Interactions of semilinear progressing waves in two or more space dimensions

Antônio Sá Barreto

doi:10.3934/ipi.2020055

Article Contents

2020, Volume 14, Issue 6: 1057-1105. Doi: 10.3934/ipi.2020055

This issue Previous Article Direct and inverse time-harmonic elastic scattering from point-like and extended obstacles Next Article Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations

Interactions of semilinear progressing waves in two or more space dimensions

Antônio Sá Barreto

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

Received: January 2020

Revised: July 2020

Early access: August 2020

Published: December 2020

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

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Abstract

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.

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Mathematics Subject Classification: 35A18, 35A21, 35B40, 35L15, 35L71.

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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

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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]

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Figure 3. 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

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Figure 4. Singularities produced by the intersection of three plane waves (3.2). An observer in $ {\mathbb{R}}^3 $ sees a conic shaped wave

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Figure 5. 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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