Assumption 8.3 In the situation above, the functions $u_{a,b}$ and $v_{a,b}$ satisfy

• (i)

  For all $a,b$, the function $u_{a,b}(x+2\pi\mathbin{\mathbb{Z}},0)$ is strictly increasing for $x$ in $(-\pi,0)$ and strictly decreasing for $x$ in $(0,\pi)$, with a maximum at $2\pi\mathbin{\mathbb{Z}}$ and a minimum at $\pi+2\pi\mathbin{\mathbb{Z}}$.

• (ii)

  For all $a,b,b^{\prime},x,y$ with $b<b^{\prime}$ we have $u_{a,b}(x+2\pi\mathbin{\mathbb{Z}},y)<u_{a,b^{\prime}}(x+2\pi\mathbin{\mathbb{Z}},y)$.

• (iii)

  For all $a,b,x\in\mathbin{\mathbb{R}}$ we have $u_{a,b}(x+2\pi\mathbin{\mathbb{Z}},0)=0$ if and only if $b+\cos x=0$.

• (iv)

  Let $b\in(-1,1)$, and write $b=\cos\beta$ for $\beta\in(0,\pi)$. Then the solution $u_{0,b},v_{0,b}$ of (32) has isolated singularities of order 1 at $(\pm\beta+2\pi\mathbin{\mathbb{Z}},0)$, in the sense of Definition 6.4.

  Near $(-\beta+2\pi\mathbin{\mathbb{Z}},0)$ 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 $(\beta+2\pi\mathbin{\mathbb{Z}},0)$ 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.

• (v)

  The solution $u_{0,-1},v_{0,-1}$ of (32) has an isolated singularity of order 2 at $(2\pi\mathbin{\mathbb{Z}},0)$, and the solution $u_{0,1},v_{0,1}$ has an isolated singularity of order 2 at $(\pi+2\pi\mathbin{\mathbb{Z}},0)$, in the sense of Definition 6.4.

• (vi)

  For all $a,b,x,y\in\mathbin{\mathbb{R}}$ we have $u_{a,-b}(x+2\pi\mathbin{\mathbb{Z}},y)=-u_{a,b}(x+\pi+2\pi\mathbin{\mathbb{Z}},y)$ and $v_{a,-b}(x+2\pi\mathbin{\mathbb{Z}},y)=-v_{a,b}(x+\pi+2\pi\mathbin{\mathbb{Z}},y)$.