Dissipative Dynamical Behavior Of The Kdv Equation In The Wavelet And Fourier Analysis

This paper studies an important kind of equation—dissipative KdV equation which belongs to nonlinear evolution equation by Wavelet analysis and Fourier analysis.Firstly,we construct a set of ODEs of seven modes by applying wavelet analysis into infinite dynamical system and build Galerkin solution.So our study of local space-time behaviour and numerical analysis are given .The numerical results show that the solution of the equation takes on complicated dynamics.Movever, Galerkin solution under Fourier bases is also given and we construct a set of ODEs of seven modes.Numerical result shows that Galerkin solution is stable under the same conditions,which proves that in reflecting dynamical characters of wavelet analysis does better than Fourier analysis .Secondly,wavelet approximate inertial manifold and the approximate inertial manifold under Fourier bases are given and made numerically. Approximate inertial manifold has complement,but Galerkin solution has no.So wavelet analysis do better than Fourier analysis in reflecting dynamical characters. Finally,we also analyze dynamics of dissipative KdV equation under Fourier bases and attain the conditions on which the bifurcation of the equation takes place.From above results,we can learn that approximate inertial manifold which is translated from PDE reflects the dynamics of original system.