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December  2020, 14(6): 1057-1105. doi: 10.3934/ipi.2020055

Interactions of semilinear progressing waves in two or more space dimensions

 Department of Mathematics, Purdue University, 150 North University Street, West Lafayette Indiana, 47907, USA

Received  January 2020 Revised  July 2020 Published  December 2020 Early access  August 2020

Fund Project: The author is supported by the Simons Foundation grant #349507, Antônio Sá Barreto

We analyze the behavior of the singularities of solutions of semilinear wave equations after the interaction of three transversal conormal waves. Our results hold for space dimensions two and higher, and for arbitrary ${{C}^{\infty }}$ nonlinearity. The case of two space dimensions in which the nonlinearity is a polynomial was studied by the author and Yiran Wang. We also indicate possible applications to inverse problems.

Citation: Antônio Sá Barreto. Interactions of semilinear progressing waves in two or more space dimensions. Inverse Problems & Imaging, 2020, 14 (6) : 1057-1105. doi: 10.3934/ipi.2020055
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, Fig.5 and Fig.4 below illustrate the higher dimensional cases">Figure 1.  The interaction of three conormal plane waves in two space dimensions. The only possible singularities created by the triple interaction appear on the surface of the light cone. Fig.3, Fig.5 and Fig.4 below illustrate the higher dimensional cases
]">Figure 2.  A swallowtail singularity formed on the light cone emanating from a point in two space dimensions. This can be due to the existence of conjugate points of the geodesic flow in the case $P = D_t^2-\Delta_g,$ $g$ a Riemannian metric in ${{\mathbb{R}}^{2}}.$ The solution to (2.4) would remain conormal to ${\mathcal{Q}}$ away from the caustic, but other singularities could be generated by the caustic. This figure resembles one after equation 5.1.24 in Duistermaat's book [15]
The dotted line represents an expanding cylindrical wave, generated by the interaction of three plane waves given by (3.1) in ${{\mathbb{R}}^{4}},$ viewed by an observer in ${\mathbb{R}}^3$ as time increases. The speed in which the radius of the wave expands is equal to one
Singularities produced by the intersection of three plane waves (3.2). An observer in ${\mathbb{R}}^3$ sees a conic shaped wave
The dotted line shows the surface (3.4) as $(x_1,x_2,x_3)$ vary for $t$ fixed. Unlike the wave formed by the interaction of three plane waves considered above, which is an infinite cylinder, three spherical waves intersect along a bounded curve for fixed time. The level sets of this surface for $\{x_3 = c\}$ are circles centered on the line $\{x_1 = a, x_2 = b\}.$
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