[Paper Review] Strong instability of standing waves for nonlinear Schrödinger equations with a delta potential

Masahito Ohta, Takahiro Yamaguchi|arXiv (Cornell University)|Nov 10, 2014

Advanced Mathematical Physics Problems19 references3 citations

TL;DR

This paper establishes a sufficient condition for strong instability (finite-time blowup) of standing wave solutions to the one-dimensional nonlinear Schrödinger equation with an attractive delta potential and $L^2$-supercritical nonlinearity. Using a variational approach and scaling arguments, it proves that if the energy of the standing wave $e^{i\omega t}\phi_\omega$ is positive, then arbitrarily small $H^1$-perturbations of the initial data lead to finite-time blowup, implying strong instability for $\omega > \omega_1(p,\gamma)$, where $\omega_1$ is defined via a nonlinear eigenvalue equation.

ABSTRACT

We study strong instability (instability by blowup) of standing wave solutions for a nonlinear Schrödinger equation with an attractive delta potential and $L^2$-supercritical power nonlinearity in one space dimension. We also compare our sufficient condition on strong instability with some known results on orbital instability.

Motivation & Objective

• To analyze the strong instability (blowup) of standing wave solutions for the nonlinear Schrödinger equation with an attractive delta potential in one dimension.
• To establish a sufficient condition for strong instability based on the sign of the energy functional $E(\phi_\omega)$ of the standing wave profile.
• To compare the new sufficient condition for strong instability with existing criteria for orbital instability, particularly those based on $\partial_\omega \|\phi_\omega\|_{L^2}^2$ and $\partial_\lambda^2 E(\phi_\omega^\lambda)$ at $\lambda=1$.
• To clarify the relationship between different instability thresholds: $\omega_0(p,\gamma)$ for orbital instability, $\omega_1(p,\gamma)$ for strong instability, and $\omega_2(p,\gamma)$ for instability via the second derivative condition.

Proposed method

• The authors define the standing wave solution $e^{i\omega t}\phi_\omega(x)$ as the unique positive solution to the stationary equation $-\partial_x^2\phi + \omega\phi - \gamma\delta(x)\phi - |\phi|^{p-1}\phi = 0$.
• They introduce a scaling transformation $\phi_\omega^\lambda(x) = \lambda^{1/2}\phi_\omega(\lambda x)$ and analyze the behavior of the energy $E(\phi_\omega^\lambda)$ and the $L^2$-norm under this scaling.
• Using the identity $P(\phi_\omega) = 0$ and the fact that $E(\phi_\omega) > 0$, they show that for $\lambda > 1$ close to 1, $E(\phi_\omega^\lambda) < E(\phi_\omega)$ and $K_\omega(\phi_\omega^\lambda) < 0$, placing $\phi_\omega^\lambda$ in the set $\mathcal{B}_\omega$ of functions that lead to blowup.
• They apply a known blowup criterion from the literature (Theorem 1.6) which guarantees finite-time blowup for initial data in $\mathcal{B}_\omega$, and use the continuity of the $H^1$-norm under scaling to conclude strong instability.
• The proof relies on the variational characterization of the standing wave and the strict decrease of the energy under scaling for $\lambda > 1$, which is derived from the derivative $\partial_\lambda K_\omega(\phi_\omega^\lambda)\big|_{\lambda=1} < 0$.
• They compare the threshold $\omega_1(p,\gamma)$ for strong instability with $\omega_0(p,\gamma)$ and $\omega_2(p,\gamma)$, showing $\omega_0 < \omega_2 < \omega_1$ for $p > 5$, and discuss the implications for the conjecture that strong instability holds for $\omega > \omega_2(p,\gamma)$.

Experimental results

Research questions

• RQ1 Under what conditions is the standing wave solution $e^{i\omega t}\phi_\omega$ of the nonlinear Schrödinger equation with a delta potential strongly unstable (i.e., leads to finite-time blowup under arbitrarily small perturbations)?
• RQ2 How does the sufficient condition $E(\phi_\omega) > 0$ for strong instability compare to the known criteria for orbital instability based on $\partial_\omega \|\phi_\omega\|_{L^2}^2 < 0$?
• RQ3 What is the relationship between the instability thresholds $\omega_0(p,\gamma)$, $\omega_1(p,\gamma)$, and $\omega_2(p,\gamma)$, defined via different instability conditions?
• RQ4 Is it possible to conjecture strong instability for $\omega > \omega_2(p,\gamma)$, given that $\omega_2(p,\gamma)$ corresponds to the second-derivative blowup criterion?

Key findings

• The standing wave solution $e^{i\omega t}\phi_\omega$ is strongly unstable if $E(\phi_\omega) > 0$, which holds when $\omega > \omega_1(p,\gamma)$, where $\omega_1(p,\gamma) = \gamma^2 / (4\xi_1(p)^2)$ and $\xi_1(p)$ solves a specific integral equation.
• The threshold $\omega_1(p,\gamma)$ for strong instability satisfies $\omega_1(p,\gamma) > \omega_2(p,\gamma) > \omega_0(p,\gamma)$ for $p > 5$, indicating that strong instability can occur even when orbital instability is not detected by the $\partial_\omega \|\phi_\omega\|_{L^2}^2$ criterion.
• The condition $E(\phi_\omega) > 0$ is both necessary and sufficient for the existence of a family of initial data $\phi_\omega^\lambda$ with $\lambda \in (1, \lambda_0)$ that lead to finite-time blowup, establishing strong instability via the variational method.
• For $p > 5$, the energy-based criterion $E(\phi_\omega) > 0$ yields a stronger instability threshold than the second-derivative condition $\partial_\lambda^2 E(\phi_\omega^\lambda)\big|_{\lambda=1} < 0$, which corresponds to $\omega > \omega_2(p,\gamma)$.
• Numerical evidence from [15] for $p=6$, $\gamma=1$, $\omega=4$ supports the theoretical findings, as $\omega_2(6,1) < 4 < \omega_1(6,1)$, placing $\omega=4$ in the region where strong instability is expected but not ruled out by orbital instability criteria.