Transverse Stability of Line Soliton and Characterization of Ground State for Wave Guide Schrödinger Equations

Abstract

In this paper, we study the transverse stability of the line Schrödinger soliton under a full wave guide Schrödinger flow on a cylindrical domain ${\mathbb {R}}\times {\mathbb {T}}$. When the nonlinearity is of power type $|\psi |^{p-1}\psi$ with $p>1$, we show that there exists a critical frequency $\omega _{p} >0$ such that the line standing wave is stable for $0<\omega < \omega _{p}$ and unstable for $\omega > \omega _{p}$. Furthermore, we characterize the ground state of the wave guide Schrödinger equation. More precisely, we prove that there exists $\omega _{*} \in (0, \omega _{p}]$ such that the ground states coincide with the line standing waves for $\omega \in (0, \omega _{*}]$ and are different from the line standing waves for $\omega \in (\omega _{*}, \infty )$.

Appendices

Density of y-Trigonometric Polynomials

In this appendix, we shall show that the set of the trigonometric polynomial on y is dense in $C_{0}^{y}({\mathbb {T}}, L_{x}^{1}({\mathbb {R}}))$ and $L_{x, y}^{q}({\mathbb {R}}\times {\mathbb {T}})$. We can prove this by a classical argument. However, we will give a proof here for the sake of completeness. Note that it suffices to prove the density in $C_y^0\left( {\mathbb {T}}, L_x^1({\mathbb {R}}) \right)$ only because the density in $L_{x, y}^{q}\left( {\mathbb {R}}\times {\mathbb {T}}\right)$ follows exactly in the same way. We start with reviewing the following definition.

Definition A.1

A family of functions $\left\{ \varphi _{n} \in C^0({\mathbb {T}}) :n \in {\mathbb {N}}\right\}$ is an approximate identity if:

$$\begin{aligned}&\varphi _{n}(y) \ge 0 \quad \hbox {for every}\, y \in {\mathbb {T}}\end{aligned}$$

(A.1)

$$\begin{aligned}&\int _{{\mathbb {T}}} \varphi _{n}(y) d y=1 \quad \hbox {for every}\, n \in {\mathbb {N}} \end{aligned}$$

(A.2)

$$\begin{aligned}&\lim _{n \rightarrow \infty } \int _{\delta \le |y| \le \pi } \varphi _{n}(y) d y=0 \quad \hbox {for every}\, \delta >0. \end{aligned}$$

(A.3)

In (A.3) we identify ${\mathbb {T}}$ with the interval ${\mathcal {C}}=[- \pi , \pi )$.

We now provide the following approximation lemma:

Lemma A.1

Let $f \in C_y^0\left( {\mathbb {T}}, L_x^1({\mathbb {R}}) \right)$ and $\left\{ \varphi _{n} \in {\mathcal {C}}^0({\mathbb {T}}) :n \in {\mathbb {N}}\right\}$ be an approximate identity. Then, $\lim _{n \rightarrow \infty }\varphi _{n} *_y f = f$ in $C_y^0\left( {\mathbb {T}}, L_x^1({\mathbb {R}}) \right)$, where

$$\begin{aligned} \varphi _{n} *_{y} f = \int _{-\pi }^{\pi } \varphi _{n}(y - t) f(t) dt. \end{aligned}$$

Proof

From (A.1) and (A.2), we write

$$\begin{aligned}&\Vert \varphi _{n} *_y f(\cdot ,y)\Vert _{L_{x}^1({\mathbb {R}})}-\Vert f(\cdot ,y)\Vert _{L_{x}^1({\mathbb {R}})}\\&\quad \le \int _{\mathbb {R}}\int _{-\pi }^{\pi } \varphi _{n} (t) |f(x, y-t)|dt\ dx - \int _{\mathbb {R}}|f(x,y)| \ dx\\&\quad \le \int _{-\pi }^{\pi } \varphi _{n} (t) \left( \int _{\mathbb {R}}|f(x,y-t)| dx - \int _{\mathbb {R}}|f(x,y)| \ dx \right) \ dt\\&\quad \le \int _{|t|\le \delta } \varphi _{n} (t) \left( \Vert f(\cdot ,y-t)\Vert _{L_{x}^1({\mathbb {R}})} - \Vert f(\cdot ,y)\Vert _{L_{x}^1({\mathbb {R}})} \right) \ dt\\&\qquad + \int _{\pi> |t|> \delta } \varphi _{n} (t) \left( \Vert f(\cdot ,y-t)\Vert _{L_{x}^1({\mathbb {R}})} - \Vert f(\cdot ,y)\Vert _{L_{x}^1({\mathbb {R}})} \right) \ dt. \end{aligned}$$

On the other hand, since $f \in C_y^0\left( {\mathbb {T}}, L_x^1({\mathbb {R}}) \right)$, we infer that the function $y\mapsto \Vert f(\cdot ,y)\Vert _{L_{x}^1({\mathbb {R}})}$ is uniformly continuous. Combining this with (A.3), we deduce that for any $\varepsilon >0$

$$\begin{aligned} \sup _{y \in {\mathbb {T}}} \biggl |\Vert \varphi _{n} *_y f(\cdot ,y)\Vert _{L_{x}^1({\mathbb {R}})}-\Vert f(\cdot ,y)\Vert _{L_{x}^1({\mathbb {R}})}\biggl | \le \varepsilon \qquad \hbox {for sufficiently large}\, n \in {\mathbb {N}}. \end{aligned}$$

This finishes the proof of Lemma A.1. $\square$

As a consequence, we obtain the density property.

Lemma A.2

The set of trigonometric polynomials on y are dense in $C_y^0\left( {\mathbb {T}}, L_x^1({\mathbb {R}}) \right)$.

Proof

For each $n\in {\mathbb {N}}$, we define a function $\varphi _{n}$ by

$$\begin{aligned} \varphi _n(y):=c_n\left( 1+\cos y \right) ^n, \end{aligned}$$

where

$$\begin{aligned} c_{n} = \left( \int _{-\pi }^{\pi } (1 + \cos y)^{n} dy \right) ^{-1}. \end{aligned}$$

Clearly, the sequence $\{\varphi _{n}\}_{n \in {\mathbb {N}}}$ satisfies (A.1) and (A.2). We claim that $\{\varphi _n\}_{{\mathbb {N}}}$ also satisfies (A.3). Putting $t = \tan \frac{y}{2}$. Then, we have

$$\begin{aligned} \int _{\delta }^{\pi } (1 + \cos y)^{n} dy \!=\! \int _{\tan \frac{\delta }{2}}^{\infty } \left( \frac{2}{1 + t^{2}} \right) ^{n+1} dt, \!\!\!\!\qquad \!\!\!\int _{0}^{\pi } (1 + \cos y)^{n} dy = \int _{0}^{\infty } \left( \frac{2}{1 + t^{2}} \right) ^{n+1} dt. \end{aligned}$$

We can easily verify that

$$\begin{aligned}&\int _{\tan \frac{\delta }{2}}^{\infty } \left( \frac{2}{1 + t^{2}} \right) ^{n+1} dt \le \left( \frac{2}{1 + (\tan \frac{\delta }{2})^{2}} \right) ^{n} \int _{0}^{\infty } \frac{2}{1 + t^{2}} dt = \left( \frac{2}{1 + (\tan \frac{\delta }{2})^{2}} \right) ^{n} \pi , \end{aligned}$$

(A.4)

$$\begin{aligned}&\int _{0}^{\infty } \left( \frac{2}{1 + t^{2}} \right) ^{n+1} dt \ge \int _{0}^{\frac{\delta }{4}} \left( \frac{2}{1 + t^{2}} \right) ^{n+1} dt \ge \left( \frac{2}{1 + \frac{\delta ^{2}}{16}} \right) ^{n} \frac{\delta }{2}. \end{aligned}$$

(A.5)

Since $\tan s \ge s$ for all $s>0$, we have, by (A.4) and (A.5), that

$$\begin{aligned} \left[ \int _{0}^{\infty } \left( \frac{2}{1 + t^{2}} \right) ^{n+1} dt \right] ^{-1} \int _{\tan \frac{\delta }{2}}^{\infty } \left( \frac{2}{1 + t^{2}} \right) ^{n+1} dt \le \frac{2\pi }{\delta } \left( \dfrac{1 + \frac{\delta ^{2}}{16}}{1 + \frac{\delta ^{2}}{4}} \right) ^{n}. \end{aligned}$$

(A.6)

Note that $\varphi _{n}$ is an even function for each $n \in {\mathbb {N}}$. This together with (A.6) yields that

$$\begin{aligned} \lim _{n \rightarrow \infty } \int _{\delta \le |y| \le \pi } \varphi _{n}(y) d y \le \lim _{n \rightarrow \infty } \frac{4\pi }{\delta } \left( \dfrac{1 + \frac{\delta ^{2}}{16}}{1 + \frac{\delta ^{2}}{4}} \right) ^{n} = 0. \end{aligned}$$

Therefore, (B.3) holds.

Hence, the sequence $\{\varphi _{n}\}_{n \in {\mathbb {N}}}$ is an approximate identity. Thus, from Lemma A.1, we infer that for any $f \in C_y^0\left( {\mathbb {T}}, L_x^1({\mathbb {R}}) \right)$, $\varphi _{n} *_y f$ converges to f in $C_y^0\left( {\mathbb {T}}, L_x^1({\mathbb {R}}) \right)$ as $n \rightarrow \infty$.

It remains then to show that $\varphi _{n} *_y f$ is a trigonometric polynomials on y. We claim that $\varphi _{n}$ is a trigonometric polynomial. By the binomial theorem, we have

$$\begin{aligned} \begin{aligned} (1 + \cos y)^{n}&= 2^{n} \left( \cos \frac{y}{2} \right) ^{2n} \\&= 2^{n} \dfrac{\left( e^{i \frac{y}{2}} + e^{- i \frac{y}{2}}\right) ^{2n}}{2^{2n}} \\&= 2^{-n} \sum _{k=0}^{2n} \begin{pmatrix} 2n \\ k \end{pmatrix} e^{\frac{ky}{2}}e^{- \frac{2n-k}{2} y} \\&= 2^{-n} \sum _{k=0}^{2n} \begin{pmatrix} 2n \\ k \end{pmatrix} e^{i (k-n)y} = 2^{-n} \sum _{k=-n}^{n} \begin{pmatrix} 2n \\ k \end{pmatrix} e^{i k y}. \end{aligned} \end{aligned}$$

(A.7)

Thus, we can write

$$\begin{aligned} \varphi _{n}(y)=\sum _{k=-n}^{n} a_{n k} e^{i k y}, \quad \hbox {where}\, a_{n k}=2^{-n} c_{n}\left( \begin{array}{c} {2 n} \\ {n+k} \end{array} \right) \end{aligned}$$

Namely, $\varphi _{n}$ is a trigonometric polynomial. This implies that

$$\begin{aligned} \begin{aligned} \varphi _{n} *_y f(x,y) = \int _{-\pi }^{\pi } \sum _{k=-n}^{n} a_{n k} e^{i k(y-t)} f(x,t) d t&=\sum _{k=-n}^{n} a_{n k} e^{i k y} \int _{-\pi }^{\pi } e^{i k t} f(x,t) d t \\&=\sum _{k=-n}^{n} b_{k}(x) e^{i k y} , \end{aligned} \end{aligned}$$

where

$$\begin{aligned} b_{k}(x) =a_{n k} e^{i k y} \int _{-\pi }^{\pi } e^{i k t} f(x,t) d t. \end{aligned}$$

This finishes the proof of this lemma. $\square$

Continuity of the Minimization Value $m_{\omega }$

Lemma B.1

Let $p\in (1,5)$. There exists a constant $C( p)>0$ such that if $0< \omega _{1}<\omega _{2} < \infty$, and $Q_{\omega _{1}}$ and $Q_{\omega _{2}}$ are minimizers of the variational problems for $m_{\omega _{1}}$ and $m_{\omega _{2}}$, respectively, then, we have

$$\begin{aligned} m_{\omega _{1}}&\le m_{\omega _{2}} - \frac{{\mathcal {M}}(Q_{\omega _{2}})}{p+1}(\omega _{2}-\omega _{1}) + C(d, p)\frac{{\mathcal {M}}(Q_{\omega _{2}})^{2}}{m_{\omega _{2}}}|\omega _{2}-\omega _{1}|^{2}, \end{aligned}$$

(B.1)

$$\begin{aligned} m_{\omega _{2}}&\le m_{\omega _{1}} + \frac{{\mathcal {M}}(Q_{\omega _{1}})}{p+1}(\omega _{2}-\omega _{1}) + C(d, p)\frac{{\mathcal {M}}(Q_{\omega _{1}})^{2}}{m_{\omega _{1}}}|\omega _{2}-\omega _{1}|^{2}. \end{aligned}$$

(B.2)

In particular, $m_{\omega }$ is continuous and strictly increasing on $(0, \infty )$.

Proof

Let us begin with a proof of (B.1). Put $Q_{\omega _{2},\lambda }(x, y) := \lambda Q_{\omega _{2}}(x, y)$ for $\lambda >0$. Since ${\mathcal {N}}_{\omega _{2}}(Q_{\omega _{2}})=0$, we see that

$$\begin{aligned} \begin{aligned} {\mathcal {N}}_{\omega _{1}}(Q_{\omega _{2},\lambda })&= \lambda ^{2}\Vert \partial _{x} Q_{\omega _{2}}\Vert _{L^{2}}^{2} + \lambda ^{2} \Vert |D|_{y}^{\frac{1}{2}} Q_{\omega _{2}}\Vert _{L^{2}}^{2} + \lambda ^{2}\omega _{1} {\mathcal {M}}(Q_{\omega _{2}}) -\lambda ^{p+1} \Vert Q_{\omega _{2}}\Vert _{L^{p+1}}^{p+1} \\&= \lambda ^{2} \Bigm \{ \Vert Q_{\omega _{2}}\Vert _{L^{p+1}}^{p+1} + (\omega _{1} -\omega _{2}) {\mathcal {M}}(Q_{\omega _{2}}) -\lambda ^{p-1} \Vert Q_{\omega _{2}}\Vert _{L^{p+1}}^{p+1} \Bigm \}. \end{aligned}\nonumber \\ \end{aligned}$$

(B.3)

We define $\lambda _{*} < 1$ by

$$\begin{aligned} \lambda _{*} := \left( 1 + (\omega _{1}-\omega _{2}) \frac{{\mathcal {M}}(Q_{\omega _{2}})}{\Vert Q_{\omega _{2}}\Vert _{L^{p+1}}^{p+1}}\right) ^{\frac{1}{p-1}}. \end{aligned}$$

(B.4)

so that ${\mathcal {N}}_{\omega _{1}}(Q_{\omega _{2},\lambda _{*}})=0$. Thus,

$$\begin{aligned} m_{\omega _{1}} \le {\mathcal {I}}_{\omega _{1}}(Q_{\omega _{2},\lambda _{*}}) = \lambda _{*}^{2} {\mathcal {I}}_{\omega _{1}}(Q_{\omega _{2}}) = \lambda _{*}^{2}m_{\omega _{2}}. \end{aligned}$$

(B.5)

The Taylor expansion yields that there exists $\theta _{*} \in (0,1)$, depending on $\omega _{1}$ and $\omega _{2}$, satisfying

$$\begin{aligned} \begin{aligned} \lambda _{*}^{2}&= 1+ \frac{2}{p-1} (\omega _{1}-\omega _{2}) \frac{{\mathcal {M}}(Q_{\omega _{2}})}{\Vert Q_{\omega _{2}}\Vert _{L^{p+1}}^{p+1}}\\&\quad + \frac{2(3-p)}{(p-1)^{2}} \Big \{ 1+ \theta _{*} (\omega _{1}-\omega _{2}) \frac{{\mathcal {M}}(Q_{\omega _{2}})}{\Vert Q_{\omega _{2}}\Vert _{L^{p+1}}^{p+1}} \Big )^{\frac{2(2-p)}{p-1}} \Big ((\omega _{1}-\omega _{2}) \frac{{\mathcal {M}}(Q_{\omega _{2}})}{\Vert Q_{\omega _{2}}\Vert _{L^{p+1}}^{p+1}} \Big )^{2}\\&\le 1 - \frac{{\mathcal {M}}(Q_{\omega _{2}})}{(p+1)m_{\omega _{2}}}(\omega _{2} -\omega _{1}) + C(d, p)\frac{{\mathcal {M}}(Q_{\omega _{2}})^{2}}{m_{\omega _{2}}^{2}}|\omega _{2}-\omega _{1}|^{2}, \end{aligned} \end{aligned}$$

(B.6)

where $C( p)>0$ is some constant depending only on and p. In the last estimate, we used

$$\begin{aligned} m_{\omega _{2}} = {\mathcal {S}}_{\omega _{2}}(Q_{\omega _{2}}) = \frac{p-1}{2(p+1)} \Vert Q_{\omega _{2}}\Vert _{L^{p+1}}^{p+1} \end{aligned}$$

which is a consequence of (4.1) since ${\mathcal {N}}_{\omega _{2}}(Q_{\omega _{2}}) = 0$. Combining (B.5) with (B.6), we obtain the desired inequality

$$\begin{aligned} m_{\omega _{1}} \le m_{\omega _{2}} - \frac{{\mathcal {M}}(Q_{\omega _{1}})}{p+1}(\omega _{2}-\omega _{1}) + C(d, p)\frac{{\mathcal {M}}(Q_{\omega _{2}})^{2}}{m_{\omega _{2}}}|\omega _{2}-\omega _{1}|^{2}. \end{aligned}$$

Thus, (B.1) holds. We can obtain (B.2) similarly. This completes the proof. $\square$

Table of Notations

Symbols	Descriptions or equation numbers
X	(1.5)
$X_{2}$	$X_2 =H^2_xL^2_y \cap L^2_xH^{1}_y({\mathbb {R}}\times {\mathbb {T}})$
$X_{k}$	$X_k =H^k_xL^2_y \cap L^2_xH^{\frac{k}{2}}_y({\mathbb {R}}\times {\mathbb {T}})$
${\mathcal {M}}$	(1.3)
${\mathcal {H}}$	(1.2)
${\mathcal {S}}_{\omega }, \widetilde{{\mathcal {S}}}_{\omega }, {\mathcal {S}}_{\omega , {\mathbb {R}}}$	(1.11), (4.13), (4.6)
${\mathcal {N}}_{\omega }, \widetilde{{\mathcal {N}}}_{\omega }, {\mathcal {N}}_{\omega , {\mathbb {R}}}$	(1.13), (4.14), (4.7)
${\mathcal {I}}_{\omega }, \widetilde{{\mathcal {I}}}_{\omega }$	(4.1), (4.16)
$R_{\omega }$	Ground state of (1.7), $(2\omega )^{\frac{1}{p-1}} {{\,\mathrm{sech}\,}}(\sqrt{\omega } x)$
$Q_{\omega }, {\widetilde{Q}}_{\omega }$	Ground state of (1.6), (4.10)
${\mathcal {S}}_{{\mathbb {R}}}, {\mathcal {N}}_{{\mathbb {R}}}, m_{{\mathbb {R}}}, R, Q$	${\mathcal {S}}_{{\mathbb {R}}} = {\mathcal {S}}_{1, {\mathbb {R}}}, \ {\mathcal {N}}_{{\mathbb {R}}} = {\mathcal {N}}_{1, {\mathbb {R}}}, \ m_{{\mathbb {R}}} = m_{1, {\mathbb {R}}}, \ R = R_{1}, \ Q = Q_{1}$
$m_{\omega }, {\widetilde{m}}_{\omega }, m_{{\mathbb {R}}}$	(1.12), (4.12), (4.5)
$\omega _{p}$	$\frac{4}{(p-1)(p+3)}$
$\omega _{*}$	Given in Theorem 1.3
$\nu _{\omega }$	$\frac{\omega }{\omega _{p}}$
$L_{\omega , +}, L_{\omega , -}$	(2.2)
$L_{\omega , +, n}, L_{\omega , -, n}$	(2.3)
$L_{\omega , \text {g}, +}$	(4.44)
$S_{\omega }(a)$	(3.9)
$A_{n}$	$L_{\omega _{p}, +, n}$
J	(3.3)
$NL(v, R_{\omega })$	(3.6)
f(z)	$f(z) = |z|^{p-1}z$
F(s)	$F(s) = f(sv + R_{\omega })$
$P_{\le k}$	(3.14)
$\lambda _{0}$	Positive eigenvalue of $- J{\mathcal {S}}_{\omega }^{\prime \prime }$, (3.15)
$\chi$	Eigenfunction of $- J{\mathcal {S}}_{\omega }^{\prime \prime }$ corresponding to $\lambda _{0}$
$\lambda _{2}(a)$	Second eigenvalue of $-\partial _{xx} + |D_{y}| + \omega (a) - p \varphi (a)^{p-1}$
$\lambda (\omega _{p})$	Second eigenvalue of $L_{\omega _{p}, +}$
${\mathcal {P}}$, ${\mathcal {Q}}$	Trigonometric polynomials on y