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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2410.14762 |
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| _version_ | 1866918179907829760 |
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| author | Li, Deke Li, Yuan Wang, Qingxuan |
| author_facet | Li, Deke Li, Yuan Wang, Qingxuan |
| contents | We investigate the thermodynamic limit for the cubic-quintic Schrödinger model as the size of the domain tends to infinity with fixed density $ρ= N/|\mathcal{D}|$, where $N$ denotes particle number and $|\mathcal{D}|$ denotes the volume of the bounded domain $\mathcal{D}\subset\mathbb{R}^d$ ($d=1,2,3$). We firstly prove the existence of thermodynamic limit, which is equal to $-\frac{3}{32}$ for \(0<ρ\leq \frac{3}{4}\), while $-\left(\frac{1}{2}-\fracρ{3}\right)\fracρ{2}$ for $\frac{3}{4}< ρ\leq 1$. When \(0<ρ<1\) and \(\mathcal{D}\) is a spherical domain, we further show that, up to a scaling, the ground state of the cubic-quintic Schrödinger energy will converge strongly to a Thomas-Fermi ground state in $L^2\cap L^6$. Finally, we obtain the $L^\infty$-convergence rate of ground states for \(0<ρ<3/4\) by developing a novel method, including some iterative techniques, uniform energy estimates and gradient estimates. We believe this method is applicable to other general nonlinearities. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_14762 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Thermodynamic limit and $L^\infty$-convergence rate for the cubic-quintic Schrödinger model Li, Deke Li, Yuan Wang, Qingxuan Analysis of PDEs We investigate the thermodynamic limit for the cubic-quintic Schrödinger model as the size of the domain tends to infinity with fixed density $ρ= N/|\mathcal{D}|$, where $N$ denotes particle number and $|\mathcal{D}|$ denotes the volume of the bounded domain $\mathcal{D}\subset\mathbb{R}^d$ ($d=1,2,3$). We firstly prove the existence of thermodynamic limit, which is equal to $-\frac{3}{32}$ for \(0<ρ\leq \frac{3}{4}\), while $-\left(\frac{1}{2}-\fracρ{3}\right)\fracρ{2}$ for $\frac{3}{4}< ρ\leq 1$. When \(0<ρ<1\) and \(\mathcal{D}\) is a spherical domain, we further show that, up to a scaling, the ground state of the cubic-quintic Schrödinger energy will converge strongly to a Thomas-Fermi ground state in $L^2\cap L^6$. Finally, we obtain the $L^\infty$-convergence rate of ground states for \(0<ρ<3/4\) by developing a novel method, including some iterative techniques, uniform energy estimates and gradient estimates. We believe this method is applicable to other general nonlinearities. |
| title | Thermodynamic limit and $L^\infty$-convergence rate for the cubic-quintic Schrödinger model |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2410.14762 |