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 \(\omega \in (\omega _{*}, \infty )\).
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Notes
Abuse of notation \(R_\omega :={}^t(R_\omega ,0)\).
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Acknowledgements
The authors would like to thank the anonymous reviewers for their useful comments. This work was done while HK was visiting at University of Victoria. HK thanks all members of the Department of Mathematics and Statistics for their warm hospitality. YB was supported by PIMS Grant and NSERC Grant (371637-2014). SI was supported by NSERC Grant (371637-2019). HK was supported by JSPS KAKENHI Grant Number JP17K14223.
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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:
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
Proof
From (A.1) and (A.2), we write
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 \(\varepsilon >0\)
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
where
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
We can easily verify that
Since \(\tan s \ge s\) for all \(s>0\), we have, by (A.4) and (A.5), that
Note that \(\varphi _{n}\) is an even function for each \(n \in {\mathbb {N}}\). This together with (A.6) yields that
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
Thus, we can write
Namely, \(\varphi _{n}\) is a trigonometric polynomial. This implies that
where
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
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
We define \(\lambda _{*} < 1\) by
so that \({\mathcal {N}}_{\omega _{1}}(Q_{\omega _{2},\lambda _{*}})=0\). Thus,
The Taylor expansion yields that there exists \(\theta _{*} \in (0,1)\), depending on \(\omega _{1}\) and \(\omega _{2}\), satisfying
where \(C( p)>0\) is some constant depending only on and p. In the last estimate, we used
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
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) |
\({\mathcal {S}}_{\omega }, \widetilde{{\mathcal {S}}}_{\omega }, {\mathcal {S}}_{\omega , {\mathbb {R}}}\) | |
\({\mathcal {N}}_{\omega }, \widetilde{{\mathcal {N}}}_{\omega }, {\mathcal {N}}_{\omega , {\mathbb {R}}}\) | |
\({\mathcal {I}}_{\omega }, \widetilde{{\mathcal {I}}}_{\omega }\) | |
\(R_{\omega }\) | Ground state of (1.7), \((2\omega )^{\frac{1}{p-1}} {{\,\mathrm{sech}\,}}(\sqrt{\omega } x)\) |
\(Q_{\omega }, {\widetilde{Q}}_{\omega }\) | |
\({\mathcal {S}}_{{\mathbb {R}}}, {\mathcal {N}}_{{\mathbb {R}}}, m_{{\mathbb {R}}}, R, Q\) | \({\mathcal {S}}_{{\mathbb {R}}} = {\mathcal {S}}_{1, {\mathbb {R}}}, \ {\mathcal {N}}_{{\mathbb {R}}} = {\mathcal {N}}_{1, {\mathbb {R}}}, \ m_{{\mathbb {R}}} = m_{1, {\mathbb {R}}}, \ R = R_{1}, \ Q = Q_{1} \) |
\(m_{\omega }, {\widetilde{m}}_{\omega }, m_{{\mathbb {R}}}\) | |
\(\omega _{p}\) | \(\frac{4}{(p-1)(p+3)}\) |
\(\omega _{*}\) | 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 |
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Bahri, Y., Ibrahim, S. & Kikuchi, H. Transverse Stability of Line Soliton and Characterization of Ground State for Wave Guide Schrödinger Equations. J Dyn Diff Equat 33, 1297–1339 (2021). https://doi.org/10.1007/s10884-020-09937-1
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DOI: https://doi.org/10.1007/s10884-020-09937-1