Standing waves of the quintic NLS equation on the tadpole graph
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Abstract
Abstract The tadpole graph consists of a circle and a half-line attached at a vertex. We analyze standing waves of the nonlinear Schrödinger equation with quintic power nonlinearity equipped with the Neumann–Kirchhoff boundary conditions at the vertex. The profile of the standing wave with the frequency $$\omega \in (-\infty ,0)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ω</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>(</mml:mo> <mml:mo>-</mml:mo> <mml:mi>∞</mml:mi> <mml:mo>,</mml:mo> <mml:mn>0</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> is characterized as a global minimizer of the quadratic part of energy constrained to the unit sphere in $$L^6$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>6</mml:mn> </mml:msup> </mml:math> . The set of standing waves includes the set of ground states, which are the global minimizers of the energy at constant mass ( $$L^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> -norm), but it is actually wider. While ground states exist only for a certain interval of masses, the standing waves exist for every $$\omega \in (-\infty ,0)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ω</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>(</mml:mo> <mml:mo>-</mml:mo> <mml:mi>∞</mml:mi> <mml:mo>,</mml:mo> <mml:mn>0</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> and correspond to a bigger interval of masses. It is proven that there exist critical frequencies $$\omega _1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ω</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math> and $$\omega _0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ω</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> with $$-\infty< \omega _1< \omega _0 < 0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>-</mml:mo> <mml:mi>∞</mml:mi> <mml:mo><</mml:mo> <mml:msub> <mml:mi>ω</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo><</mml:mo> <mml:msub> <mml:mi>ω</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo><</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> such that the standing waves are the ground state for $$\omega \in [\omega _0,0)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ω</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>[</mml:mo> <mml:msub> <mml:mi>ω</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mn>0</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , local constrained minima of the energy for $$\omega \in (\omega _1,\omega _0)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ω</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>ω</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>ω</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> and saddle points of the energy at constant mass for $$\omega \in (-\infty ,\omega _1)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ω</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>(</mml:mo> <mml:mo>-</mml:mo> <mml:mi>∞</mml:mi> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>ω</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . Proofs make use of the variational methods and the analytical theory for differential equations.
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Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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