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| Main Authors: | , , |
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| Format: | Preprint |
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2021
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2110.05277 |
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| _version_ | 1866914654764138496 |
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| author | Meng, Fanfei Wang, Sheng Xu, Chengbin |
| author_facet | Meng, Fanfei Wang, Sheng Xu, Chengbin |
| contents | In this article, we consider the dynamics of the energy-critical quadratic nonlinear Schrödinger system $\[ \left\{ \begin{aligned} & i u^1_t + κ_1 Δu^1 = -\overline{u^2}u^3, \\ & i u^2_t + κ_2 Δu^2 = -\overline{u^1}u^3, \\ & i u^3_t + κ_3 Δu^3 = -u^1u^2, \\ \end{aligned} \right. \qquad (t, x) \in \R \times \R^6 \] in energy-space $ {\dot H}^1 \times {\dot H}^1\times{\dot H}^1 $, where the sign of potential energy can not be determined. We prove the scattering theory with mass-resonance (or with radial initial data) below ground state via concentration compactness method. We discover a family of new physically conserved quantities with mass-resonance which play an important role in the proof of scattering. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2110_05277 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | On the energy-critical quadratic nonlinear Schrödinger system with three waves Meng, Fanfei Wang, Sheng Xu, Chengbin Analysis of PDEs In this article, we consider the dynamics of the energy-critical quadratic nonlinear Schrödinger system $\[ \left\{ \begin{aligned} & i u^1_t + κ_1 Δu^1 = -\overline{u^2}u^3, \\ & i u^2_t + κ_2 Δu^2 = -\overline{u^1}u^3, \\ & i u^3_t + κ_3 Δu^3 = -u^1u^2, \\ \end{aligned} \right. \qquad (t, x) \in \R \times \R^6 \] in energy-space $ {\dot H}^1 \times {\dot H}^1\times{\dot H}^1 $, where the sign of potential energy can not be determined. We prove the scattering theory with mass-resonance (or with radial initial data) below ground state via concentration compactness method. We discover a family of new physically conserved quantities with mass-resonance which play an important role in the proof of scattering. |
| title | On the energy-critical quadratic nonlinear Schrödinger system with three waves |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2110.05277 |