The Long Time Behavior Of Long-Short Wave Equations

The long-short wave equations is a kind of nonlinear resonance equations. It is an importrepresentative of resonance theory in infinite-dimensional dynamic systems and is applied to manyphysical models. Studying long-short wave equations is of great theory meanings and applicationvalues.In this paper, the long time behavior of generalized (2+1)-dimensional dissipative long-shortwave equations and non-autonomous dissipative long-short wave equations were studied. Theexistence of the global attractor for the generalized (2+1)-dimensional dissipative long-short waveequations, and that of the compact uniform attractor for the non-autonomous long-short waveequations were obtained. Besides, one approximate inertial manifold for the non-autonomous long-short wave equations was constructed.The paper is organized as follows.Chapter1, some basic theories on autonomous and non-autonomous infinite dimensional dy-namic systems, which is the very fundamental of this article, were reviewed. Some diferencesbetween autonomous systems and non-autonomous ones were presented in this chapter. The back-grounds in which long-short wave equations arise were stated briefly and the purposes for thisresearch were presented in this chapter as well.Chapter2, a kind of generalized (2+1)-dimensional dissipative long-short wave equationswere studied. By the Energy methods (i.e. the weak convergence method called in some otherpublications), the existence of the compact global attractor for the equations were established.Chapter3, non-autonomous long-short wave equations with translation compact forces werestudied. The uniform a prior estimates and the Gal′ekin method were applied to get the uniqueexistence of the inertial boundary value problem. The existence of uniform absorbing sets and theweakly compact uniform attractor were obtained afterwards.Chapter4, non-autonomous long-short wave equations with quasi-periodic forces were studied.By means of extending phase space and operator projections we constructed an approximate inertialmanifold for this non-autonomous system.