具有不连续驻波的可显式求解NLS模型
An explicitly solvable NLS model with discontinuous standing waves
AI总结：
本文研究带点相互作用的NLS方程，证明次临界和临界情形下基态与激发态的存在性及唯一性，给出最优常数并显式描述驻波分支，构成可求解模型。
摘要：
我们研究直线上的NLS方程，该方程具有由吸引δ势与偶极相互作用叠加而成的点相互作用，考虑$L^2$-次临界和$L^2$-临界非线性情形。对于次临界非线性，我们证明在任何质量下基态的存在性和唯一性。如果质量超过显式阈值，则还存在正激发态。对于临界非线性，我们证明基态仅存在于特定的质量区间内，而在不同的区间内存在激发态。我们给出了Gagliardo-Nirenberg估计中最优常数的值，并描述了偶极情形下驻波分支随相互作用强度的变化。由于所有驻波均可显式计算，我们的模型是一个可求解模型，涉及非线性与点相互作用的非标准相互作用。
英文摘要：
We study the NLS Equation on the line with a point interaction given by the superposition of an attractive delta potential with a dipole interaction, in the cases of $L^2$-subcritical and $L^2$-critical nonlinearity. For a subcritical nonlinearity we prove the existence and the uniqueness of Ground States at any mass. If the mass exceeds an explicit threshold, then there exists a positive excited state too. For the critical nonlinearity we prove that Ground States exist only in a specific interval of masses, while in a different interval excited states exist. We provide the value of the optimal constant in the Gagliardo-Nirenberg estimate and describe in the dipole case the branches of the stationary states as the strength of the interaction varies. Since all stationary states are explicitly computed, ours is a solvable model involving a non-standard interplay of a nonlinearity with a point interaction.
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