Assumption 7.3 In the situation above, the functions $u_{a,b}$ and $v_{a,b}$ satisfy

• (i)

  For all $a,b$, the function $u_{a,b}(x,0)$ is strictly increasing for $x<0$ and strictly decreasing for $x>0$, with a maximum at 0.

• (ii)

  For all $a,b,b^{\prime},x,y$ with $b<b^{\prime}$ we have $u_{a,b}(x,y)<u_{a,b^{\prime}}(x,y)$.

• (iii)

  Let $b>0$, and write $b=\beta^{2}$ for $\beta>0$. Then $u_{a,b}(\beta,0)=u_{a,b}(-\beta,0)=0$ for all $a$. The solution $u_{0,b},v_{0,b}$ of (32) has isolated singularities of order 1 at $(\pm\beta,0)$, in the sense of Definition 6.4.

  Near $(-\beta,0)$ for small $a$, the functions $u_{a,b},v_{a,b}$ are approximately equal to the functions $u,v$ constructed from $N_{a,-\beta}$ in Proposition 6.7.

  Near $(\beta,0)$ for small $a$, the functions $u_{a,b},v_{a,b}$ are approximately equal to the functions $u,v$ constructed from $N^{\prime}_{a,\beta}$ in Proposition 6.8.

• (iv)

  For all $a$ we have $u_{a,0}(0,0)=0$, and the solution $u_{0,0},v_{0,0}$ of (32) has an isolated singularity of order 2 at $(0,0)$, in the sense of Definition 6.4.

• (v)

  For all $a,x\in\mathbin{\mathbb{R}}$ and $b<0$, we have $u_{a,b}(x,0)<0$.