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Consider Helmholtz equation with two spatial variables (and later we replace it by simpler models). Assume that ''initially'' solution is \begin{gather} u(x,y) = A(x,y) e^{ik\phi (x,y)} \label{eq-7.3.1} \end{gather} with eikonal satisfying $|\nabla \phi|=1$. What hapens then?
Obviously wave front $\{(x,y)\colon \phi (x,y)=c\}$ morphes (it moves normally in the oprthogonsal direction) and at some moment it developes singularities (despite Lagrangian manifold remains smooth but its projection on the coordinate space develops singularities). The most extreme case is focusing when $\phi(x,y)= \sqrt{x^2+y^2}$ and wave fronts are concentrated circumferences. One can prove that near the center $u_k(x,y) \asymp k^{-\frac{1}{2}}$
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| Focusing |
In the general case singularities (caustics) appear in the centers of the curvature of wave fronts. What can happen in the generic case? Let us parametrize $\{(x,y)\colon \phi(x,y)=0\}$ by a parameter $\lambda$ and denote $\kappa(\lambda)$ the curvature.
If $\kappa '(\lambda)$ does not vanish we get fold like on the red curves in the picture below except of vicinities of the spikes, which are pleats (a.k.a. Whitney's cusps).
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| Parabola; curvature reaches maximun on its axis |
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| Ellipse; curvature reaches maxima on its large axis and minima on its small exis; see direction of spikes | |
The toy–model would be \begin{gather} u(x,y)=k^{\frac{1}{2}} \int_{-\infty}^\infty e^{ik\bigl(t^3 +xt\bigr)}\,dt= k^{\frac{1}{6}}A (k^{\frac{2}{3}}x) \label{eq-7.3.2} \end{gather} where \begin{gather} A(x)= \int_{-\infty}^\infty e^{i\bigl(t^3+ xt\bigr)}\,dt \label{eq-7.3.3} \end{gather} is the Airy function. In this case fold is described by \begin{gather*} \Phi_t (x,y,t)=0, \qquad \Phi_{tt} (x,y,t)=0 \end{gather*} with \begin{gather*} \Phi (x,y,t)= t^3 +xt \end{gather*} that is $\{x=0,\ t=0\}$.
Obviously, $u\asymp k^{\frac{1}{6}}$ as $|x |\le \epsilon k^{-\frac{2}{3}}$. So, near fold solution is $\asymp k^{\frac{1}{6}}$.
The toy–model would be \begin{gather} u(x,y)=k^{\frac{1}{2}}\int_{-\infty}^\infty e^{ik\bigl(t^4+y t^2+xt\bigr)}\,dt= k^{\frac{1}{4}} P(k^{\frac{1}{2}}y, k^{\frac{3}{4}}x) \label{eq-7.3.4} \end{gather} where \begin{gather} P(x,y)= \int_{-\infty}^\infty e^{i\bigl(t^4+y t^2+xt\bigr)}\,dt \label{eq-7.3.5} \end{gather} is the Pearcey integral. In this case fold is described by \begin{gather*} \Phi_t (x,y,t)=0, \qquad \Phi_{tt} (x,y,t)=0 \end{gather*} with \begin{gather*} \Phi (x,y,t)= t^4+y t^2+xt \end{gather*} that is \begin{gather*} y=-2 t^2,\qquad x= 8 t^3 \iff y=- \frac{3}{2} x^{\frac{2}{3}}. \end{gather*}
Obviously, $u\asymp k^{\frac{1}{4}}$ as $|x|\le \epsilon k^{-\frac{3}{4}}$, $|y|\le \epsilon k^{-\frac{1}{2}}$. So, near pleat solution is $\asymp k^{\frac{1}{4}}$.
There are some studies of more more rare and stronger caustics, especially in a larger spatial dimension and involving more variables of integration. See Corollaries 1 and 2 in