Characteristic fold for sinusoidal data in a forced Burgers equation ($y_{\eta }=1+x\cos \eta ,u_y=\frac{\cos \eta }{1+x\cos \eta }$)

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 surfaceThe 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 $\eta \ge 0$, 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 $x\uparrow 1$, and at fixed $y=2k\pi -7/6$ as $x\downarrow -1$, using $k\ge 0$ and $k\ge 1$, respectively, to stay in the half-line initial domain. At these first singularities $y_{\eta \eta }=0$ and $y_{\eta \eta \eta }\ne 0$; this is the onset of loss of invertibility, rather than an ordinary nondegenerate fold at the initial singular time.