Recovery of a Null Form in the Wave Equation from Scattering Data
Abstract
We use highly oscillatory geometric optics solutions to solve the inverse problem for the system □ u(1) u(2) vdots u(n) = ∑k≥ l=1n (Q0(u(k),u(l)) q1kl q2kl vdots qnkl + ∑i>j=1d Qij(u(k),u(l)) p1ijkl p2ijkl vdots pnijkl ), where Q0 and Qij are the symmetric and anti-symmetric bilinear null forms. We present solutions in both the linear and weakly nonlinear regimes. In the linear regime, we show that the coefficients of order h3 determine an injective light-ray transform of a vector field which depends on the coefficients qrkl, prijkl. In the weakly nonlinear regime, we see that the coefficients of order h determine the non-abelian light ray transform for matrices associated with the coefficients. While we do not have an injectivity result for this case, we do have one if we assume the coefficients do not depend on the time variable x0, as our coefficients instead determine an injective non-abelian X-ray transform.
Related
- Global existence for a scalar quasilinear wave equation in two space dimensions beyond the cubic null condition
- The Direct Scattering Transform for the Intermediate Long Wave Equation
- The Strauss conjecture for the wave equation with scattering damping
- An FFT-Accelerated Boundary Integral Equation Method for Wave Scattering by Smooth Surfaces in Three Dimensions
- Null polynomials over a finite ring need not form a two-sided ideal
BibTeX
@misc{nathe2026recovery,
title = {{Recovery of a Null Form in the Wave Equation from Scattering Data}},
author = {Joel Nathe},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.28917},
eprint = {2607.28917},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.28917},
}
RIS
TY - UNPB
TI - Recovery of a Null Form in the Wave Equation from Scattering Data
AU - Joel Nathe
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2607.28917
UR - https://arxiv.org/abs/2607.28917
AB - We use highly oscillatory geometric optics solutions to solve the inverse problem for the system □ u^(1) u^(2) vdots u^(n) = ∑_k≥ l=1^n (Q_0(u^(k),u^(l)) q_1kl q_2kl vdots q_nkl + ∑_i>j=1^d Q_ij(u^(k),u^(l)) p_1ijkl p_2ijkl vdots p_nijkl ), where Q_0 and Q_ij are the symmetric and anti-symmetric bilinear null forms. We present solutions in both the linear and weakly nonlinear regimes. In the linear regime, we show that the coefficients of order h^3 determine an injective light-ray transform of a vector field which depends on the coefficients q_rkl, p_rijkl. In the weakly nonlinear regime, we see that the coefficients of order h determine the non-abelian light ray transform for matrices associated with the coefficients. While we do not have an injectivity result for this case, we do have one if we assume the coefficients do not depend on the time variable x_0, as our coefficients instead determine an injective non-abelian X-ray transform.
ER -