Orbital stability for NLS

Let $n=3$, and take . Some of the material we will present for general $n$ when it doesn’t simplify our work to use $n=3$.

The (defocusing) nonlinear Schrödinger equation is

$$i{\varphi }_t+\mathrm{\Delta }\varphi +{|\varphi |}^{p-1}\varphi =0.$$

$\varphi (x,0)={\varphi }_0\in H^1$.

For a function $\psi$ on ${ℝ}^n$, the orbit of the function under the symmetries of NLS is

$${𝒢}_{\psi }=\{\psi (\cdot +x_0)e^{i\gamma }:(x_0,\gamma )\in {ℝ}^n\times 𝕋\}.$$

We say that $\psi$ is orbitally stable if initial data being near it implies that the solution of NLS is near it always.

We define

$$\rho (\varphi (t),{𝒢}_{\psi })=\underset{(x_0,\gamma )\in {ℝ}^n\times 𝕋}{inf}\parallel \varphi (\cdot +x_0,t)e^{i\gamma }-\psi {\parallel }_{H^1}.$$

The ground state equation is

$$\mathrm{\Delta }u-u+{|u|}^{p-1}u=0.$$

The ground state equation comes from the solution $\varphi (x,t)=e^{it}u(x)$ of NLS. It is a fact that there is a positive bounded solution $R$ of the ground state equation, which we call a ground state.

Theorem 1.

The ground state $R$ is orbitally stable: for any $ϵ\mathrm{>}\mathrm{0}$ there is a $\delta \mathit{}\mathrm{(}ϵ\mathrm{)}\mathrm{>}\mathrm{0}$ such that if

then for all $t\mathrm{>}\mathrm{0}$

We define the energy functional $ℰ$ by

$$ℰ[\varphi ]=\int {|\nabla \varphi |}^2+{|\varphi |}^2-\frac{2}{p+1}{|\varphi |}^{p+1}dx,$$

so $ℰ[\varphi ]$ is a function of time but not of space.

It is a fact that for each $t$ there are $x_0=x_0(t)$ and $\gamma =\gamma (t)$ such that

$$\parallel \varphi (\cdot +x_0,t)e^{i\gamma }-R{\parallel }_{H^1}=\rho (\varphi (t),{𝒢}_R).$$

Let $w=\varphi (\cdot +x_0,t)e^{i\gamma }-R$; so ${\parallel w(t)\parallel }_{H^1}=\rho (\varphi (t),{𝒢}_R)$.

Let $\mathrm{\Delta }ℰ=ℰ[{\varphi }_0]-ℰ[R]$. We have

$\mathrm{\Delta }ℰ$ $=$ $ℰ[\varphi (\cdot ,t]-ℰ[R]$

$=$ $ℰ[\varphi (\cdot +x_0,t)e^{i\gamma }]-ℰ[R]$

$=$ $ℰ[R+w]-ℰ[R].$

We shall express $ℰ[R+w]$ as a Taylor expansion about $R$. We compute the first variation as follows:

$dℰ[R]w$ $=$ ${\displaystyle \int \nabla w\nabla \overline{R}}+\nabla R\nabla \overline{w}+w\overline{R}+R\overline{w}-{|R|}^{p-1}(w\overline{R}+R\overline{w})$

$=$ $2\mathrm{\Re }{\displaystyle \int \nabla w\nabla R}+wR-w{|R|}^{p-1}R$

$=$ $2\mathrm{\Re }{\displaystyle \int w(-\mathrm{\Delta }R+R-{|R|}^{p-1}R)}$

$=$ $0,$

where we used the fact that $R$ is real valued, integration by parts, and the fact that $R$ is a solution of the ground state equation. So the first variation of $ℰ$ at $R$ is $0$.

We now compute the second variation of $ℰ$.

$d^2ℰ[R][w]$ $=$ $2\mathrm{\Re }{\displaystyle \int }-w\mathrm{\Delta }\overline{w}+{|w|}^2-{\displaystyle \frac{p-1}{2}}{R}^{p-1}{w}^2-{\displaystyle \frac{p-1}{2}}{R}^{p-1}{|w|}^2$

$-R^{p-1}{|w|}^2$

$=$ $2\mathrm{\Re }{\displaystyle \int }-w\mathrm{\Delta }\overline{w}+{|w|}^2-{\displaystyle \frac{p-1}{2}}{R}^{p-1}{w}^2-{\displaystyle \frac{p+1}{2}}{R}^{p-1}{|w|}^2$

Write $w=u+iv$. Then we have

$d^2ℰ[R][w]$ $=$ $2{\displaystyle \int }-u\mathrm{\Delta }u-v\mathrm{\Delta }v+u^2+v^2-R^{p-1}(p{u}^2+v^2)$

Define

$$L_{+}=-\mathrm{\Delta }+1-p{R}^{p-1}\mathit{\hspace{1em}\hspace{1em}}{L}_{-}=-\mathrm{\Delta }+1-R^{p-1},$$

which gives

$${(L_{+}u,u)}_{L^2}=\int -u\mathrm{\Delta }u+u^2-p{u}^2{R}^{p-1}$$

and

$${(L_{-}v,v)}_{L^2}=\int -v\mathrm{\Delta }v+v^2-v^2{R}^{p-1}.$$

Thus

$$d^2ℰ[R][w]=2{(L_{+}u,u)}_{L^2}+2{(L_{-}v,v)}_{L^2}.$$

And we assert that the remainder term of the Taylor series is $O(\int {|w|}^3)$, because $R$ is bounded. Therefore

$$\mathrm{\Delta }ℰ={(L_{+}u,u)}_{L^2}+{(L_{-}v,v)}_{L^2}+O\left(\int {|w|}^3\right).$$

We can bound $\int {|w|}^3$ using the Gagliardo-Nirenberg inequality, which gives us (for $n=3$)

$${\parallel w\parallel }_{L^3}^3\le C_0{\parallel \nabla w\parallel }_{L^2}^{3/2}{\parallel w\parallel }_{L^2}^{3/2}\le C_0{\parallel w\parallel }_{H^1}^3,$$

for some $C_0$ that doesn’t depend on $w$. Therefore

$$\mathrm{\Delta }ℰ={(L_{+}u,u)}_{L^2}+{(L_{-}v,v)}_{L^2}+O({\parallel w\parallel }_{H^1}^3),$$

so there is some $C$ such that

$$\mathrm{\Delta }ℰ\ge {(L_{+}u,u)}_{L^2}+{(L_{-}v,v)}_{L^2}-C{\parallel w\parallel }_{H^1}^3.$$