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On Traveling Waves of the Nonlinear Schrödinger Equation Escaping a Potential Well

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Abstract

In this paper, we consider the NLS equation with focusing nonlinearities in the presence of a potential. We investigate the compact soliton motions that correspond to a free soliton escaping the well created by the potential. We exhibit the dynamical system driving the exiting trajectory and construct associated nonlinear dynamics for untrapped motions. We show that the nature of the potential/soliton is fundamental, and two regimes may exist: one where the tail of the potential is fat and dictates the motion, and one where the tail is weak and the soliton self-interacts with the potential defects, hence leading to different motions.

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Acknowledgements

Both authors are supported by the ERC-2014-CoG 646650 SingWave.

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Correspondence to Ivan Naumkin.

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Appendix

Appendix

1.1 Proof of Lemma 2.5

Recall that the kernel \(G\left( x\right) \) of the Bessel potential \(\left( 1-\Delta \right) ^{-1}\) behaves asymptotically as (see pages 416-417 of [3])

$$\begin{aligned} \left. G\left( x\right) =2^{\frac{d+1}{2}}\pi ^{\frac{d-1}{2}}\left| x\right| ^{-\frac{d-1}{2}}e^{-\left| x\right| }\left( 1+o\left( 1\right) \right) ,\text { as }\left| x\right| \rightarrow \infty ,\right. \end{aligned}$$
(7.1)

and

$$\begin{aligned} G\left( x\right) =\left\{ \begin{array} [c]{ll} \frac{1}{2\pi }\ln \frac{1}{\left| x\right| }\left( 1+o\left( 1\right) \right) ,&{}\quad d=2,\\ \frac{\Gamma \left( \frac{d-2}{2}\right) }{4\pi \left| x\right| ^{d-2}}\left( 1+o\left( 1\right) \right) ,&{}\quad d\ge 3, \end{array} \right. \text { as }\left| x\right| \rightarrow 0, \end{aligned}$$
(7.2)

where \(\Gamma \) denotes the gamma function. For \(0<\delta <1,\) let \(G_{\delta }\left( x\right) =e^{\delta \left| x\right| }G\left( x\right) \). Using (7.1) and (7.2), we estimate

$$\begin{aligned} \left| G_{\delta }\left( x\right) \right| \le Ce^{-\left( 1-\delta \right) \left| x\right| }\left\langle x\right\rangle ^{-\frac{d-1}{2}}\left\langle \left| x\right| ^{-\left( d-2\right) -\nu }\right\rangle ,\quad \nu >0. \end{aligned}$$
(7.3)

We decompose

$$\begin{aligned} \left| e^{-\delta \left| y\right| }T\left( y\right) \right| \le C_{0}\left( I_{1}+I_{2}+I_{3}\right) , \end{aligned}$$
(7.4)

where

$$\begin{aligned} I_{1}= & {} \int _{{\mathbb {R}}^{d}}G_{\delta }\left( y-z\right) e^{-\delta \left| z\right| }Q^{p-1}\left( z\right) \left| T\left( z\right) \right| \mathrm{d}z,\\ I_{2}= & {} \int _{{\mathbb {R}}^{d}}G_{\delta }\left( y-z\right) e^{-\delta \left| z\right| }V\left( \left| z+\frac{\chi }{\lambda }\right| \right) \left| T\left( z\right) \right| \mathrm{d}z \end{aligned}$$

and

$$\begin{aligned} I_{3}=\int _{{\mathbb {R}}^{d}}G_{\delta }\left( y-z\right) e^{-\delta \left| z\right| }\left| f\left( z\right) \right| \mathrm{d}z, \end{aligned}$$

for some \(C_{0}>0\). Let \(\psi \in C^{\infty }\) be such that \(\left\| \psi \right\| _{L\infty }\le 1,\)\(\psi =1\) for \(\left| x\right| \le 1\) and \(\psi =0\) for \(\left| x\right| \ge 2\). For \(a>0\), we decompose

$$\begin{aligned} I_{1}=I_{11}+I_{22}, \end{aligned}$$

with

$$\begin{aligned} I_{11}=\int _{{\mathbb {R}}^{d}}G_{\delta }\left( y-z\right) e^{-\delta \left| z\right| }Q^{p-1}\left( z\right) \psi \left( \frac{z}{a}\right) \left| T\left( z\right) \right| \mathrm{d}z \end{aligned}$$

and

$$\begin{aligned} I_{12}=\int _{{\mathbb {R}}^{d}}G_{\delta }\left( y-z\right) e^{-\delta \left| z\right| }Q^{p-1}\left( z\right) \left( 1-\psi \left( \frac{z}{a}\right) \right) \left| T\left( z\right) \right| \mathrm{d}z. \end{aligned}$$

By (2.54)

$$\begin{aligned} \left\| \psi T\right\| _{H^{1}}^{2}\le C_{1}\left( \left\| {\mathcal {L}}_{V}\left( \psi T\right) \right\| _{H^{-1}}^{2}+\left| \left( \psi T,\nabla Q\right) \right| ^{2}\right) ,\quad C_{1}>0. \end{aligned}$$

Noting that \([{\mathcal {L}}_{V},\psi \left( \frac{\cdot }{a}\right) ]=-a^{-2}\left( \Delta \psi \right) \left( \frac{\cdot }{a}\right) -a^{-1}\left( \nabla \psi \right) \left( \frac{\cdot }{a}\right) \nabla ,\) as \(\left( T,\nabla Q\right) =0,\) we have

$$\begin{aligned} \left\| \psi T\right\| _{H^{1}}^{2}&\le C_{1}\left( \left\| \psi f\right\| ^{2}+a^{-1}\left\| \left( \nabla \psi \right) \nabla T\right\| ^{2}+a^{-2}\left\| \Delta \psi T\right\| ^{2}+\left| \left( \left( 1-\psi \right) T,\nabla Q\right) \right| ^{2}\right) \\&\le C_{1}\left( \left\| \psi f\right\| ^{2}+C_{2}a^{-1}\left\| T\right\| _{H^{1}}^{2}\right) ,\quad C_{2}>0. \end{aligned}$$

Then, taking \(a\ge A_{0}^{-2}\left( \left| \chi \right| \right) +1\) and using (2.58), we get

$$\begin{aligned} \left\| \psi T\right\| _{H^{1}}^{2}\le C_{1}\left( \left\| \psi e^{\delta \left| z\right| }e^{-\delta \left| z\right| }f\right\| ^{2}+C_{2}A_{0}^{2}\left( \left| \chi \right| \right) \left\| T\right\| _{H^{1}}^{2}\right) \le C_{3}A_{0}^{2}\left( \left| \chi \right| \right) \end{aligned}$$

with \(C_{3}>0\). Thus, by (7.3) and Young’s inequality we get

$$\begin{aligned} \left\| I_{11}\right\| _{L^{2}}\le C_{0}\left\| G_{\delta }\left( \cdot \right) \right\| _{L^{1}}\left\| \psi T_{1}\right\| _{L^{2} }\le C_{4}\left( \delta \right) A_{0}\left( \left| \chi \right| \right) ,\quad C_{4}\left( \delta \right) >0. \end{aligned}$$

Moreover, we have

$$\begin{aligned} \left\| I_{12}\right\| _{L^{2}}\le Ca^{-1}\left\| G_{\delta }\left( \cdot \right) \right\| _{L^{1}}\left\| \left| z\right| Q^{p-1}\left( z\right) \right\| _{L^{\infty }}\left\| T\right\| _{L^{2}}\le Ca^{-1}\le CC_{4}\left( \delta \right) A_{0}^{2}\left( \left| \chi \right| \right) . \end{aligned}$$

Hence, \(\left\| I_{1}\right\| _{L^{2}}\le CC_{4}\left( \delta \right) A_{0}\left( \left| \chi \right| \right) \). By (2.31) and (2.70)%

$$\begin{aligned} \left\| I_{2}\right\| _{L^{2}}&\le C\left\| T\right\| _{L^{\infty }}\left\| e^{-\delta \left| \cdot \right| }V\left( \left| \cdot +\frac{\chi }{\lambda }\right| \right) \right\| _{L^{2} }\left\| \left| \cdot \right| ^{\frac{d-1}{2}}e^{-\left( 1-\delta \right) \left| \cdot \right| }\right\| _{L^{2}}\\&\le C\left\| T\right\| _{L^{\infty }}\left\| V\left( \left| \cdot +\frac{\chi }{\lambda }\right| \right) Q\left( \cdot \right) \right\| _{L^{\infty }}^{\delta ^{\prime }}\left\| \left| \cdot \right| ^{\delta ^{\prime }\frac{d-1}{2}}e^{-\left( \delta -\delta ^{\prime }\right) \left| \cdot \right| }\right\| _{L^{2} }\left\| \left| \cdot \right| ^{\frac{d-1}{2}}e^{-\left( 1-\delta \right) \left| \cdot \right| }\right\| _{L^{2}}\\&\le CC_{5}\left( \delta ,\delta ^{\prime }\right) \left\| T\right\| _{L^{\infty }}\Theta \left( \left| {\tilde{\chi }}\right| \right) ^{\delta ^{\prime }},\quad C_{5}\left( \delta ,\delta ^{\prime }\right) >0, \end{aligned}$$

for any \(\delta ^{\prime }<\delta \). By using (2.58), we control \(\left\| I_{3}\right\| _{L^{2}}\le C_{6}\left( \delta \right) A_{0}\left( \left| \chi \right| \right) ,\)\(C_{6}\left( \delta \right) >0\). Using the estimates for \(I_{1},I_{2},I_{3}\) in (7.4), we deduce

$$\begin{aligned} \left\| e^{-\delta \left| \cdot \right| }T\left( \cdot \right) \right\| _{L^{2}}\le C\left( \delta ,\delta ^{\prime }\right) \left( A_{0}\left( \left| \chi \right| \right) +\left\| T\right\| _{L^{\infty }}\Theta \left( \left| {\tilde{\chi }}\right| \right) ^{\delta ^{\prime }}\right) . \end{aligned}$$
(7.5)

Since \(G_{\delta }\in L^{p},\) for any \(1<p<\frac{d}{d-2},\) from Eq. (2.57), via Young’s inequality, we control the \(L^{\frac{2p}{2-p}}\) norm of \(e^{-\delta \left| \cdot \right| }T\left( \cdot \right) \) by the right-hand side of (7.5). Note that \(\frac{2p}{2-p}>2\). Iterating the last argument a finite number of times, we attain (2.59). Lemma 2.5 is proved.

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Naumkin, I., Raphaël, P. On Traveling Waves of the Nonlinear Schrödinger Equation Escaping a Potential Well. Ann. Henri Poincaré 21, 1677–1758 (2020). https://doi.org/10.1007/s00023-020-00897-2

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