Wavelet basis analysis in perturbed periodic KdV equation
This paper studies the perturbed periodic KdV equation given by \[ u_t+\varepsilon u_{xxxx}+u_{xxx}-\eta u_{xx}+\gamma u+uu_x=f,\quad u(x,t)= u(x+1,t), \quad u(x,0)= u_0(x)\in H^3(\Omega) \] where \(\Omega=[0,1]\), \(\eta, \gamma>0\) and \(f\in H^3(\Omega)\) is time independent. The purpose of the present paper is to study the long time behavior of perturbed periodic KdV equation by using a spline wavelet basis to construct the approximate inertial manifold. The essential part is the construction of the inertial manifold by using the spline wavelet basis. The method to study the approximate inertial manifold by using the spline wavelet basis originates from a paper by Goubet on the self-adjoint nonlinear evolutionary equation. But the question and the result in the present paper are different from that of Goubet's paper.
scientific article; zbMATH DE number 1322258