On Orbital Instability of Spectrally Stable Vortices of the NLS in the Plane

Abstract

We explain how spectrally stable vortices of the nonlinear Schrödinger equation in the plane can be orbitally unstable. This relates to the nonlinear Fermi golden rule, a mechanism which exploits the nonlinear interaction between discrete and continuous modes of the NLS.

Appendix: Proof of Lemma 6.1

First of all, it is equivalent to consider Eq. 6.2 for h. Let $X_c = M ^{-1}L^2_c(\omega _1)$ and by an abuse of notation let us set $\widetilde{P}_c = M ^{-1}P_c(\omega _1) M$, where $P_c$ is introduced under Lemma 2.3. Set also $\mathcal {K}= \mathcal {K}_{\omega _1}$ The following three lemmas are Lemma 3.1–3.3 in Cuccagna and Tarulli (2009).

Lemma 8.1

(Strichartz estimate) There exists a positive number C such that for any $k\in [0,2]$:

• (a) for any $h= \widetilde{P}_c h$ and any admissible all pair (p, q),

  $$\begin{aligned} \Vert e^{-\mathrm{i}t\mathcal {K} } h\Vert _{L_t^pW_x^{k,q } }\le C\Vert h\Vert _{H^k}; \end{aligned}$$

• (b) for any $g(t,x)\in S({\mathbb {R}}^2)$ and any couple of admissible pairs $(p_{1},q_{1})\;(p_{2},q_{2})$ we have

  $$\begin{aligned} \left\| \int _{0}^te^{-\mathrm{i}(t-s)\mathcal {K} } \widetilde{P}_c g(s,\cdot )ds \right\| _{L_t^{p_1}W_x^{k,q_1 } } \le C\Vert g\Vert _{L_t^{p_2'}W_x^{k,q_2' } }. \end{aligned}$$

Lemma 8.2

Let $s>1$. $\exists \,C=C$ such that:

• (a) for any $f\in S({\mathbb {R}}^2 )$,

  $$\begin{aligned} \Vert e^{-\mathrm{i}t \mathcal {K}}\widetilde{P}_c f\Vert _{ L^2_tL_x^{2,-s}} \le C\Vert f\Vert _{L^2} ; \end{aligned}$$

• (b) for any $g(t,x)\in {S}({\mathbb {R}}^2)$

  $$\begin{aligned} \left\| \int _{{\mathbb {R}}} e^{\mathrm{i}t\mathcal {K}}\widetilde{P}_c g(t,\cdot )dt\right\| _{L^2_x} \le C\Vert g\Vert _{ L_t^2L_x^{2,s}}. \end{aligned}$$

Lemma 8.3

Let $s>1$. $\exists \;C$ such that $\forall \;g(t,x)\in {S}({\mathbb {R}}^2)$ and $t\in {\mathbb {R}}$:

$$\begin{aligned} \left\| \int _0^t e^{-\mathrm{i}(t-s)\mathcal {K}}\widetilde{P}_c g(s,\cdot )ds\right\| _{ L^2_tL_x^{2,-s}} \le C\Vert g\Vert _{ L^2_tL_x^{2, s}} . \end{aligned}$$

Lemma 8.4

Let (p, q) be an admissible pair and let $s>1$. $\exists$ a constant $C>0$ such that $\forall \,g(t,x)\in {S}({\mathbb {R}}^2)$ and $t\in {\mathbb {R}}$:

$$\begin{aligned} \left\| \int _0^t e^{-\mathrm{i}(t-s)\mathcal {K}}\widetilde{P}_cg(s,\cdot )ds \right\| _{L_t^pL_x^q} \le C\Vert g\Vert _{ L^2_tL_x^{2, s}}. \end{aligned}$$

The following is Proposition 1.2 in Cuccagna and Tarulli (2009).

Lemma 8.5

The following limits are well-defined isomorphism, inverse of each other:

$$\begin{aligned} \begin{aligned}&W u= \lim _{t\rightarrow +\infty } e^{ \mathrm{i}t\mathcal {K} } e^{ \mathrm{i}t \sigma _3(\Delta -\omega _1) }u \text { for any }u\in L^2 \\&Z u= \lim _{t\rightarrow +\infty } e^{ \mathrm{i}t(-\Delta +\omega _1) } e^{ -\mathrm{i}t \mathcal {K} } \text { for any }u=\widetilde{P}_cu. \end{aligned} \end{aligned}$$

For any $p\in (1,\infty )$ and any k the restrictions of W and Z to $L^2\cap W^{k,p}$ extend into operators such that for a constant C we have

$$\begin{aligned} \Vert W\Vert _{ W^{k,p}({\mathbb {R}}^2), W^{k,p}_c}+\Vert Z\Vert _{ W^{k,p}_c, W^{k,p}({\mathbb {R}}^2) }<C \end{aligned}$$

with $W^{k,p} _c$ the closure in $W^{k,p} ({\mathbb {R}}^2)$ of $W^{k,p}({\mathbb {R}}^2)\cap \widetilde{P}_c L^2 _c$.

The following is Cuccagna and Tarulli (2009, Lemma 3.5).

Lemma 8.6

Consider the diagonal matrices $E_+=\text {diag}(1 , 0)\,E_-=\text {diag}(0 , 1).$ Set $P_\pm =Z E_\pm W$ with Z and W the wave operators associated with $\mathcal {K}$. Then we have for $u =\widetilde{P}_cu$

$$\begin{aligned} \begin{aligned}&P_+ u =\lim _{\epsilon \rightarrow 0^+} \frac{1}{2\pi \mathrm{i}} \lim _{M \rightarrow +\infty } \int _\omega ^M \left[ R_{ \mathcal {K} }(\lambda +\mathrm{i}\epsilon )- R_{\mathcal {K} }(\lambda -\mathrm{i}\epsilon ) \right] ud\lambda \\&P_- u =\lim _{\epsilon \rightarrow 0^+} \frac{1}{2\pi \mathrm{i}} \lim _{M \rightarrow +\infty } \int _{-M }^{-\omega } \left[ R_{\mathcal {K} }(\lambda +\mathrm{i}\epsilon )- R_{\mathcal {K} }(\lambda -\mathrm{i}\epsilon ) \right] ud\lambda \end{aligned} \end{aligned}$$

and for any $s_1$ and $s_2$ and for $C=C (s_1,s_2 )$ we have

$$\begin{aligned} \Vert (P_+ -P_- -\widetilde{P}_c \sigma _3) f\Vert _{L^{2,s_1} }\le C \Vert f\Vert _{L^{2,s_2} }. \end{aligned}$$

Now we look at the term $\mathbf {E}$ in (6.2).

Lemma 8.7

For any preassigned s and for $\epsilon _0 >0$ small enough we have

$$\begin{aligned} \begin{aligned}&\mathbf {E} = R_1+R_2 \text { with } \Vert R_1 \Vert _{L^1_t([0,T],H^{ 1 }_x)} +\Vert R_2 \Vert _{L^{2 }_t([0,T],H^{ 1 , s}_x)}\le C( s,C_0 ) \epsilon ^2 . \end{aligned} \end{aligned}$$

(8.1)

Furthermore for a fixed constant c we have

$$\begin{aligned} \begin{aligned}&\Vert A \Vert _{L^\infty ( (0,T), {\mathbb {R}})}\le c C_0^2 \epsilon ^2. \end{aligned} \end{aligned}$$

(8.2)

Proof

The estimate on $A=A'+{A} ^{\prime \prime }$ follows from the definitions of $A'$ in (5.2) and of $A^{\prime \prime }$ in (5.3).

$\mathbf {E}$ is a sum of various terms. For example we have

$$\begin{aligned} \begin{aligned}&\Vert z^\mu \overline{z}^\nu M^{-1} [{G}_{\mu \nu } (Q,0) - {G}_{\mu \nu } (Q,Q(f)) ] \Vert _{L^{2 }_t([0,T],H^{ 1 , s}_x)} \\&\quad \le \Vert z^\mu \overline{z}^\nu \Vert _{L^{2 }_t [0,T] } \Vert {G}_{\mu \nu } (Q,0) - {G}_{\mu \nu } (Q,Q(f)) \Vert _{L^{\infty }_t([0,T],H^{ 1 , s}_x)} \lesssim C_0^3 \epsilon ^3. \end{aligned} \end{aligned}$$

So this term can be absorbed in $R_2$. Another example is $\beta (|f|^2)f =\chi _{|f|\le 1}\beta (|f|^2)f +\chi _{|f|\ge 1}\beta (|f|^2)f$. The first term can be bounded, schematically, by

$$\begin{aligned} \begin{aligned}&\Vert |f |^2 f \Vert _{ L^{1 }_t([0,T],H^{ 1 }_x) } \lesssim \left\| \Vert f \Vert _{W^{1,6}_x} \Vert f \Vert ^2_{L^{ 6}_x} \right\| _{L^1_t[0,T] } \le \Vert f \Vert _{L^3_t([0,T],W^{1,6}_x)}^3 \ \lesssim C_0^3 \epsilon ^3 \end{aligned} \end{aligned}$$

(8.3)

while the 2nd term can be bounded by

$$\begin{aligned} \begin{aligned}&\Vert f ^{{L}} \Vert _{ L_t^1H^1_x } \lesssim \left\| \Vert f \Vert _{W^{1,2{L}}_x} \Vert f \Vert ^{{L}-1}_{L^{ 2{L}}_x} \right\| _{L^1_t } \le \Vert f \Vert _{L^{\frac{ 2{L}}{{L}-1}}_tW^{1,2{L}}_x} \Vert f \Vert ^{{L}-1}_{L^{ 2{L} \frac{{L}-1}{{L}+1} }_tW^{1,2L}_x} \lesssim C_0^L\epsilon ^{L}, \end{aligned} \end{aligned}$$

(8.4)

where in the last step we use $\Vert f \Vert _{L^{ 2{L} \frac{{L}-1}{{L}+1} }_tW^{1,2{L}}_x}\lesssim \Vert f \Vert ^{\alpha }_{L^{\frac{ 2{L}}{{L}-1}}_tL^{ 2{L}}_x} \Vert f \Vert _{ L_{t }^\infty H^1_x } ^{1-\alpha }$ for some $0<\alpha <1$ by ${L}>3$ (which we can always assume), interpolation and Sobolev embedding.

Notice that by $\nabla _{f} {\mathcal {R}}^{1,2}_{k,m}(Q ,\varrho , f) _{|\varrho =Q (f)}= S^{1,1}_{k,m-1}(Q ,Q (f), f)$ we have by (3.16)

$$\begin{aligned} \begin{aligned}&\Vert \nabla _{f} {\mathcal {R}}^{1,2}_{k,m}(Q ,\varrho , f) _{|\varrho =Q (f)} \Vert _{ L_t^2H ^{1,s}_x } \le \Vert \Vert f \Vert _{L^{2,-\sigma }_x} \Vert _{ L_t^2 } (\Vert f\Vert _{L^2}+|z | +|Q (f) |) \Vert _{ L_t^2 }\\ {}&\le \Vert f \Vert _{L_t^2L^{2,-\sigma }_x} (\Vert z \Vert _{ L_t^\infty } + \Vert f \Vert _{ L_t^\infty }L^2) \le 2 C_0 ^2 \epsilon ^2. \end{aligned} \end{aligned}$$

Consider for example the contribution of

$$\begin{aligned} \begin{aligned}&\nabla _{f} \int _{\mathbb {R}^2} B_L (x, f(x), Q , z,\varrho , f ) f^{ L}(x) \hbox {d}x _{|\varrho =Q(f)}\\&\quad \sim B_L (x, f(x), Q , z,Q(f) , f ) f^{ L-1}(x)\\&\quad + \int _{\mathbb {R}^2} \partial _{6} B_L (x, f(x), Q , z,Q(f) , f ) f^{ L}(x) \hbox {d}x \\&\quad + \partial _{2} B_L (x, f(x), Q , z,Q(f) , f ) f^{ L}(x) . \end{aligned} \end{aligned}$$

(8.5)

The last term can be treated like $f^{ L}$ above, since $\Vert B_L (x, f(x), Q , z,Q(f) , f )\Vert _{ L _{tx} ^\infty } \le C$ by (3.20). We can use (8.3) or (8.4) for the first term of the r.h.s., since $L-1\ge 3$. Finally let us consider the 2nd term in the r.h.s. If we take $g\in L ^{\infty }_tH ^{-1}_x$ we need to bound

$$\begin{aligned} \begin{aligned}&\int _{0}^{T} \hbox {d}t \int _{\mathbb {R}^2} | \langle g, \partial _{6} B_L (x , f(x ), Q , z,Q(f) , f )\rangle _{L_{x'}^2} f^{ L}(x) \hbox {d}x | \\&\quad \le \int _{0}^{T} dt \Vert | \langle g, \partial _{6} B_L (x , f(x ), Q , z,Q(f) , f )\rangle _{L_{x'}^2}\Vert _{L^2_x} \Vert f^{ L}\Vert _{L^2_x}\\&\quad \le \Vert | \langle g, \partial _{6} B_L (x , f(x ), Q , z,Q(f) , f )\rangle _{L_{x'}^2}\Vert _{L^\infty _tL^2_x}\Vert f^{ L}\Vert _{L^1 _tL^2_x} \end{aligned} \end{aligned}$$

and we bound the last factor by (8.4). We have for fixed t

$$\begin{aligned} \begin{aligned}&\Vert | \langle g, \partial _{6} B_L (x , f(x ), Q , z,Q(f) , f )\rangle _{L_{x'}^2}\Vert _{ L^2_x}\le \Vert \partial _{6} B_L \Vert _{B(\Sigma _{-k},\Sigma _k) } \Vert g \Vert _{\Sigma _{-k}} \end{aligned} \end{aligned}$$

so that by (3.20), or by its analogue for the $B_L$ in Lemma 4.1, we have that the last quantity is bounded by $C\Vert g \Vert _{H^ {-1}}$. This yields a bound $\Vert \text {first term 2nd line (A.5)}\Vert _{L^1_tL^2_x}\lesssim C_0^L \epsilon ^L$.

Proof of Lemma 6.1

We rewrite (6.2) as

$$\begin{aligned} \mathrm{i}\dot{h}= & {} [\mathcal {K} h + A (P_+ -P_-) ] h+ A [\widetilde{P}_c \sigma _3-P_+ +P_-] \sigma _3h\\&+ \sum _{\mathbf {e} \cdot (\mu -\nu )\in \sigma _e(\mathcal {L} _{\omega _1 })} z^\mu \overline{z}^\nu \mathbf {G}_{\mu \nu } + \widetilde{P}_c\mathbf {E} . \end{aligned}$$

Then we have

$$\begin{aligned} \begin{aligned} h (t) =&\, \mathcal {U}(t,0) e^{\mathrm{i}t\mathcal {K}}h(0) +\int _0^t\mathcal {U}(t,s)e^{\mathrm{i}(t-s)\mathcal {K}}\left[ A [\widetilde{P}_c \sigma _3-P_+ +P_-] \sigma _3h \right. \\&\left. + \sum _{\mathbf {e} \cdot (\mu -\nu )\in \sigma _e(\mathcal {L} _{\omega _1 })} z^\mu \overline{z}^\nu \mathbf {G}_{\mu \nu } + \widetilde{P}_c\mathbf {E} \right] ds, \end{aligned} \end{aligned}$$

(8.6)

where the following operator commutes with $\mathcal {K}$:

$$\begin{aligned} \mathcal {U}(t,s)= ^{\mathrm{i}\int _s^t A(s') ds'(P_+ -P_-) }. \end{aligned}$$

Then

$$\begin{aligned} \begin{aligned} \Vert h \Vert _{L^p_t W^{ 1 ,q}_x \cap L^2_t H^{ 1 ,-s}_x }&\lesssim \Vert h (0)\Vert _{H^1} +\sum _{\mu \nu }\Vert z^\mu \overline{z}^\nu \Vert _{L^2_t }\Vert \mathbf {G}_{\mu \nu }\Vert _{L^\infty _t H^{ 1 , s}_x}\\&\quad + \Vert A \Vert _{L^\infty _t } \Vert h \Vert _{ L^2_t H^{ 1 ,-s}_x } + \Vert R_1 \Vert _{L^1_t H^{ 1 }_x } +\Vert R_2 \Vert _{L^{2 }_t H^{ 1 , s}_x } . \end{aligned} \end{aligned}$$

The terms on the second line are $O(\epsilon ^2)$, and the r.h.s. is bounded by the r.h.s. of (6.1), proving Lemma 6.1. $\square$