# Question: Solving a differential equation (involving phase space variables) and expanding to get an answer an explicit integral

February 18 2013 by
false
Maple 15

1

Hi, I am looking for some help for a computation on Maple. I have the following objects

\begin{align} U_z&=\frac{i}{4}\left(\pi- \frac{\partial}{\partial x} \phi\right) \nonumber \\ U_+ &= \frac{im \lambda}{4} \left(1-\frac{1}{\lambda^2 e^{i \phi}}\right) \nonumber \\ U_- &= \frac{im \lambda}{4} \left(1-\frac{e^{i\phi}} {\lambda^2}\right) \nonumber \end{align}

where lambda is a parameter, with pi and phi are functions of phase space (functions of x), m is just a constant (mass) and i is just the imaginary number.
I would then like to substitute these into

$\frac{\partial v}{\partial x}+v^2= \frac{\partial}{\partial x} \left(U_z -\frac{1}{2}\frac{\partial}{\partial x} \log(U_+)\right)+\left(U_z - \frac{1}{2} \frac{\partial}{\partial x} \log(U_+)\right)^2 +U_-U_+$
and solve for v=v(x). Finally, I wish to expand v(x) in increasing powers of 1/lambda. So it is of the form
$v(x)=\frac{im}{4} \lambda + v_{-1} \lambda^{-1} + v_{-2} \lambda^{-2} + ...$
And I would like to read off these 'coefficients' v_i. So I need a code that allows me to to pick an index and find out what the coefficients expressioni s. The answer (for odd indices I believe) will be an integral in terms of phase space variables. Here is an example which I believe (should) be right, perhaps up to some constants.
$v_{-3}=\int_{-\infty}^{+\infty} dx \left[\left(\pi(x)-\partial_x \phi(x)\right)\sin(\phi)-\frac{1}{2m}\left(\partial_x \pi(x)-\partial_x^2 \phi(x)\right)(\left(\pi(x)-\partial_x\phi(x)\right)^2 -2m^2\cos(\phi))\right]$

Many thanks.
edit: I believe thelast expression was cut off early.  The image is here http://latex.codecogs.com/gif.latex?v_{-3}=\int_{-\infty}^{+\infty}%20dx%20\left[\left(\pi(x)-\partial_x%20\phi(x)\right)\sin(\phi)-\frac{1}{2m}\left(\partial_x%20\pi(x)-\partial_x^2%20\phi(x)\right)(\left(\pi(x)-\partial_x\phi(x)\right)^2%20-2m^2\cos(\phi))\right]