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so that
The expression
is always nonnegative, since θ _{0} is a minimum f . Similarly,
is positive, since $\gamma \left(t\right)$ is positive and ϕ _{θ} is somewhere nonzero by assumption. Thus $E\left(\varphi \right)-E\left(\tilde{\varphi}\right)>0$ . $\square $
In section 2.3, we prove that there is a unique $\varphi (\theta ,t):[0,2\pi ]\times [0,1]\to \mathbb{R}$ that is periodic in θ and satisfies $\Delta \varphi +\frac{sin\left(2\varphi \right)}{2}=0$ with given boundary conditions. (The proof actually holds for $t\in [0,h]$ , where $0<h<\sqrt{8}$ .) Furthermore, at least with horizontal boundary conditions, the short cylinder admits a vector field which attains the minimum energy. However, this is not the case on a sufficiently tall cylinder: $\varphi (\theta ,t)=0$ is an unstable critical function.
Given a cylinder of height h , consider ${\varphi}_{\u03f5,h}(\theta ,t)=\u03f5t(h-t)$ . (Since this satisfies ${\varphi}_{\u03f5,h}(\theta ,0)={\varphi}_{\u03f5,h}(\theta ,h)=0$ , it describes a vector field with horizontal boundary conditions.) We can compute
Plotting $E\left({\varphi}_{\u03f5,h}\right)-2\pi h$ with respect to ϵ and h yields the graph shown in the following figure:
There is a clear region on which $E\left(0\right)=2\pi h$ is greater than $E\left({\varphi}_{\u03f5,h}\right)$ . It is simple to construct specific examples where ${\varphi}_{\u03f5,h}$ has lower energy than the horizontal field; a more difficult task is to find the lowest h _{0} that admits a lower-energy field. By Theorem 2, such an h _{0} is bounded below by $\sqrt{8}$ . A series of examples bounds it above by $\sqrt{10}$ , but the possibility remains that a better bound is available.
Even without an exact value for h _{0} , we may discuss the significance of its existence. On a cylinder less than a certain height, the horizontal vector field described by $\varphi (\theta ,t)=0$ uniquely minimizes energy; on any tall cylinder, a slight turn up or down will show an improvement. An energy-minimizing function on the tall cylinder, if it exists, must still satisfy the differential equation $\Delta \varphi +\frac{sin\left(2\varphi \right)}{2}=0$ , but $\varphi =0$ is a trivial solution to this. We are left with two possible situations.
Perhaps solutions $\varphi :[0,2\pi ]\times [0,h]\to \mathbb{R},\varphi (\theta ,0)=\varphi (\theta ,h)=0$ to the differential equation are unique for $0<h<{h}_{0}$ and non-unique for ${h}_{0}<h$ . This conclusion is plausible, if slightly uncomfortable. Alternatively, it is not obvious that on a cylinder of height greater than h _{0} , there exists a vector field that attains the minimal energy. The compactness properties of our space have not been adequately explored to say if such a situation makes sense.
We have seen that the partial differential equation
$\Delta \varphi +\frac{sin\left(2\varphi \right)}{2}=0$ does not yield to any of the standard analytical solving techniques. This motivates us to seek out numerical methods to use a computer to approximate the solution. MATLAB is an excellent environment in which to pursue this goal, as it has a powerful
fsolve
command which rapidly and accurately solves partial differential equations.
When a computer “solves" a differential equation, it actually only assures that the differential equation is satisfied at a finite number of points. Between these points the computer uses polynomial interpolation to create a smooth solution. The question then arises as to how well we can trust the interpolation between these points. There is a classic example, called Runge's Phenomenon, of how interpolating using an evenly spaced grid leads to disastrous results. Suppose we would like to approximate the function
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