This paper deals with various applications of two basic theorems in orderpreserving systems under a group action -monotonicity theorem and convergence theorem. Among other things we show symmetry properties of stable solutions of semilinear elliptic equations and systems. Next we apply our theory to traveling waves and pseudo-traveling waves for a certain class of quasilinear diffusion equations and systems, and show that stable traveling waves and pseudo-traveling waves have monotone profiles and, conversely, that monotone traveling waves and pseudotraveling waves are stable with asymptotic phase. We also discuss pseudo-traveling waves for equations of surface motion.2000 Mathematics Subject Classification. 35K40, 35K57. Key words and phrases. order-preserving systems, semilinear elliptic equations, traveling waves and pseudo-traveling waves, quasilinear diffusion equations.

Given an equation with a certain symmetry, such as symmetry with respect to rotation or translation, one of the most fundamental questions to ask is whether or not the symmetry of the equation is inherited by its solutions. We first discuss this question in a general framework of order-preserving dynamical systems under a group action and establish a theory concerning symmetry or monotonicity properties of stable equilibrium points. We then apply this general theory to nonlinear partial differential equations. Among other things, we prove the rotational symmetry of solutions for a class of nonlinear elliptic equations and the monotonicity of travelling waves of some nonlinear diffusion equations. We also discuss the stability of stationary or periodic solutions for equations of surface motion.

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