We shall now focus on the behaviour of solutions of (32) near points $(x,0)$ with $u(x,0)=0$.

We call a point $(x,0)$ in $U$ with $u(x,0)=0$ a singularity of the solution $u,v$. We call a singularity $(x,0)$ isolated if there exists $\epsilon>0$ such that the open disc $B_{\epsilon}(x,0)$ of radius $\epsilon$ about $(x,0)$ lies in $U$, and the only point $(x^{\prime},y^{\prime})$ in $B_{\epsilon}(x,0)$ with $u(x^{\prime},y^{\prime})=0$ and $v(x^{\prime},y^{\prime})=v(x,0)$ is $(x,0)$.

Let $(x,0)$ be an isolated singularity of $u,v$, and let $\epsilon$ be as above. Consider the map $\gamma:{\mathcal{S}}^{1}\rightarrow\mathbin{\mathbb{C}}$ given by

$$\gamma:{\rm e}^{i\theta}\mapsto u\bigl(x+{\textstyle\frac{1}{2}}\epsilon\cos\theta,{\textstyle\frac{1}{2}}\epsilon\sin\theta\bigr)-iv\bigl(x+{\textstyle\frac{1}{2}}\epsilon\cos\theta,{\textstyle\frac{1}{2}}\epsilon\sin\theta\bigr)+iv(x,0).$$

As $(x,0)$ is isolated we see that $\gamma$ is smooth and maps ${\mathcal{S}}^{1}\rightarrow\mathbin{\mathbb{C}}\setminus\{0\}$. Define the order of the isolated singularity $(x,0)$ to be the winding number of $\gamma$ about 0 in $\mathbin{\mathbb{C}}$. It is easy to show that the order is independent of $\epsilon$, provided $\epsilon>0$ is sufficiently small.

Not all singularities are isolated. For instance, if we put $u(x,y)=\alpha y$ and $v(x,y)=\alpha x+c$ for $\alpha,c\in\mathbin{\mathbb{R}}$, as in Example 6.3, then $(x,0)$ is a nonisolated singularity for all $x$. However, the author believes that nonisolated singularities are rather nongeneric, and so not of much interest in this paper. Also, by analogy with the Identity Theorem of complex analysis, the author conjectures that if $(x,0)$ is a singularity of $u,v$ and $x$ is an isolated zero of $u(x^{\prime},0)$, then $(x,0)$ is isolated.

The motivation for this definition is as follows. In §6.1 we saw that equation (32) is a nonlinear version of the Cauchy–Riemann equations for $u-iv$ to be a holomorphic function of $x+iy$. So it seems reasonable for singularities of $u,v$ to be a bit like zeros of holomorphic functions.

But zeros of holomorphic functions have an order, which is a positive integer. Definition 6.4 mimics the definition of this. In particular, if $u-iv$ were really holomorphic near $(x,0)$ then for $(x^{\prime},y^{\prime})$ near $(x,0)$ we would expect

$$u(x^{\prime},y^{\prime})-iv(x^{\prime},y^{\prime})\approx\alpha(x^{\prime}-x+iy^{\prime})^{k}+ic$$

for $\alpha\in\mathbin{\mathbb{C}}\setminus\{0\}$, $c\in\mathbin{\mathbb{R}}$ and $k>0$ in $\mathbin{\mathbb{Z}}$, and the order of $(x,0)$ would be $k$.

Our next result follows from Propositions 6.7 and 6.8. In particular, parts (b) and (c) of each imply that the singularity is isolated and of order 1.

Here is a conjecture on isolated singularities.

Note that if $u,v$ are solutions of (25), then so are $u^{\prime},v^{\prime}$, where $u^{\prime}=-u$ and $v^{\prime}=-v$. If $u(x,0)$ is strictly increasing, as in (c), then $u^{\prime}(x,0)$ is strictly decreasing. Similarly, if $u(x,0)$ is strictly increasing for $x>0$ and decreasing for $x<0$, then $u^{\prime}(x,0)$ is strictly decreasing for $x>0$ and increasing for $x<0$. The author speculates that there are essentially only two kinds of isolated singularity at (0,0) of order $k$, those in which $u(x,0)$ increases or decreases near $x=0$ as in the conjecture, and those in which it does the opposite.

The author does not yet know how to prove Conjecture 6.14. However, by Proposition 6.5 the conjecture can be reduced to a statement about singular solutions of the second-order nonlinear p.d.e.

$$\frac{\partial^{2}f}{\partial x^{2}}+2\Bigl(\Bigl(\frac{\partial f}{\partial x}\Bigr)^{2}+y^{2}\Bigr)^{1/2}\frac{\partial^{2}f}{\partial y^{2}}=0$$

(46)

on $\mathbin{\mathbb{R}}^{2}$. This is a fairly simple equation, and it seems likely that the conjecture could be proved (or disproved) using existing results. If any reader knows how to do this, the author would be glad to be told.

As supporting evidence for Conjecture 6.14, consider the related linear problem of functions $f:\mathbin{\mathbb{R}}^{2}\rightarrow\mathbin{\mathbb{R}}$ satisfying

$$\frac{\partial^{2}f}{\partial x^{2}}+2(x^{4}+y^{2})^{1/2}\frac{\partial^{2}f}{\partial y^{2}}=0.$$

(47)

This equation has singular behaviour at $(0,0)$ that is somewhat similar to that of (46) at points $(x,0)$ with $\frac{\partial f}{\partial x}(x,0)=0$. It also has a useful scaling property: if $f(x,y)$ is a solution to (47) then so is $f(tx,t^{2}y)$ for any $t>0$.

Therefore we may look for solutions $f$ of (47) which are homogeneous of order $\alpha$ under this scaling, so that $f(tx,t^{2}y)=t^{\alpha}f(x,y)$ for some $\alpha>2$. Then $f$ is determined by $\alpha$ and its values on the circle $x^{2}+y^{2}=1$, and (47) reduces to a linear o.d.e. on the circle.

For generic values of $\alpha$ this o.d.e. has no nonzero solutions, but for a discrete set of values of $\alpha$ there do exist nontrivial solutions, which give solutions of (47). By studying these homogeneous solutions, the author is able to prove an analogue of Conjecture 6.14 for the equations

$$\frac{\partial u}{\partial x}=-2\bigl(x^{4}+y^{2}\bigr)^{1/2}\frac{\partial v}{\partial y}\quad\text{and}\quad\frac{\partial u}{\partial y}=\frac{\partial v}{\partial x}.$$