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Coordinates at Small Energy and Refined Profiles for the Nonlinear Schrödinger Equation

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In this paper we give a new and simplified proof of the theorem on selection of standing waves for small energy solutions of the nonlinear Schrödinger equations (NLS) that we gave in [6]. We consider a NLS with a Schrödinger operator with several eigenvalues, with corresponding families of small standing waves, and we show that any small energy solution converges to the orbit of a time periodic solution plus a scattering term. The novel idea is to consider the “refined profile”, a quasi–periodic function in time which almost solves the NLS and encodes the discrete modes of a solution. The refined profile, obtained by elementary means, gives us directly an optimal coordinate system, avoiding the normal form arguments in [6], giving us also a better understanding of the Fermi Golden Rule.

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C. was supported by a FIRB of the University of Trieste. M.M. was supported by the JSPS KAKENHI Grant Number 19K03579, G19KK0066A, JP17H02851 and JP17H02853.

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Correspondence to Masaya Maeda.

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A Proofs of Lemma 1.5 and of Proposition 1.10

A Proofs of Lemma 1.5 and of Proposition 1.10

Proof of Lemma 1.5

For \(j,k\in \{1,\cdots ,N\}\), \(j< k\), set \(n_{jk}\) to be the smallest integer satisfying \(n_{jk}( \omega _k-\omega _j)+\omega _k>0\). Then, for \(\mathbf {m}^{(jk)}=(m_1^{(jk)},\cdots ,m_N^{(jk)})\) defined by

$$\begin{aligned} m_j^{(jk)}= -n_{jk},\ m_k^{(jk)}=n_{jk}+1\text { and }m_l^{(jk)}=0\ (l\ne j,k), \end{aligned}$$

we have \(\mathbf {m}^{(jk)} \in \mathbf {R}_{\mathrm {min}}\). Suppose \(\mathbf {R}_{\mathrm {min}}\) is an infinite set. Then, there exists \(j\in \{1,\cdots ,N\}\) and \(\{\mathbf {m}_k\}_{k=1}^\infty \subset \mathbf {R}_{\mathrm {min}}\) s.t. \(|m_{kj}| \xrightarrow {k\rightarrow \infty } \infty \). If there exists \(M>0\) s.t. for all \(l\ne j\), \(|m_{kj}|\le M\), then \(\mathbf {m}_k\) cannot satisfy \(\sum \mathbf {m}_k=1\). Therefore, if necessary taking a subsequence, there exists \(l\ne j\) s.t. \(|m_{kl}| \xrightarrow {k\rightarrow \infty } \infty \). However, for k sufficiently large, we have \(|\mathbf {m}^{(jl)}|\prec |\mathbf {m}_k|\) with \(\mathbf {m}^{(jl)} \in \mathbf {R}_{\mathrm {min}}\) defined by (A.1). This, by the definition of \(\mathbf {R}_{\mathrm {min}}\) in (1.8), implies \(\mathbf {m}_k\not \in \mathbf {R}_{\mathrm {min}}\), contradicting the hypothesis \(\mathbf {m}_k \in \mathbf {R}_{\mathrm {min}}\).

Let \(\mathbf {m}\in \mathbf {NR}_1\). It is elementary, by the definition of \(\mathbf {NR}_1\) (1.9), that for all \(\mathbf {n} \in \mathbf {R}_{\mathrm {min}}\), either there exists j s.t. \(|n_j|>|m_j|\) or \( | \mathbf {n} | = | \mathbf {m} |\). So, for \(\mathbf {n}=\mathbf {m}^{(jk)}\) in (A.1), we have either \(|\mathbf {m}|=|\mathbf {m}^{(jk)}|\) or \(|m_l|<|m_l^{(jk)}|\) for \(l=j\) or k. Since there are finitely many \(\mathbf {m}\in \mathbf {NR}_1\) s.t. \(|\mathbf {m}|=|\mathbf {m}^{(jk)}|\), we can assume \(|\mathbf {m}|\ne |\mathbf {m}^{(jk)}|\) for all \(j<k\). Thus, for all \(j<k\), we have \(|m_l|< m^{(jk)}_l\) for at least one of \(l \in \{j,k\}\). It is easy to conclude that \(|m_j|\le \max _{1\le k<l\le N} \left( |n_{kl}|+1\right) \) for all j except for at most one. However, from \(\sum \mathbf {m} =1\) it is immediate that this special j must satisfy \(|m_j| \le N \max _{1\le k<l\le N} \left( |n_{kl}|+1\right) \). Thus, \(\mathbf {m}\) is in a fixed bounded set. Hence \(\mathbf {NR}_1\) is a finite set.

Proof of Proposition 1.10

The simple proof is analogous to Bambusi and Cuccagna [2, p.1444]. For \(\mathbf {m} \in \mathbf {R}_{\mathrm {min}}\) set \({\mathbb N}\ni L_{\mathbf {m}} := \frac{\Vert \mathbf {m} \Vert -1}{2}\). Then from (1.13)–(1.14), for any \(\mathbf {m} \in \mathbf {R}_{\mathrm {min}}\) we have

$$\begin{aligned}&G_{\mathbf {m}}= N _{\mathbf {m}} \frac{g^{\left( L_{\mathbf {m}} \right) }(0)}{L_{\mathbf {m}} !} \phi ^{\mathbf {m}} + K _{\mathbf {m}}, \end{aligned}$$

where \(N _{\mathbf {m}} \in {\mathbb N}\) is the number of elements of \( A(L_{\mathbf {m}},\mathbf {m})\), which in this particular case is given by the set

$$\begin{aligned}&A(L_{\mathbf {m}},\mathbf {m})= \left\{ \{\mathbf {e}_{\ell _j}\}_{j=1}^{\Vert \mathbf {m} \Vert }\in (\mathbf {NR}_0)^{\Vert \mathbf {m} \Vert }\ |\ \sum _{j=0} ^{ L_{\mathbf {m}}} \mathbf {e}_{\ell _{2j+1}}-\sum _{j=1}^{ L_{\mathbf {m}}} \mathbf {e}_{\ell _{2j }}= \mathbf {m} \right\} , \end{aligned}$$

and where

$$\begin{aligned}&K _{\mathbf {m}} :=\sum _{1\le m < L_{\mathbf {m}} } \frac{1}{m!}g^{(m)}(0)\sum _{(\mathbf {m}_1,\cdots ,\mathbf {m}_{2m+1})\in A(m,\mathbf {m})}{\widetilde{\phi }}_{\mathbf {m}_1}(0)\cdots {\widetilde{\phi }}_{\mathbf {m}_{2m+1}}(0). \end{aligned}$$

So, expanding we have on the sphere \(S _{\mathbf {m}}=\{ \xi : |\xi |^2 = \mathbf {m}\cdot {\varvec{\omega }} \}\) we obtain

$$\begin{aligned} \Vert \widehat{G} _{\mathbf {m}} \Vert ^{2}_{L^2(S _{\mathbf {m}})} =&\left( N _{\mathbf {m}} \frac{g^{\left( L_{\mathbf {m}} \right) }(0)}{L_{\mathbf {m}} !} \right) ^2 \Vert \widehat{\phi ^ \mathbf {m}} \Vert ^{2}_{L^2(S _{\mathbf {m}})} \\&+ 2 N _{\mathbf {m}} \frac{g^{\left( L_{\mathbf {m}} \right) }(0)}{L_{\mathbf {m}} !} \left\langle \widehat{\phi ^ \mathbf {m}} , \widehat{K _{\mathbf {m}}} \right\rangle _{L^2(S _{\mathbf {m}})} + \Vert \widehat{K} _{\mathbf {m}} \Vert ^{2}_{L^2(S _{\mathbf {m}})} . \end{aligned}$$

Equating the above to 0 we obtain, in view of (1.16), a quadratic equation for \(g^{\left( L_{\mathbf {m}} \right) }(0)\) which expresses it in terms of \((g' (0),...., g ^{(L_{\mathbf {m}}-1)} (0))\). This proves Proposition 1.10. \(\square \)

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Cuccagna, S., Maeda, M. Coordinates at Small Energy and Refined Profiles for the Nonlinear Schrödinger Equation. Ann. PDE 7, 16 (2021).

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