Ginzburg-Landau-Schrodinger equation: Difference between revisions
No edit summary |
|||
| (8 intermediate revisions by 3 users not shown) | |||
| Line 1: | Line 1: | ||
{{stub}} | |||
[[Category: | The '''Ginzburg-Landau-Schrodinger''' equation is | ||
<center><math>i u^\epsilon_t + \Delta u^\epsilon = \frac{1}{\epsilon^2} (|u^\epsilon|^2 - 1)u^\epsilon.</math></center> | |||
The main focus of study for this equation is the formation of vortices and their dynamics in the limit <math>\epsilon \to 0</math>. | |||
The [http://en.wikipedia.org/wiki/Ginzburg-Landau_theory Ginzburg-Landau theory] is briefly surveyed on [http://en.wikipedia.org Wikipedia]. | |||
==Perturbative Approach== | |||
The limit <math>\epsilon\to 0</math> can be treated with the same methods given in [[Perturbation theory]]. To see this we note that an exact solution can be written as | |||
<math> u^\epsilon = \sqrt{n_0}e^{-i(n_0-1)\frac{t}{\epsilon^2}} </math> | |||
<math>n_0</math> being a real constant. Then, if we rescale time as <math>\tau=t/\epsilon^2</math> and take the solution series | |||
<math> u^\epsilon = u_0+\epsilon^2 u_1+\epsilon^4 u_2+\ldots </math> | |||
one has the non trivial set of equations | |||
<math>i\dot u_0=u_0(|u_0|^2-1)</math> | |||
<math>i\dot u_1+\Delta u_0=u_1(|u_0|^2-1)+(u_1^*u_0+u_0^*u_1)u_0</math> | |||
<math>i\dot u_2+\Delta u_1=u_2(|u_0|^2-1)+(u_1^*u_0+u_0^*u_1)u_1+(|u_1|^2+u_2^*u_0+u_0^*u_2)u_0</math> | |||
<math>\ldots</math>. | |||
where dot means derivation with respect to <math>\tau</math>. The leading order solution is easily written down as | |||
<math>u_0 = \sqrt{n_0(x)}e^{-i[n_0(x)-1]\tau} </math>. | |||
With this expression we can write down the next order correction as | |||
<math>u_1 = \phi(x,\tau)e^{-i[2n_0(x)-1]\tau}</math> | |||
<math>i\dot\phi=n_0(x)\phi^*e^{-i[2n_0(x)-1]\tau}-(\Delta u_0)e^{-i[2n_0(x)-1]\tau}</math> | |||
<math>-i\dot\phi^*=n_0(x)\phi e^{i[2n_0(x)-1]\tau}-(\Delta u_0^*)e^{i[2n_0(x)-1]\tau}</math>. | |||
This set is easy to solve. The most important point to notice is the limit surface <math>n_0(x)=1/2</math> that denotes a change into the stability of the solution of GL equation. It should also be pointed out the appearence at this order of secular terms going like <math>\tau</math> | |||
and <math>\tau^2</math>. These terms can be treated with several known techniques. | |||
[[Category:Schrodinger]] | |||
[[Category:Equations]] | [[Category:Equations]] | ||
Latest revision as of 07:45, 2 October 2007
The Ginzburg-Landau-Schrodinger equation is
The main focus of study for this equation is the formation of vortices and their dynamics in the limit Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \epsilon \to 0} .
The Ginzburg-Landau theory is briefly surveyed on Wikipedia.
Perturbative Approach
The limit Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \epsilon\to 0} can be treated with the same methods given in Perturbation theory. To see this we note that an exact solution can be written as
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle u^\epsilon = \sqrt{n_0}e^{-i(n_0-1)\frac{t}{\epsilon^2}} }
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n_0} being a real constant. Then, if we rescale time as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \tau=t/\epsilon^2} and take the solution series
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle u^\epsilon = u_0+\epsilon^2 u_1+\epsilon^4 u_2+\ldots }
one has the non trivial set of equations
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle i\dot u_0=u_0(|u_0|^2-1)}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle i\dot u_1+\Delta u_0=u_1(|u_0|^2-1)+(u_1^*u_0+u_0^*u_1)u_0}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle i\dot u_2+\Delta u_1=u_2(|u_0|^2-1)+(u_1^*u_0+u_0^*u_1)u_1+(|u_1|^2+u_2^*u_0+u_0^*u_2)u_0}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \ldots} .
where dot means derivation with respect to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \tau} . The leading order solution is easily written down as
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle u_0 = \sqrt{n_0(x)}e^{-i[n_0(x)-1]\tau} } .
With this expression we can write down the next order correction as
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle u_1 = \phi(x,\tau)e^{-i[2n_0(x)-1]\tau}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle i\dot\phi=n_0(x)\phi^*e^{-i[2n_0(x)-1]\tau}-(\Delta u_0)e^{-i[2n_0(x)-1]\tau}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle -i\dot\phi^*=n_0(x)\phi e^{i[2n_0(x)-1]\tau}-(\Delta u_0^*)e^{i[2n_0(x)-1]\tau}} .
This set is easy to solve. The most important point to notice is the limit surface Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n_0(x)=1/2} that denotes a change into the stability of the solution of GL equation. It should also be pointed out the appearence at this order of secular terms going like Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \tau} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \tau^2} . These terms can be treated with several known techniques.
