具有竞争色散的双调和非线性薛定谔方程的基态

中文摘要 AI 辅助

我们通过椭圆剖面方程 $\Delta^2 Q + 2a \Delta Q + bQ - |Q|^\alpha Q=0$ 研究具有混合色散的四阶非线性薛定谔方程的基态解，重点关注在 $a=1$, $b=1+\epsilon$, $0<\epsilon \ll 1$ 区域内二维最小作用基态的对称性破缺，其中非径向性由 Lenzmann 和 Weth [26] 证明。我们建立了一个商-质量关系，将 Weinstein 商的小 $\epsilon$ 渐近性转化为质量渐近性，该关系适用于任意维数和对称类。在一维中，我们证明了对于每个 $\alpha>0$，商的尖锐小 $\epsilon$ 速率，并推导出质量、作用和势范数的速率。在二维和更高维中，相同的关系与 Lenzmann-Weth [26] 和 Mandel-Oliveira e Silva [27] 的商渐近性相结合，给出了 Knapp 型非径向和径向基态的质量速率。由此得到的阈值 $\alpha_K(d)=8/(d+1)$ 在 $d\leq3$ 时与 Fernández-Jeanjean-Mandel-Mariş [16] 的归一化解存在性阈值一致，将最小作用公式和质量约束公式联系起来。我们通过详细的数值研究对此进行补充。对于三次非线性，我们从 Knapp 型例子构造了一个非径向分支，将其与径向和角模态分支进行比较，并验证了预测的速率。对于四次非线性，在径向或双峰和四峰非径向解的质量-能量图中存在转向点和分支。在此，尽管最小作用基态对于小 $\epsilon$ 是非径向的，但相同质量的归一化基态表现为径向，这是分支的结果。对于 $\alpha\geq4$，我们只发现径向基态，这与 Stein-Tomas 不等式在 $\mathbb S^1$ 上的猜想尖锐形式一致，该形式等价于径向和非径向质量具有相同的前导系数。

英文摘要

We study ground-state solutions of the 4th order nonlinear Schrödinger equation with mixed dispersion via the elliptic profile equation $Δ^2 Q + 2a ΔQ + bQ - |Q|^αQ=0$, focusing on symmetry breaking of least-action ground states in 2D in the regime $a=1$, $b=1+ε$, $0<ε\ll 1$, where nonradiality was proved by Lenzmann and Weth [26]. We establish a quotient-to-mass relation converting small-$ε$ asymptotics of the Weinstein quotient into mass asymptotics, in any dimension and symmetry class. In 1D we prove the sharp small-$ε$ rate for the quotient for every $α>0$ and deduce the rates for the mass, action and potential norm. In two and higher dimensions the same relation, combined with the quotient asymptotics of Lenzmann-Weth [26] and Mandel-Oliveira e Silva [27], gives the mass rates for the Knapp-type nonradial and the radial ground states. The resulting threshold $α_K(d)=8/(d+1)$ coincides for $d\leq3$ with the existence of normalized minimizers threshold of Fernández-Jeanjean-Mandel-Mariş [16], connecting the least-action and mass-constrained formulations. We complement this with a detailed numerical study. For the cubic nonlinearity we construct a nonradial branch from a Knapp-type example, compare it with radial and angular-mode branches, and verify the predicted rates. For the quartic nonlinearity, turning points and branching are present in the mass-energy diagrams in radial or two- and four-peak nonradial solutions. Here, although least-action ground states are nonradial for small $ε$, the normalized ground states of the same mass appear radial, a consequence of branching. For $α\geq4$ we find only radial ground states, consistent with the conjectured sharp form of the Stein-Tomas inequality on $\mathbb S^1$, equivalent to the radial and nonradial masses having the same leading coefficient.

发表机构

• Université Bourgogne Europe, CNRS, IMB UMR 5584（勃艮第欧洲大学）
• Institut Universitaire de France（法国高等研究院）
• Department of Mathematics & Statistics Florida International University（佛罗里达国际大学数学与统计系）

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