Well, I can't seem to find an error in my derivation and I can't decipher your writing, so I will assume the derivation I posted is correct. As someone already mentioned in this thread, it seems impossible that the equation can be solved in a closed form. So, I'm afraid you are left with solving it numerically.
Last time we ended at this step:
Dickfore said:
<br />
\left\{\begin{array}{rcl}<br />
\frac{d v}{d x} & = & p \\<br />
<br />
\frac{d p}{d x} & = & \frac{v}{a} \, \left[p + \frac{n}{2 \, b} \, v^{2} \, (v^{2} - c) \right]<br />
\end{array}\right.<br />
where we also made the transformation p/2 \rightarrow p again.
The boundary conditions that you gave translate to the following:
<br />
\begin{array}{l}<br />
x \rightarrow -\infty \Rightarrow v = \sqrt{c} \\<br />
<br />
x = 0 \Rightarrow v = \sqrt{b + c}<br />
\end{array}<br />
The connection with the old variables is given by:
<br />
u(x) = [v(x)]^{2}<br />
When working with numerics, it is best to get rid of as many parameters as possible. Let's scale everything:
<br />
\begin{array}{l}<br />
x = x_{0} \, \bar{x} \\<br />
<br />
v = v_{0} \, \bar{v} \\<br />
<br />
p = p_{0} \, \bar{p}<br />
\end{array}<br />
Then the equations become:
<br />
\begin{array}{rcl}<br />
\frac{v_{0}}{x_{0}} \, \frac{d \bar{v}}{d \bar{x}} & = & p_{0} \, \bar{p} \\<br />
<br />
\frac{p_{0}}{x_{0}} \, \frac{d \bar{p}}{d \bar{x}} & = & \frac{v_{0}}{a} \, \left[ p_{0} \, \bar{p} + \frac{n}{2 b} \, v^{2}_{0} \, \bar{v}^{2} \left(v^{2}_{0} \, \bar{v}^{2} - c \right) \right]<br />
\end{array}<br />
I think it is most convenient to make this choice:
<br />
v^{2}_{0} = c, \; p_{0} = \frac{n}{2 b} \,v^{4}_{0}. \; \frac{p_{0}}{x_{0}} = \frac{v_{0} \, p_{0}}{a}<br />
<br />
x_{0} = \frac{a}{\sqrt{c}}, p_{0} = \frac{n \, c^{2}}{2 b}, \; v_{0} = \sqrt{c} \Rightarrow \frac{x_{0} \, p_{0}}{v_{0}} = \frac{a n c}{2 b} \equiv k<br />
Also, let's get rid of the bars above the symbols again:
<br />
\begin{array}{rcl}<br />
\frac{d v}{d x} & = & k \, p \\<br />
<br />
\frac{d p}{d x} & = & v \, \left[ p + v^{2} \, (v^{2} - 1) \right]<br />
\end{array}<br />
With the boundary conditions being:
<br />
\left\{\begin{array}{l}<br />
\sqrt{c} \, v = \sqrt{c}, \; x \rightarrow -\infty \\<br />
<br />
\sqrt{c} \, v = \sqrt{b + c}, \; x = 0<br />
\end{array}\right. \Rightarrow \left\{\begin{array}{l}<br />
v = 1, \; x \rightarrow -\infty \\<br />
<br />
v = \sqrt{1 + \frac{b}{c}} = M, \; x = 0<br />
\end{array}\right.<br />
Instead of having this limit as x \rightarrow -\infty, let us make the simultaneous substitution x \rightarrow -x, p \rightarrow -p. The equations become:
<br />
\begin{array}{rcl}<br />
\frac{d v}{d x} & = & k \, p \\<br />
<br />
\frac{d p}{d x} & = & v \, \left[ p + v^{2} \, (v^{2} - 1) \right]<br />
\end{array}<br />
with the boundary conditions:
<br />
\left\{\begin{array}{l}<br />
v = \sqrt{1 + \frac{b}{c}} = M, \; x = 0 \\<br />
<br />
v \rightarrow 1, \; x \rightarrow \infty \\<br />
\end{array}\right.<br />
Can you find the stationary points for this autonomous non-linear system. What is their type?
