Web bonus problem : Week 3 (#4)

[Victor Ivrii]

http://www.math.toronto.edu/courses/apm346h1/20159/PDE-textbook/Chapter2/S2.3.P.html#problem-2.3.P.8 Problem 8

[Jeremy Li 2]

I didn't really make it that far on this problem, but I thought I ought to post as far as I got in case anyone has any ideas.

Plugging $u(x,t)=\phi(x-vt)$ into the given PDEs:
\begin{equation}
v^2\phi''-\phi''+\phi-2\phi^3=0\\
v^2\phi''-\phi''-\phi+2\phi^3=0
\end{equation}

And so we get
\begin{equation}
(v^2-1)\phi''=2\phi^3-\phi\\
(v^2-1)\phi''=\phi-2\phi^3
\end{equation}

This ODE looks very difficult - any ideas?

[Chi Ma]

We look for solutions such that $u_{tt}-u_{xx}=u(1-2u^2)=0$.
$u_{tt}-u_{xx}=0$ implies that $u(x,t) = f(x \pm t)$ for some function $f$.
$u(1-2u^2)=0$ implies that either $u = 0$ or $u = \pm \frac{1}{\sqrt 2}$.

A kink may be described by
\begin{equation}
u(x,t) = \left\{\begin{array}{21}
 &\pm \frac{1}{\sqrt 2} \qquad & x \ge t\\
 & 0 & x < t \end{array} \right.
\end{equation}

A soliton may be described by
\begin{equation}
u(x,t) = \left\{\begin{array}{21}
 &\pm \frac{1}{\sqrt 2} \qquad & x = t\\
 & 0 & x \neq t \end{array} \right.
\end{equation}

[Victor Ivrii]

We have 2nd order ODE with no explicit $x$. Could be reduced to 1st order by the standard $\phi'=\psi$, $\phi''=\psi'=\frac{d\psi}{d\phi}\phi'$.