On the stationary solutions of the 2D Doi-Onsager model
This paper deals with the \(2D\) Doi-Onsager model \[ F = Z^{-1}e^{-U(F)} \text{ on } S^1 \tag{1} \] with the potential \[ U(F)(\phi) := U_l (F)(\phi) := \beta \int_{S^1} | \sin (\theta - \phi) |^l F(\theta) \;d \theta, \tag{2} \] where \(l\) is a positive integer, \(\beta\) is a parameter, \(F\) is a probability distribution function, i.e., \(F > 0, \;\int_{S^1} F =1\) and \(Z = \int_{S^1} e^{-U(F)} \;d \theta.\) First, the authors show that, for \(l=1,\) any pair of solutions \(F\) and \(U(F)\) of equations (1) and (2) are axially symmetric. If \(l\) is any odd positive integer, then, for any \(\beta > 0,\) the pair of equations (1) and (2) possesses non-constant solutions. In addition, in the case \(l=1,\) for any positive integer \(k\), there exists at least one pair of \(\frac{\pi}{k}\)-periodic non-constant solutions. Finally, if \(l=2k\) is an even integer, then for some appropriate values of \(\beta,\) the pair of equations (1) and (2) possesses solutions of the form \(A_0 + \sum_{m=0}^{k} (A_m \cos 2m\phi + B_m \sin 2m \phi)\) as well as axial symmetric solutions of the form \(A_0 + \sum_{m=0}^k A_m \cos 2m\phi.\) In the proof, the authors derive differential equations equivalent to (1) and (2) and, then, they use variational methods.
scientific article; zbMATH DE number 5774956