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Defocusing cubic nonlinear Schrödinger equation in two dimensions

Harmonic-oscillator energy space

{c}
{title2=$-\Delta v+|x|^2v-\omega v+v^3=0$}

Critical points of the trapped defocusing cubic energy
$$
J(v)=\frac12\|v\|_\Sigma^2-\frac\omega2\|v\|_2^2+\frac14\|v\|_4^4
$$
satisfy $-\Delta v+|x|^2v-\omega v+v^3=0$.


Schrödinger trapped defocusing stationary equation

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 154 / 1 / 4 / Solution

Schrödinger trapped defocusing stationary equation ( $- Δ v + ∣ x ∣^{2} v - ω v + v^{3} = 0$ )

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Schrödinger trapped defocusing stationary equation (−Δv+∣x∣2v−ωv+v3=0)

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Schrödinger trapped defocusing stationary equation ( $- Δ v + ∣ x ∣^{2} v - ω v + v^{3} = 0$ )

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