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| Format: | Preprint |
| Published: |
2024
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| Online Access: | https://arxiv.org/abs/2410.14300 |
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| _version_ | 1866911156866646016 |
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| author | Li, Deke Wang, Qingxuan |
| author_facet | Li, Deke Wang, Qingxuan |
| contents | We focus on the ground state of the cubic-quintic nonlinear Schrödinger energy functional \begin{gather*}
\begin{aligned}
{E}(φ)=\frac{1}{2}\int_{\mathbb{R}^d}\left(|\nabla φ|^2+V(x)|φ|^2\right)\,dx
\pm\frac{1}{4}\int_{\mathbb{R}^d}|φ|^4\,dx
+\frac{1}{6}\int_{\mathbb{R}^d}|φ|^6\,dx, (d=1,2,3)
\end{aligned} \end{gather*}
under the mass constraint $\int_{\mathbb{R}^d}|φ|^2\,dx=N$, where $N$ can be viewed as particle number, and $V(x)$ behaves like $C|x|^p (p\geq 2)$ as $|x|\rightarrow +\infty$, including the harmonic potential. When $N\rightarrow +\infty$, we show that up to a suitable scaling the ground state $φ_N$ would convergence strongly in some $L^q(\mathbb{R}^d)$ space to a Thomas-Fermi minimizer, this limit can be referred to as the \emph{Thomas-Fermi limit}. The limit Thomas-Fermi profile has compact support, given by $u^{TF}(x)=\left[μ^{TF}-C_0|x|^p\right]^{\frac{1}{4}}_{+}$, where $μ^{TF}$ is a suitable Lagrange multiplier with exact value. We find that, similar to the asymptotic analysis in [J. Funct. Anal. 260 (2011), 2387-2406.] and [Arch. Ration. Mech. Anal. 217 (2015), 439-523.] for Gross-Pitaevskii energy in the Thomas-Fermi limit where a small parameter $\varepsilon$ tends to 0, there also has a steep \emph{corner layer} near the boundary of compact support of $u^{TF}(x)$, in which the ground state has irregular behavior as $N\rightarrow +\infty$. Finally, we establish a new energy method to obtain the $L^\infty$-convergence rates of ground states $φ_N$ inside the corner layer and outside corner layer respectively, this method may be applicable to other general nonlinearities. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_14300 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Ground State of a Cubic-quintic Nonlinear Schrödinger Equation with Radial Potential in the Thomas-Fermi Limit Li, Deke Wang, Qingxuan Analysis of PDEs We focus on the ground state of the cubic-quintic nonlinear Schrödinger energy functional \begin{gather*} \begin{aligned} {E}(φ)=\frac{1}{2}\int_{\mathbb{R}^d}\left(|\nabla φ|^2+V(x)|φ|^2\right)\,dx \pm\frac{1}{4}\int_{\mathbb{R}^d}|φ|^4\,dx +\frac{1}{6}\int_{\mathbb{R}^d}|φ|^6\,dx, (d=1,2,3) \end{aligned} \end{gather*} under the mass constraint $\int_{\mathbb{R}^d}|φ|^2\,dx=N$, where $N$ can be viewed as particle number, and $V(x)$ behaves like $C|x|^p (p\geq 2)$ as $|x|\rightarrow +\infty$, including the harmonic potential. When $N\rightarrow +\infty$, we show that up to a suitable scaling the ground state $φ_N$ would convergence strongly in some $L^q(\mathbb{R}^d)$ space to a Thomas-Fermi minimizer, this limit can be referred to as the \emph{Thomas-Fermi limit}. The limit Thomas-Fermi profile has compact support, given by $u^{TF}(x)=\left[μ^{TF}-C_0|x|^p\right]^{\frac{1}{4}}_{+}$, where $μ^{TF}$ is a suitable Lagrange multiplier with exact value. We find that, similar to the asymptotic analysis in [J. Funct. Anal. 260 (2011), 2387-2406.] and [Arch. Ration. Mech. Anal. 217 (2015), 439-523.] for Gross-Pitaevskii energy in the Thomas-Fermi limit where a small parameter $\varepsilon$ tends to 0, there also has a steep \emph{corner layer} near the boundary of compact support of $u^{TF}(x)$, in which the ground state has irregular behavior as $N\rightarrow +\infty$. Finally, we establish a new energy method to obtain the $L^\infty$-convergence rates of ground states $φ_N$ inside the corner layer and outside corner layer respectively, this method may be applicable to other general nonlinearities. |
| title | The Ground State of a Cubic-quintic Nonlinear Schrödinger Equation with Radial Potential in the Thomas-Fermi Limit |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2410.14300 |