007/978-3-642-83219-2}}
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| ==External links==
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| * E. W. Weisstein, [http://scienceworld.wolfram.com/physics/TodaLattice.html Toda Lattice] at ScienceWorld
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| * G. Teschl, [http://www.mat.univie.ac.at/~gerald/ftp/book-jac/toda.html The Toda Lattice]
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| [[Category:Exactly solvable models]]
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| [[Category:Solitons]]
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