# Coupled partial differential equation...how to find a solution?

• Jun 1st 2012, 06:03 AM
climbon
Coupled partial differential equation...how to find a solution?
I have four coupled pde's I need to find solutions to, they are;

$\displaystyle \hbar \frac{\partial}{\partial t}W_{(x,y,t)} = g_1 \left( sin(\omega t) \frac{\partial}{\partial x} - cos(\omega t) \frac{\partial}{\partial y} \right) \Gamma_{(x,y,t)} +g_2 \left( sin(\omega t) \frac{\partial}{\partial x} - cos(\omega t) \frac{\partial}{\partial y} \right) Z_{(x,y,t)} + 4g_3 \left( x cos(\omega t) - y sin(\omega t) \right) \left( sin(\omega t) \frac{\partial}{\partial x} - cos(\omega t) \frac{\partial}{\partial y} \right) Z_{(x,y,t)}$

$\displaystyle \hbar \frac{\partial}{\partial t}Z_{(x,y,t)} = -4g_1 \left( x cos(\omega t) - y sin(\omega t) \right) \Lambda_{(x,y,t)} + g_2 \left( sin(\omega t) \frac{\partial}{\partial x} - cos(\omega t) \frac{\partial}{\partial y} \right) W_{(x,y,t)} + 4g_3 \left( x cos(\omega t) - y sin(\omega t) \right) \left( sin(\omega t) \frac{\partial}{\partial x} - cos(\omega t) \frac{\partial}{\partial y} \right) W_{(x,y,t)}$

$\displaystyle \hbar \frac{\partial}{\partial t} \Gamma_{(x,y,t)} = \hbar \Omega \Lambda_{(x,y,t)} + g_1 \left( sin(\omega t) \frac{\partial}{\partial x} - cos(\omega t) \frac{\partial}{\partial y} \right) W_{(x,y,t)} + 4g_2 \left( x cos(\omega t) - y sin(\omega t) \right) \Lambda_{(x,y,t)} + 8g_3 \left( x^2 cos^2 (\omega t) + y^2 sin^2 (\omega t) - 2xy cos(\omega t) sin(\omega t) \right) \Lambda_{(x,y,t)} + \frac{1}{2} g_3 \left( cos^2 (\omega t) \frac{\partial ^2}{\partial x^2} + sin^2 (\omega t) \frac{\partial ^2}{\partial y^2} \right) \Lambda_{(x,y,t)}$

$\displaystyle \hbar \frac{\partial}{\partial t} \Lambda_{(x,y,t)} = -\hbar \Omega \Gamma_{(x,y,t)} + 4g_1 \left( xcos(\omega t) - ysin(\omega t) \right) Z_{(x,y,t)} - 4g_2 \left( x cos(\omega t) - y sin(\omega t) \right) \Gamma_{(x,y,t)} - 8g_3 \left( x^2 cos^2 (\omega t) + y^2 sin^2 (\omega t) - 2xy cos(\omega t) sin(\omega t) \right) \Gamma_{(x,y,t)} - \frac{1}{2} g_3 \left( cos^2 (\omega t) \frac{\partial ^2}{\partial x^2} + sin^2 (\omega t) \frac{\partial ^2}{\partial y^2} \right) \Gamma_{(x,y,t)}$

I want to try and solve these analytically as best I can. Is it possible to use perturbation theory or characteristics...I'm not sure where to begin??

Any help would be great...cheers :D