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.
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Acknowledgments
S.C. was partially funded by grants FIRB 2012 (Dinamiche Dispersive) from the Italian Government, FRA 2013 and FRA 2015 from the University of Trieste. M.M. was supported by the Japan Society for the Promotion of Science (JSPS) with the Grant-in-Aid for Young Scientists (B) 15K17568.
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Communicated by George Haller.
Appendix: Proof of Lemma 6.1
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 \(t\in {\mathbb {R}}\):
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 \(t\in {\mathbb {R}}\):
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:
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
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 \(u =\widetilde{P}_cu\)
and for any \(s_1\) and \(s_2\) and for \(C=C (s_1,s_2 )\) we have
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
Furthermore for a fixed constant c we have
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
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
while the 2nd term can be bounded by
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)
Consider for example the contribution of
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
and we bound the last factor by (8.4). We have for fixed t
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
Then we have
where the following operator commutes with \(\mathcal {K}\):
Then
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. \(\square \)
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Cuccagna, S., Maeda, M. On Orbital Instability of Spectrally Stable Vortices of the NLS in the Plane. J Nonlinear Sci 26, 1851–1894 (2016). https://doi.org/10.1007/s00332-016-9322-9
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DOI: https://doi.org/10.1007/s00332-016-9322-9