[Paper Review] Normalized solutions to the Chern-Simons-Schrödinger system

Tianxiang Gou, Zhitao Zhang|arXiv (Cornell University)|Mar 18, 2019

Advanced Mathematical Physics Problems51 references4 citations

TL;DR

This paper establishes the existence, uniqueness, and stability properties of normalized solutions to the Chern-Simons-Schrödinger system in two spatial dimensions via variational methods under an $L^2$-norm constraint. It proves compactness of minimizing sequences in the mass subcritical case, orbital stability of minimizers, and existence of ground states and infinitely many radially symmetric solutions in the supercritical regime, with orbital instability of ground states demonstrated.

ABSTRACT

In this paper, we study normalized solutions to the Chern-Simons-Schrödinger system, which is a gauge-covariant nonlinear Schöridnger system with a long-range electromagnetic field, arising in nonrelativistic quantum mechanics theory. The solutions correspond to critical points of the underlying energy functional subject to the $L^2$-norm constraint. Our research covers several aspects. Firstly, in the mass subcritical case, we establish the compactness of any minimizing sequence to the associated global minimization problem. As a by-product of the compactness of any minimizing sequence, the orbital stability of the set of minimizers to the problem is achieved. In addition, we discuss the radial symmetry and uniqueness of minimizer to the problem. Secondly, in the mass critical case, we investigate the existence and nonexistence of normalized solution. Finally, in the mass supercritical case, we prove the existence of ground state and infinitely many radially symmetric solutions. Moreover, the instability of ground states is explored

Motivation & Objective

• To study normalized solutions to the Chern-Simons-Schrödinger system, which models nonrelativistic quantum systems with long-range electromagnetic fields.
• To analyze the existence and stability of standing wave solutions under an $L^2$-norm constraint, corresponding to fixed particle number.
• To classify solutions based on the mass-critical, subcritical, and supercritical regimes of the nonlinearity exponent $p$.
• To establish orbital stability of minimizers in the mass subcritical case and orbital instability of ground states in the supercritical case.
• To investigate radial symmetry and uniqueness of minimizers in the mass subcritical regime.

Proposed method

• Formulate the stationary Chern-Simons-Schrödinger system as a constrained variational problem on $H^1(\mathbb{R}^2)$ with fixed $L^2$-norm.
• Use the Coulomb gauge condition to express gauge fields $A_0, A_1, A_2$ in terms of the wave function $u$ via convolution with singular kernels $G_j(x) = -\frac{1}{2\pi}\frac{x_j}{|x|^2}$.
• Apply concentration-compactness principles to prove compactness of minimizing sequences in the mass subcritical case ($p < 4$).
• Employ the mountain pass lemma and symmetric criticality to construct ground states and infinitely many radially symmetric solutions in the mass supercritical case ($p > 4$).
• Analyze orbital stability and instability via the sharp threshold method and second variation of the energy functional.
• Use test functions $\xi$ with compact support and compute the time derivative of a weighted $L^2$-norm to derive instability criteria.

Experimental results

Research questions

• RQ1 Under what conditions does a normalized solution exist for the Chern-Simons-Schrödinger system in two dimensions?
• RQ2 Is the set of minimizers for the constrained energy functional orbitally stable in the mass subcritical regime?
• RQ3 What is the role of radial symmetry and uniqueness in the minimization problem for normalized solutions?
• RQ4 Does a ground state exist in the mass supercritical case, and is it orbitally unstable?
• RQ5 What are the conditions for nonexistence of normalized solutions in the mass critical case ($p = 4$)?

Key findings

• In the mass subcritical case ($p < 4$), any minimizing sequence for the constrained energy functional is relatively compact in $H^1(\mathbb{R}^2)$, implying the existence of a minimizer.
• The set of minimizers is orbitally stable in the mass subcritical regime due to the compactness of minimizing sequences.
• The minimizer of the constrained energy problem is unique and radially symmetric in the mass subcritical case.
• In the mass critical case ($p = 4$), the paper establishes both existence and nonexistence results for normalized solutions depending on the coupling constant $\lambda$.
• In the mass supercritical case ($p > 4$), the existence of a ground state and infinitely many radially symmetric solutions is proven via variational methods.
• The ground state solutions in the supercritical regime are orbitally unstable, as shown by analyzing the second variation of the energy functional and constructing a suitable test function.