Blow-up and strong instability of standing waves for the NLS-\(\delta\) equation on a star graph. (English) Zbl 1464.35322
The present paper is devoted to study the strong instability of the standing wave solutions the NLS [N. Fukaya and M. Ohta, Osaka J. Math. 56, No. 4, 713–726 (2019; Zbl 1431.35172)] equation with \(\delta\)-interaction on a star graph \(\Gamma\)
\[
i\partial_{t}U(t,x)-HU(t,x)+F(U(t,x))U(t,x)=0,\tag{1}
\]
where \(U(t,x)=(u_{j}(t,x))_{j=1}^{N}:\mathbb{R}\times\mathbb{R}_{+}\rightarrow\mathbb{C}^{N}\), \(F(U(t,x)))=(f_{ij})\), \(i,j=1,\dots,N\) is a diagonal matrix with \(f_{ii}=\left\vert u_{i}\right\vert^{p-1}\), \(p>1\), and \(f_{ij}=0\) if \(i\neq j\), and \(H\) is the self-adjoint operator on \(L^{2}(\Gamma)\) given by
\[
\begin{aligned}
(HU)(x)& =(-\partial_{x}^{2} u_{j}(x))_{j=1}^{N},\ x>0,\\
D(H)& =\{U\in H^{2}(\Gamma):u_{1}(0)=\dots=u_{N}(0),\sum_{j=1}^{N}u_{j}^{\prime}(0)=\alpha u_{1}(0)\}.
\end{aligned}\tag{2}
\]
Condition (2) is an analog of \(\delta\)-interaction condition for the Schrödinger operator on the line. Let \(U(t,x)=e^{i\omega t}\Phi(x)\) be the standing wave solution, where \(\Phi\) satisfies the stationary equation
\[
H\Phi+\omega\Phi-F(\Phi)\Phi=0.
\]
It is known that the last equation has \([\frac{N-1}{2}]+1\) explicit vector solutions \(\Phi_{k}^{\alpha}=(\varphi_{k,j}^{\alpha})_{j=1}^{N}\), \(k=0,\dots,[\frac{N-1}{2}]\). The authors prove strong instability for \(\Phi_{0}^{\alpha}\). More precisely, in Theorem 1.3 they show that if \(\alpha>0\), \(\omega>\frac{\alpha^{2}}{N^{2}}\) and \(p\geq5\), then \(e^{i\omega t}\Phi_{0}^{\alpha}(x)\) is strongly unstable. Moreover, Theorem 1.4 shows that if \(\alpha<0\), \(p>5\), \(\omega>\frac{\alpha^{2}}{N^{2}}\), \(e^{i\omega t}\Phi_{0}^{\alpha}(x)\) is strongly unstable for all \(\omega\in\lbrack\omega_{1},\infty)\), where \(\omega_{1}=\omega_{1}(p,\alpha)\) can be computed from a given integral equation. In order to prove the above theorems the authors use the ideas of [N. Fukaya and M. Ohta, Osaka J. Math. 56, No. 4, 713–726 (2019; Zbl 1431.35172); M. Ohta, São Paulo J. Math. Sci. 13, No. 2, 465–474 (2019; Zbl 1433.35368)]. The essential ingredients are the virial identity for the NLS-\(\delta\) equation on \(\Gamma\) and a novel variational technique applied to the standing wave solutions being minimizers of a specific variational problem.
Reviewer: Ivan Naumkin (Nice)
MSC:
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 81Q35 | Quantum mechanics on special spaces: manifolds, fractals, graphs, lattices |
| 37K40 | Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems |
| 37K45 | Stability problems for infinite-dimensional Hamiltonian and Lagrangian systems |
| 47E05 | General theory of ordinary differential operators |
| 35B44 | Blow-up in context of PDEs |
| 35B35 | Stability in context of PDEs |
Keywords:
\(\delta\)- and \(\delta^\prime\)-interaction; nonlinear Schrödinger equation; strong instability; standing wave; star graph; virial identityReferences:
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