Exact Meromorphic Stationary Solutions of the Cubic-Quintic Swift-Hohenberg Equation

Abstract

In this paper, we study an ODE of the form$$b_0 u^{(4)} + b_1 u'' + b_2 u + b_3 u^3 + b_4 u^5 = 0, \  ' = \frac{d}{d z},$$ which includes, as a special case,  the stationary case of the cubic-quintic Swift-Hohenberg equation. Based on Nevanlinna theory and Painlevé analysis, we first show that all its meromorphic solutions are elliptic or degenerate elliptic. Then we obtain them all explicitly by the method introduced in [7].