Inverse scattering problem for the third-order equation on the line
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| author | Aktosun, Tuncay Toledo, Ivan Unlu, Mehmet |
| author_facet | Aktosun, Tuncay Toledo, Ivan Unlu, Mehmet |
| contents | We consider the third-order linear differential equation $$\displaystyle\frac{d^3ψ}{dx^3}+Q(x)\,\displaystyle\frac{dψ}{dx}+P(x)\,ψ=k^3\,ψ,\qquad x\in\mathbb R,$$ where the complex-valued potentials $Q$ and $P$ are assumed to belong to the Schwartz class. We describe the basic solutions, the scattering coefficients, and the bound-state information, and we introduce the dependency constants and the normalization constants at the bound states. When the secondary reflection coefficients are zero, we provide a method to solve the corresponding inverse scattering problem, where the goal is to recover the two potentials $Q$ and $P$ from the scattering data set consisting of the transmission and primary reflection coefficients and the bound-state information. We formulate the corresponding inverse scattering problem as a Riemann--Hilbert problem on the complex $k$-plane and describe how the potentials are recovered from the solution to the Riemann--Hilbert problem. In the absence of bound states, we introduce a linear integral equation, which is the analog of the Marchenko integral equation used in the inverse scattering theory for the full-line Schrödinger equation. We describe the recovery of the two potentials from the solution to the aforementioned linear integral equation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_09346 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Inverse scattering problem for the third-order equation on the line Aktosun, Tuncay Toledo, Ivan Unlu, Mehmet Mathematical Physics Exactly Solvable and Integrable Systems 34A55 34L25 34M50 47A40 We consider the third-order linear differential equation $$\displaystyle\frac{d^3ψ}{dx^3}+Q(x)\,\displaystyle\frac{dψ}{dx}+P(x)\,ψ=k^3\,ψ,\qquad x\in\mathbb R,$$ where the complex-valued potentials $Q$ and $P$ are assumed to belong to the Schwartz class. We describe the basic solutions, the scattering coefficients, and the bound-state information, and we introduce the dependency constants and the normalization constants at the bound states. When the secondary reflection coefficients are zero, we provide a method to solve the corresponding inverse scattering problem, where the goal is to recover the two potentials $Q$ and $P$ from the scattering data set consisting of the transmission and primary reflection coefficients and the bound-state information. We formulate the corresponding inverse scattering problem as a Riemann--Hilbert problem on the complex $k$-plane and describe how the potentials are recovered from the solution to the Riemann--Hilbert problem. In the absence of bound states, we introduce a linear integral equation, which is the analog of the Marchenko integral equation used in the inverse scattering theory for the full-line Schrödinger equation. We describe the recovery of the two potentials from the solution to the aforementioned linear integral equation. |
| title | Inverse scattering problem for the third-order equation on the line |
| topic | Mathematical Physics Exactly Solvable and Integrable Systems 34A55 34L25 34M50 47A40 |
| url | https://arxiv.org/abs/2506.09346 |