On Selection of Standing Wave at Small Energy in the 1D Cubic Schrödinger Equation with a Trapping Potential

Abstract

Combining virial inequalities by Kowalczyk, Martel and Munoz and Kowalczyk, Martel, Munoz and Van Den Bosch with our theory on how to derive nonlinear induced dissipation on discrete modes, and in particular the notion of Refined Profile, we show how to extend the theory by Kowalczyk, Martel, Munoz and Van Den Bosch to the case when there is a large number of discrete modes in the cubic NLS with a trapping potential which is associated to a repulsive potential by a series of Darboux transformations. Even though, by its non translation invariance, our model avoids some of the difficulties related to the effect that translation has on virial inequalities of the kink stability problem for wave equations, it still is a classical model and it retains some of the main difficulties.

A Appendix: Proof of Lemma 9.3

It is equivalent to show that there is a constant $C>0$ such that for all v

$$\begin{aligned}&\left\| {\mathrm {sech}}\left( \frac{a x}{10} \right) \prod _{j=1}^{N}R _{H}( \omega _j) P_c \mathcal {A} \left\langle \mathrm{i}\varepsilon \partial _x \right\rangle ^{ N} v\right\| _{L^2({\mathbb {R}})} \le C \left\| {\mathrm {sech}}\left( \frac{a x}{20} \right) v \right\| _{L^2({\mathbb {R}})} . \end{aligned}$$

(A.1)

By (8.1), for $x<y$ we have the formula

$$\begin{aligned} R _{H}(z^2) (x,y)&= \frac{T(z)}{2\mathrm{i}z} f_- (x, z) f_+ (y, z) \nonumber \\&\quad = \frac{1}{z^2+\omega _j} \frac{ f_- (x, \mathrm{i}\sqrt{|\omega _j|}) f_+ (y, \mathrm{i}\sqrt{|\omega _j|}) }{ \int _{{\mathbb {R}}} f_-(x',\mathrm{i}\sqrt{|\omega _j|}) f_+(x',\mathrm{i}\sqrt{|\omega _j|}) dx'} + \widetilde{R} _{H}(z^2) (x,y) , \end{aligned}$$

(A.2)

where $\frac{T(z)}{2\mathrm{i}z}= \frac{1}{[ f_+ (x, z) , f_- (x, z)]}$, where in the denominator in the r.h.s. we have the Wronskian, where $\widetilde{R} _{H}(z^2) (x,y)$ is not singular in $z=\mathrm{i}\sqrt{|\omega _j|}$. On the other hand,

$$\begin{aligned}&T(z) = \frac{\text {Res}(T,\mathrm{i}\sqrt{|\omega _j|}) }{z-\mathrm{i}\sqrt{|\omega _j|} }+ \widetilde{T}(z) , \end{aligned}$$

with $\widetilde{T}(z)$ non singular and with residue, see p. 146 [16],

$$\begin{aligned}&\text {Res}(T,\mathrm{i}\sqrt{|\omega _j|}) = \mathrm{i}\left( \int _{{\mathbb {R}}} f_-(x',\mathrm{i}\sqrt{|\omega _j|}) f_+(x',\mathrm{i}\sqrt{|\omega _j|}) dx' \right) ^{-1} . \end{aligned}$$

It is elementary to conclude, comparing the terms in (A.2), that

$$\begin{aligned}&\widetilde{R} _{H}( \omega _j) (x,y) = K_j(x,y) +C(\omega _j) \phi _j (x) \phi _j (y) \text { with}\nonumber \\ {}&K_j(x,y)= \frac{1}{2\mathrm{i}\sqrt{|\omega _j|}} \frac{ \left. \partial _{z}\left( f_-(x,z) f_+(y,z) \right) \right| _{z=\mathrm{i}\sqrt{|\omega _j|}} }{ \int _{{\mathbb {R}}} f_-(x',\mathrm{i}\sqrt{|\omega _j|}) f_+(x',\mathrm{i}\sqrt{|\omega _j|}) dx'} . \end{aligned}$$

(A.3)

for some constant $C(\omega _j)$. For $x>y$ we obtain the same formula, interchanging x and y. Denoting by $K_j$ the operator with the kernel (A.3) for $x<y$ and the formula obtained from (A.3) interchanging x and y if $x>y$, we notice that

$$\begin{aligned}&\prod _{j=1}^{N}R _{H}( \omega _j) P_c = K_1...K_N . \end{aligned}$$

It is also easy to check, following the discussion in p. 134 [16], that there is a fixed $C>0$ s.t. $|K_j(x,y) |\le C\left\langle x-y\right\rangle e^{-\sqrt{|\omega _j|} |x-y|}$. Then, for any value $a\in [ 0 , \sqrt{|\omega _N|}]$ we have

$$\begin{aligned}&\Vert {\mathrm {sech}}\left( \frac{a x}{10} \right) \prod _{j=1}^{N}R _{H}( \omega _j) P_c \mathcal {A} \left\langle \mathrm{i}\varepsilon \partial _x \right\rangle ^{ N} v \Vert _{L^2}\\ {}&\quad \lesssim \Vert \prod _{j=1}^{N}R _{H}( \omega _j) P_c {\mathrm {sech}}\left( \frac{a x}{10} \right) \mathcal {A} \left\langle \mathrm{i}\varepsilon \partial _x \right\rangle ^{ N} v \Vert _{L^2}. \end{aligned}$$

We have

$$\begin{aligned}&{\mathrm {sech}}\left( \frac{a x}{10} \right) \mathcal {A} =P_{N}(x, \mathrm{i}\partial _x ) {\mathrm {sech}}\left( \frac{a x}{10} \right) , \end{aligned}$$

for an N–th order differential operator with smooth and bounded coefficients.

Next, we write

$$\begin{aligned}&{\mathrm {sech}}\left( \frac{a x}{10} \right) \left\langle \mathrm{i}\varepsilon \partial _x \right\rangle ^{ N} = \left\langle \mathrm{i}\varepsilon \partial _x \right\rangle ^{ N} {\mathrm {sech}}\left( \frac{a x}{10} \right) + \left\langle \mathrm{i}\varepsilon \partial _x \right\rangle ^{ N} \left\langle \mathrm{i}\varepsilon \partial _x \right\rangle ^{ -N} \left[ {\mathrm {sech}}\left( \frac{a x}{10 } \right) , \left\langle \mathrm{i}\varepsilon \partial _x \right\rangle ^{ N} \right] ,\end{aligned}$$

so that

$$\begin{aligned}&\left\| {\mathrm {sech}}\left( \frac{a x}{10} \right) \prod _{j=1}^{N}R _{H}( \omega _j) P_c \mathcal {A} \left\langle \mathrm{i}\varepsilon \partial _x \right\rangle ^{ N} v\right\| _{L^2({\mathbb {R}})} \\ {}&\quad \lesssim \left\| \prod _{j=1}^{N}R _{H}( \omega _j) P_c P_{N}(x, \mathrm{i}\partial _x ) \left\langle \mathrm{i}\varepsilon \partial _x \right\rangle ^{ N} {\mathrm {sech}}\left( \frac{a x}{10} \right) v \right\| _{L^2({\mathbb {R}})} \\ {}&\qquad + \left\| \prod _{j=1}^{N}R _{H}( \omega _j) P_c P_{N}(x, \mathrm{i}\partial _x ) \left\langle \mathrm{i}\varepsilon \partial _x \right\rangle ^{ N} \left\langle \mathrm{i}\varepsilon \partial _x \right\rangle ^{ -N} \left[ {\mathrm {sech}}\left( \frac{a x}{10} \right) , \left\langle \mathrm{i}\varepsilon \partial _x \right\rangle ^{ N} \right] v \right\| _{L^2({\mathbb {R}})} \\ {}&\quad =:I+II . \end{aligned}$$

We have

$$\begin{aligned} I&\le \left\| \prod _{j=1}^{N}R _{H}( \omega _j) P_c P_{N}(x, \mathrm{i}\partial _x ) \left\langle \mathrm{i}\varepsilon \partial _x \right\rangle ^{ N} \right\| _{L^2\rightarrow L^2} \left\| {\mathrm {sech}}\left( \frac{a x}{10} \right) v \right\| _{L^2({\mathbb {R}})} \\&\quad \le C \left\| {\mathrm {sech}}\left( \frac{a x}{10} \right) v \right\| _{L^2({\mathbb {R}})} \end{aligned}$$

with a fixed constant C independent from $\varepsilon \in (0,1)$. Next, we have

$$\begin{aligned}&II \le \left\| \left\langle \mathrm{i}\varepsilon \partial _x \right\rangle ^{ -N} \left[ {\mathrm {sech}}\left( \frac{a x}{10} \right) , \left\langle \mathrm{i}\varepsilon \partial _x \right\rangle ^{ N} \right] v \right\| _{L^2({\mathbb {R}})} \le C \left\| {\mathrm {sech}}\left( \frac{a x}{20} \right) v \right\| _{L^2({\mathbb {R}})} \end{aligned}$$

by Lemma 5.5, because $\int e^{-\mathrm{i}kx} {\mathrm {sech}}(x) dx = \pi \ {\mathrm {sech}}\left( \frac{\pi }{2} k \right)$ (which can be proved by an elementary application of the Residue Theorem) so that in the strip $k=k_1+ \mathrm{i}k_2$ with $|k_2|\le \mathbf {b}:=a/20$, then ${\mathrm {sech}}\left( \frac{\pi }{2} \ \frac{10}{a} k \right)$ satisfies the estimates required on $\widehat{\mathcal {V}}$ in (5.11). This completes the proof of (A.1).