Steady state solutions of the Satsuma-Mimura diffusion equation (Q1113010)

\textit{J. Satsuma} and \textit{M. Mimura} [Exact treatment of nonlinear diffusion equations with singular integral terms, J. Phys. Soc. Japan 54, 894-900 (1985); see also \textit{M. Mimura}, Contemp. Math. 17, 343-351 (1983; Zbl 0547.35055)] investigated a class of nonlinear nonlocal diffusion equations involving singular integral terms. They developed an exact linearization method for these equations and derived some interesting particular solutions in explicit form by this method. In the present paper we rederive the steady state solutions of Satsuma and Mimura in systematic way by reducing the steady equations to complex differential equations of first order. We find that apart from an obvious invariant shifting transformation the steady state solutions found by Satsuma and Mimura are the only ones with a prescribed asymptotic behaviour at infinity. Besides we obtain new periodic steady state solutions for some corresponding equations with variable diffusion coefficient.