Nonlinear Schrödinger‐type field equation for the description of dissipative systems. I. Derivation of the nonlinear field equation and one‐dimensional example

1983 ◽  
Vol 24 (6) ◽  
pp. 1652-1660 ◽  
Author(s):  
D. Schuch ◽  
K.‐M. Chung ◽  
H. Hartmann
Keyword(s):  
Field Equation ◽  
Dissipative Systems ◽  
Nonlinear Field ◽  
One Dimensional ◽  
Nonlinear Schrödinger

Approximate Solution of a Nonlinear Field Equation

1963 ◽  
Vol 4 (3) ◽  
pp. 334-338 ◽  
Author(s):  
D. D. Betts ◽  
H. Schiff ◽  
W. B. Strickfaden
Keyword(s):  
Approximate Solution ◽  
Field Equation ◽  
Nonlinear Field

Uniqueness and Symmetry for the Mean Field Equation on Arbitrary Flat Tori

2020 ◽  
Author(s):  
Guangze Gu ◽  
Changfeng Gui ◽  
Yeyao Hu ◽  
Qinfeng Li
Keyword(s):  
Field Equation ◽  
Level Sets ◽  
Mean Field ◽  
Flat Torus ◽  
One Dimensional ◽  
Uniqueness Of The Solution ◽  
Flat Tori ◽  
The Mean ◽  
Symmetry Result ◽  
Mean Field Equation

Abstract We study the following mean field equation on a flat torus $T:=\mathbb{C}/(\mathbb{Z}+\mathbb{Z}\tau )$: $$\begin{equation*} \varDelta u + \rho \left(\frac{e^{u}}{\int_{T}e^u}-\frac{1}{|T|}\right)=0, \end{equation*}$$where $ \tau \in \mathbb{C}, \mbox{Im}\ \tau>0$, and $|T|$ denotes the total area of the torus. We first prove that the solutions are evenly symmetric about any critical point of $u$ provided that $\rho \leq 8\pi $. Based on this crucial symmetry result, we are able to establish further the uniqueness of the solution if $\rho \leq \min{\{8\pi ,\lambda _1(T)|T|\}}$. Furthermore, we also classify all one-dimensional solutions by showing that the level sets must be closed geodesics.

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