Bridged T delay equaliser - Revision history

en>Omnipaedista: noref-tag

2013-10-31T16:25:45Z

noref-tag

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			In [[solid-state physics]], the '''Landau–Lifshitz equation''' ('''LLE'''), named for [[Lev Landau]] and [[Evgeny Lifshitz]], is a [[partial differential equation]] describing time evolution of [[magnetism]] in solids, depending on 1 time variable and 1, 2, or 3 space variables.

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			The LLE describes an [[anisotropic]] magnet. The equation is described in {{harv\|Faddeev\|Takhtajan\|2007\|loc=chapter 8}} as follows: It is an equation for a [[vector field]] '''S''', in other words a function on '''R'''<sup>1+''n''</sup> taking values in '''R'''<sup>3</sup>. The equation depends on a fixed symmetric 3 by 3 [[matrix (mathematics)\|matrix]] ''J'', usually assumed to be [[Diagonal matrix\|diagonal]]; that is, <math>J=\operatorname{diag}(J_{1}, J_{2}, J_{3})</math>. It is given by Hamilton's equation of motion for the [[Hamiltonian mechanics\|Hamiltonian]]

			:<math>H=\frac{1}{2}\int \left[\sum_i\left(\frac{\partial \mathbf{S}}{\partial x_i}\right)^{2}-J(\mathbf{S})\right]\, dx\qquad (1)</math>

			(where ''J''('''S''') is the quadratic form of ''J'' applied to the vector '''S''')
			which is

			:<math> \frac{\partial \mathbf{S}}{\partial t} = \mathbf{S}\wedge \sum_i\frac{\partial^2 \mathbf{S}}{\partial x_i^{2}} + \mathbf{S}\wedge J\mathbf{S}.\qquad (2)</math>

			In 1+1 dimensions this equation is

			:<math> \frac{\partial \mathbf{S}}{\partial t} = \mathbf{S}\wedge \frac{\partial^2 \mathbf{S}}{\partial x^{2}} + \mathbf{S}\wedge J\mathbf{S}.\qquad (3)</math>

			In 2+1 dimensions this equation takes the form

			:<math> \frac{\partial \mathbf{S}}{\partial t} = \mathbf{S}\wedge \left(\frac{\partial^2 \mathbf{S}}{\partial x^{2}} + \frac{\partial^2 \mathbf{S}}{\partial y^{2}}\right)+ \mathbf{S}\wedge J\mathbf{S}\qquad (4)</math>

			which is the (2+1)-dimensional LLE. For the (3+1)-dimensional case LLE looks like

			:<math> \frac{\partial \mathbf{S}}{\partial t} = \mathbf{S}\wedge \left(\frac{\partial^2 \mathbf{S}}{\partial x^{2}} + \frac{\partial^2 \mathbf{S}}{\partial y^{2}}+\frac{\partial^2 \mathbf{S}}{\partial z^{2}}\right)+ \mathbf{S}\wedge J\mathbf{S}.\qquad (5)</math>

			==Integrable reductions==
			In general case LLE (2) is nonintegrable. But it admits the two integrable reductions:
			: a) in the 1+1 dimensions, that is Eq. (3), it is integrable
			: b) when <math>J=0</math>. In this case the (1+1)-dimensional LLE (3) turns into the [[continuous classical Heisenberg ferromagnet equation]] (see e.g. [[Heisenberg model (classical)]]) which is already integrable.

			==See also==
			* [[Nonlinear Schrödinger equation]]
			* [[Heisenberg model (classical)]]
			* [[Spin wave]]
			* [[Micromagnetism]]
			* [[Ishimori equation]]
			* [[Magnet]]
			* [[Ferromagnetism]]

			==References==
			*{{citation\|mr=2348643\|last= Faddeev\|first= Ludwig D.\|last2= Takhtajan\|first2= Leon A.\|title= Hamiltonian methods in the theory of solitons\|series= Classics in Mathematics\|publisher= Springer\|place= Berlin\|year= 2007\|pages= x+592 \| ISBN= 978-3-540-69843-2 }}
			*{{citation\|ISBN= 978-981-277-875-8
			\|title=Landau-Lifshitz Equations \|series=Frontiers of Research With the Chinese Academy of Sciences
			\|first=Boling \|last=Guo \|first2= Shijin \|last2=Ding\|year=2008
			\|publisher=World Scientific Publishing Company}}
			* [[Arnold Kosevich\|Kosevich A.M.]], Ivanov B.A., Kovalev A.S. Nonlinear magnetization waves. Dynamical and topological solitons. – Kiev: [[Naukova Dumka]], 1988. – 192 p.

			{{DEFAULTSORT:Landau-Lifshitz model}}
			[[Category:Magnetic ordering]]
			[[Category:Partial differential equations]]

en>ChrisGualtieri: /* Design */Typo fixing, typos fixed: ie → i.e. using AWB

2012-07-30T04:55:18Z

Design: Typo fixing, typos fixed: ie → i.e. using AWB

New page

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