We propose and study several inverse problems associated with the nonlinear progressive waves that arise in infrasonic inversions. The nonlinear progressive equation is of a quasilinear form <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\partial _t^2 u=\Delta f(x, u)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>∂</mi> <mi>t</mi> <mn>2</mn> </msubsup> <mi>u</mi> <mo>=</mo> <mi mathvariant="normal">Δ</mi> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$f(x, u)=c_1(x) u+c_2(x) u^n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msub> <mi>c</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mi>u</mi> <mo>+</mo> <msub> <mi>c</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <msup> <mi>u</mi> <mi>n</mi> </msup> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$n\ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, and can be derived from the hyperbolic system of conservation laws associated with the Euler equations. We establish unique identifiability results in determining f(x, u) as well as the associated initial data by the boundary measurement. Our analysis relies on high-order linearisation and construction of proper Gaussian beam solutions for the underlying wave equations. In addition to its theoretical interest, we connect our study to applications of practical importance in infrasound waveform inversion.

Inverse problems for nonlinear progressive waves

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2025年3月24日

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Yan Jiang,
Hongyu Liu,
Tianhao Ni,
Kai Zhang

摘要

We propose and study several inverse problems associated with the nonlinear progressive waves that arise in infrasonic inversions. The nonlinear progressive equation is of a quasilinear form $\partial _t^2 u=\Delta f(x, u)$ $\partial_{t}^{2} u = Δ f (x, u)$ with $f(x, u)=c_1(x) u+c_2(x) u^n$ $f (x, u) = c_{1} (x) u + c_{2} (x) u^{n}$ , $n\ge 2$ $n \geq 2$ , and can be derived from the hyperbolic system of conservation laws associated with the Euler equations. We establish unique identifiability results in determining f(x, u) as well as the associated initial data by the boundary measurement. Our analysis relies on high-order linearisation and construction of proper Gaussian beam solutions for the underlying wave equations. In addition to its theoretical interest, we connect our study to applications of practical importance in infrasound waveform inversion.