[Paper Review] Instability of bound states of a nonlinear Schr\"odinger equation with a Dirac potential

Stefan Le Coz, Reika Fukuizumi+3 more|arXiv (Cornell University)|Jul 17, 2007

Advanced Mathematical Physics Problems52 references97 citations

TL;DR

This paper analyzes the stability of standing wave solutions to a nonlinear Schrödinger equation with a Dirac delta potential, using perturbation theory and continuation arguments to compute the number of negative eigenvalues of the linearized operator. It establishes that for repulsive defects (γ < 0), radial bound states are stable in H1_rad(R) but unstable in H1(R) under subcritical nonlinearity, with instability manifesting as finite-width narrowing or lateral drift, while critical/supercritical nonlinearities lead to blowup via a virial theorem and constrained minimization.

ABSTRACT

We study analytically and numerically the stability of the standing waves for a nonlinear Schr\"odinger equation with a point defect and a power type nonlinearity. A main difficulty is to compute the number of negative eigenvalues of the linearized operator around the standing waves, and it is overcome by a perturbation method and continuation arguments. Among others, in the case of a repulsive defect, we show that the standing wave solution is stable in $\hurad$ and unstable in $\hu$ under subcritical nonlinearity. Further we investigate the nature of instability: under critical or supercritical nonlinear interaction, we prove the instability by blowup in the repulsive case by showing a virial theorem and using a minimization method involving two constraints. In the subcritical radial case, unstable bound states cannot collapse, but rather narrow down until they reach the stable regime (a {\em finite-width instability}). In the non-radial repulsive case, all bound states are unstable, and the instability is manifested by a lateral drift away from the defect, sometimes in combination with a finite-width instability or a blowup instability.

Motivation & Objective

• To analyze the stability of standing wave solutions to the nonlinear Schrödinger equation with a point defect modeled by a Dirac delta potential.
• To overcome the challenge of computing the number of negative eigenvalues of the linearized operator around standing waves, using perturbation and continuation techniques.
• To classify the nature of instability—blowup, finite-width narrowing, or lateral drift—depending on nonlinearity (subcritical, critical, supercritical) and defect type (attractive or repulsive).
• To provide a systematic numerical and analytical description of bound state dynamics under non-radial and momentum-nonconserving perturbations.

Proposed method

• Employing a perturbation method and continuation arguments to determine the Morse index (number of negative eigenvalues) of the linearized operator around standing wave solutions.
• Using a virial theorem and a minimization method with two constraints to prove blowup instability under critical or supercritical nonlinearity in the repulsive case (γ < 0).
• Applying a quantitative numerical approach to classify instability types: finite-width narrowing, lateral drift, or blowup.
• Constructing approximate solutions via a sequence of regularized potentials V_a(x) = γa e^{-πa^2 x^2} converging to δ(x) in the H^{-1} norm, and proving convergence of solutions to the original equation.
• Using the Gagliardo-Nirenberg inequality and energy/charge conservation to control H^1 norms and establish uniform bounds for the approximation sequence.
• Analyzing the time evolution of the second moment ∫ x^2 |u|^2 dx to derive the virial identity and detect blowup behavior.

Experimental results

Research questions

• RQ1 What is the stability of standing wave solutions to the nonlinear Schrödinger equation with a Dirac delta potential under radial and non-radial perturbations?
• RQ2 How does the sign and strength of the delta potential (attractive γ > 0 or repulsive γ < 0) affect the stability and dynamics of bound states?
• RQ3 What mechanisms drive instability—blowup, finite-width narrowing, or lateral drift—and under what conditions do they occur?
• RQ4 How can the number of negative eigenvalues of the linearized operator be computed analytically for such non-smooth potentials?

Key findings

• For repulsive defects (γ < 0), standing waves are stable in H^1_rad(R) but unstable in H^1(R) under subcritical nonlinearity (1 < p < 5).
• In the subcritical radial case, unstable bound states do not collapse but narrow down to a finite width, indicating a finite-width instability mechanism.
• In the non-radial repulsive case, all bound states are unstable, with instability primarily driven by lateral drift away from the defect, sometimes combined with finite-width narrowing or blowup.
• Under critical or supercritical nonlinearity (p ≥ 5), instability in the repulsive case is proven to be of the blowup type via a virial theorem and a two-constraint minimization method.
• The explicit formula for the ground state ϕω,γ(x) is derived, showing that γ < 0 leads to a "∨"-shaped profile and γ > 0 to a "∧"-shaped profile, with width and amplitude controlled by ω.
• The approximation method using regularized potentials V_a converges strongly in C([0,T], H^1(R)) to the solution of the original delta-potential equation, ensuring the validity of the limiting argument.