Resonance-induced nonlinear bound states
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915569961271296 |
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| author | Turner, Jackson C. Weinstein, Michael I. |
| author_facet | Turner, Jackson C. Weinstein, Michael I. |
| contents | We study nonlinear bound states -- time-harmonic and spatially decaying ($L^2$) solutions -- of the nonlinear Schrödinger / Gross--Pitaevskii equations (NLS/GP) with a compactly supported linear potential. Such solutions are known to bifurcate from the $L^2$ bound states of an underlying Schrödinger operator $H_V=-\partial_x^2+V$. In this article we prove an extension of this result: for the 1D NLS/GP, nonlinear bound states also arise via bifurcation from the scattering resonance states and transmission resonance states of $H_V$, associated with the poles and zeros, respectively, of the reflection coefficients, $r_\pm(k)$, of $H_V$. The corresponding resonance states are non-decaying and only $L^2_{\rm loc}$. In contrast to nonlinear states arising from $L^2$ bound states of $H_V$, these resonance bifurcations initiate at a strictly positive $L^2$ threshold which is determined by the position of the complex scattering resonance pole or transmission resonance zero. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_19538 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Resonance-induced nonlinear bound states Turner, Jackson C. Weinstein, Michael I. Mathematical Physics Analysis of PDEs Pattern Formation and Solitons We study nonlinear bound states -- time-harmonic and spatially decaying ($L^2$) solutions -- of the nonlinear Schrödinger / Gross--Pitaevskii equations (NLS/GP) with a compactly supported linear potential. Such solutions are known to bifurcate from the $L^2$ bound states of an underlying Schrödinger operator $H_V=-\partial_x^2+V$. In this article we prove an extension of this result: for the 1D NLS/GP, nonlinear bound states also arise via bifurcation from the scattering resonance states and transmission resonance states of $H_V$, associated with the poles and zeros, respectively, of the reflection coefficients, $r_\pm(k)$, of $H_V$. The corresponding resonance states are non-decaying and only $L^2_{\rm loc}$. In contrast to nonlinear states arising from $L^2$ bound states of $H_V$, these resonance bifurcations initiate at a strictly positive $L^2$ threshold which is determined by the position of the complex scattering resonance pole or transmission resonance zero. |
| title | Resonance-induced nonlinear bound states |
| topic | Mathematical Physics Analysis of PDEs Pattern Formation and Solitons |
| url | https://arxiv.org/abs/2510.19538 |