On inverse scattering for the one-dimensional nonlinear Dirac equation
Abstract
The inverse scattering problem for the one-dimensional nonlinear Dirac equation i(ut+ux)+v=𝒩1(u,v), i(vt−vx)+u=𝒩2(u,v) is studied. We assume that the unknown nonlinearities 𝒩j(u,v) (j=1,2) of the equation belong to C∞(ℂ2;ℂ) and satisfy (∂zα𝒩j)(z)=O(|z|max{ 5−|α|,0 }) (|z|→ 0) for any multi-index α∈ (ℕ ∪ { 0 })4. Here, z=(z1,z2)∈ ℂ2, $ \partial_\mathbf{z}^α= (\partial_{z_1},\partial_{\overline{z_1}},\partial_{z_2},\partial_{\overline{z_2}})^α, $ and ∂z1,∂z1̅,∂z2,∂z2̅ are the Wirtinger differential operators. Under some additional assumptions on 𝒩j, we establish a reconstruction formula for (∂zα𝒩j)(0) (|α|≥ 5) using the knowledge of the scattering operator for the equation. Although such results were established for the nonlinear Schrödinger equation [Sasaki 2024] and the nonlinear Klein-Gordon equation [Sasaki 2025], the methods in [Sasaki 2024, 2025] are not directly applicable to the nonlinear Dirac equation due to its system nature. Specifically, it is quite difficult to prove the invertibility of a matrix associated with the input data. To overcome this difficulty, we introduce a new method based on the massless limit for the free solutions.
Related
- Inverse scattering method for nonlinear negative first order coupled Klein-Gordon equation
- An inverse scattering problem for the nonlinear Schrödinger equation under partial harmonic confinement
- Scattering, bound states, and resonances in the one-dimensional Dirac equation via supersymmetric quantum mechanics
- Threshold Scattering for the Three-Dimensional Focusing Energy-Critical Nonlinear Schrödinger Equation
- Global well-posedness and scattering of energy-critical defocusing nonlinear Schrödinger equation on high-dimensional hyperbolic spaces
BibTeX
@misc{sasaki2026inverse,
title = {{On inverse scattering for the one-dimensional nonlinear Dirac equation}},
author = {Hironobu Sasaki},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.23628},
eprint = {2609.23628},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23628},
}
RIS
TY - UNPB
TI - On inverse scattering for the one-dimensional nonlinear Dirac equation
AU - Hironobu Sasaki
PY - 2026
DA - 2026/09/20
PB - arXiv
DO - 10.48550/arxiv.2609.23628
UR - https://arxiv.org/abs/2609.23628
AB - The inverse scattering problem for the one-dimensional nonlinear Dirac equation i(u_t+u_x)+v=𝒩_1(u,v), i(v_t−v_x)+u=𝒩_2(u,v) is studied. We assume that the unknown nonlinearities 𝒩_j(u,v) (j=1,2) of the equation belong to C^∞(ℂ^2;ℂ) and satisfy (∂_z^α𝒩_j)(z)=O(|z|^max{ 5−|α|,0 }) (|z|→ 0) for any multi-index α∈ (ℕ ∪ { 0 })^4. Here, z=(z_1,z_2)∈ ℂ^2, $ \partial_\mathbf{z}^α= (\partial_{z_1},\partial_{\overline{z_1}},\partial_{z_2},\partial_{\overline{z_2}})^α, $ and ∂_z_1,∂_z_1̅,∂_z_2,∂_z_2̅ are the Wirtinger differential operators. Under some additional assumptions on 𝒩_j, we establish a reconstruction formula for (∂_z^α𝒩_j)(0) (|α|≥ 5) using the knowledge of the scattering operator for the equation. Although such results were established for the nonlinear Schrödinger equation [Sasaki 2024] and the nonlinear Klein-Gordon equation [Sasaki 2025], the methods in [Sasaki 2024, 2025] are not directly applicable to the nonlinear Dirac equation due to its system nature. Specifically, it is quite difficult to prove the invertibility of a matrix associated with the input data. To overcome this difficulty, we introduce a new method based on the massless limit for the free solutions.
ER -