= Characteristic fold for sinusoidal data in a forced Burgers equation
{title2=$y_\eta=1+x\cos\eta,\quad u_y=\frac{\cos\eta}{1+x\cos\eta}$}
For the <partial differential equation> $u_x+u u_y=x$ with initial profile $u(0,\eta)=1+\sin\eta$, the <method of characteristics> gives the parametrized surface
$$
u=1+\sin\eta+\frac{x^2}{2},\qquad y=\eta+x(1+\sin\eta)+\frac{x^3}{6}.
$$
The projection to the $(x,y)$ plane has <Jacobian determinant> $1+x\cos\eta$. It is positive for $|x|<1$, so the <inverse function theorem> makes the surface a unique local solution graph there. For $\eta\ge0$, strict monotonicity and divergence to infinity give a global inverse onto $y\ge x+x^3/6$ at each such $x$.
The <derivative> $u_y=\cos\eta/(1+x\cos\eta)$ becomes singular when that projection first degenerates, although $u$ itself stays finite. At $x=1$ the degenerate labels are odd multiples of $\pi$, and at $x=-1$ they are even multiples. At either boundary the limiting map remains strictly increasing, with isolated zero <derivatives>, so fixed image values determine convergent labels. Consequently the gradient diverges at fixed $y=(2k+1)\pi+7/6$ as $x\uparrow1$, and at fixed $y=2k\pi-7/6$ as $x\downarrow-1$, using $k\ge0$ and $k\ge1$, respectively, to stay in the half-line initial domain. At these first singularities $y_{\eta\eta}=0$ and $y_{\eta\eta\eta}\ne0$; this is the onset of loss of invertibility, rather than an ordinary nondegenerate fold at the initial singular time.
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