High-Velocity Inverse Scattering for Nonlinear Schrödinger Equations with Spatially Dependent Nonlinearities
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| Format: | Preprint |
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2026
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| _version_ | 1866916072320401408 |
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| author | Masaki, Satoshi Zhao, Yaxin |
| author_facet | Masaki, Satoshi Zhao, Yaxin |
| contents | We study a high-velocity inverse scattering problem for nonlinear Schrödinger equations with spatially dependent nonlinearities in dimensions $d\ge3$. We consider the whole mass-supercritical and energy-subcritical range, including the endpoint cases. By introducing a moving frame adapted to highly boosted initial data, we construct the scattering operator for a class of large incoming states generated by Galilean boosts. The key observation is that, although the boosted data become large in Sobolev norms, the nonlinear interaction becomes effectively weak at high velocity due to rapid spatial separation. Using the resulting high-velocity asymptotics, we derive a reconstruction formula for the X-ray transform of the coefficient. As a consequence, we prove that the scattering operator uniquely determines both the nonlinearity exponent and the spatial coefficient. Our results extend previous work of Watanabe to all dimensions $d \ge 3$, include the endpoint nonlinearities, and replace the repulsiveness and radial monotonicity assumptions on the coefficient by suitable decay conditions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2606_01762 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | High-Velocity Inverse Scattering for Nonlinear Schrödinger Equations with Spatially Dependent Nonlinearities Masaki, Satoshi Zhao, Yaxin Analysis of PDEs Primary 35P25, Secondary 35Q55, 35R30, 35B40 We study a high-velocity inverse scattering problem for nonlinear Schrödinger equations with spatially dependent nonlinearities in dimensions $d\ge3$. We consider the whole mass-supercritical and energy-subcritical range, including the endpoint cases. By introducing a moving frame adapted to highly boosted initial data, we construct the scattering operator for a class of large incoming states generated by Galilean boosts. The key observation is that, although the boosted data become large in Sobolev norms, the nonlinear interaction becomes effectively weak at high velocity due to rapid spatial separation. Using the resulting high-velocity asymptotics, we derive a reconstruction formula for the X-ray transform of the coefficient. As a consequence, we prove that the scattering operator uniquely determines both the nonlinearity exponent and the spatial coefficient. Our results extend previous work of Watanabe to all dimensions $d \ge 3$, include the endpoint nonlinearities, and replace the repulsiveness and radial monotonicity assumptions on the coefficient by suitable decay conditions. |
| title | High-Velocity Inverse Scattering for Nonlinear Schrödinger Equations with Spatially Dependent Nonlinearities |
| topic | Analysis of PDEs Primary 35P25, Secondary 35Q55, 35R30, 35B40 |
| url | https://arxiv.org/abs/2606.01762 |