128.220.159.1 at 19:18, 20 September 2014

2014-09-20T19:18:34Z

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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~~
	~~described by~~ the ~~equations of motion~~
	~~:<math> \begin{align}~~
	~~\frac{d}{dt} p(n,t) &= e^{-(q(n,t) - q(n-1,t))} - e^{-(q(n+1,t) - q(n,t))}, \\~~
	~~\frac{d}{dt} q(n,t) &= p(n,t),~~
	~~\end{align} </math>~~
	~~where <math>q(n,t)</math>~~ is the ~~displacement of the <math>n</math>-th particle from its equilibrium position,~~
	and ~~<math>p(n,t)</math> is its momentum (mass <math>m=1</math>)~~.

	~~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~~
	~~:<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>~~
	~~such that the Toda lattice reads~~
	~~:<math> \begin{align}~~
	~~\dot{~~a~~}(n,t) &= a(n,t) \Big(b(n+1,t)-b(n,t)\Big), \\~~
	~~\dot{b}(n,t) &= 2 \Big(a(n,t)^2-a(n-1,t)^2\Big)~~.
	~~\end{align}</math>~~
	~~Then one can verify that the Toda lattice~~ is ~~equivalent to the Lax equation~~
	~~:<math>\frac{d}{dt} L(t) = [P(t), L(t)]</math>~~
	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
	~~:<math> \begin{align}~~
	~~L(t) f(n) &= a(n,t) f(n+1) + a(n-1,t) f(n-1) + b(n,t) f(n), \\~~
	~~P(t) f(n) &= a(n,t) f(n+1) - a(n-1,t) f(n-1)~~.
	~~\end{align}~~<~~/math~~>
	~~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.~~
	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.

	~~==References==~~

	*{{citation\|id={{MR\|2493113}}\|last=Krüger\|first=Helge\|last2=Teschl\|first2= Gerald
	~~\|title=Long-time asymptotics of the Toda lattice for decaying initial data revisited~~
	~~\|journal=Rev. Math. Phys.\|volume= 21 \|year=2009\|issue= 1\|pages= 61-109\|doi=10.1142/S0129055X0900358X}}~~
	*{{citation\|id={{MR\|1711536}}\|title=Jacobi Operators and Completely Integrable Nonlinear Lattices
	~~\|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}}~~
	*{{citation\|id={{MR\|1879178}}\|last=Teschl\|first= Gerald
	~~\|title=Almost everything you always wanted to know about the Toda equation~~
	~~\|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}}~~
	*Integrable Hamiltonians with Exponential Potential, Eugene Gutkin, Physica 16D (1985) 398-404.
	*{{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}}
	*{{citation\|id={{MR\|0971987}}\|title=Theory of Nonlinear Lattices
	~~\|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}}~~

	~~==External links==~~
	* E. W. Weisstein, [http://~~scienceworld~~.~~wolfram~~.com/~~physics/TodaLattice.html Toda Lattice] at ScienceWorld~~
	* G. Teschl, [http://www.mat.univie.ac.at/~gerald/~~ftp~~/~~book-jac~~/~~toda.html The Toda Lattice]~~

	~~[[Category:Exactly solvable models]]~~
	~~[[Category:Solitons]~~]

en>Pinethicket: Reverted edits by 2001:980:707D:1:41C6:E764:C25C:749C (talk) to last version by Addbot

2014-01-27T12:50:28Z

Reverted edits by 2001:980:707D:1:41C6:E764:C25C:749C (talk) to last version by Addbot

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en>AdmiralHood at 00:28, 17 December 2011

2011-12-17T00:28:33Z

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Coriolis frequency - Revision history

128.220.159.1 at 19:18, 20 September 2014

en>Pinethicket: Reverted edits by 2001:980:707D:1:41C6:E764:C25C:749C (talk) to last version by Addbot

en>AdmiralHood at 00:28, 17 December 2011