On a Fractional Schrödinger equation in the presence of Harmonic potential

Abstract

In this paper, we establish the existence of ground state solutions for a fractional Schrödinger equation in the presence of a harmonic trapping potential. We also address the orbital stability of standing waves. Additionally, we provide interesting numerical results about the dynamics and compare them with other types of Schrödinger equations. Our results explain the effect of each term of the Schrödinger equation : The fractional power, the power of the nonlinearity and the harmonic potential.

Figures (11)

• Figure 1: Ground state solution and time dynamics of standing waves with $s=0.8$, $\sigma=1$, $L=10$ and $J=5000$
• Figure 2: Ground state solutions with $\sigma=1$ and different $s$
• Figure 3: $L^2$ distance between ground state solutions of $s<1$ and $s=1$ when $\sigma=1$
• Figure 4: Energy and $\lambda_c$
• Figure 5: Ground state solutions with non-symmetric potential

Theorems & Definitions (10)

• Theorem 2.1
• Remark 2.1
• Theorem 2.2
• Lemma 3.1
• Lemma 3.2
• Remark 3.1
• Remark 3.2
• Lemma 4.1
• Definition 4.1
• Theorem 4.1