The wild solutions of the induced form under the spline wavelet basis in weakly damped forced KdV equation (Q1292922)

The authors study the wild solution of the induced form using a spline wavelet basis to the weakly damped forced KdV equation given by \[ \begin{aligned} & u_t+u_{xxx}- \eta u_{xx}+ \gamma u+uu_x=f\;(\eta,\gamma>0),\\ & u(x,t)= u(x+1,t),\\ & u(x,0)= u_0(x)\in H^2(\Omega) \cap H\end{aligned} \] where \(\Omega= [0,1]\), \(f\in H^3(\Omega)\) (time independent). The equation is one of the nonlinear evolutionary equations of nonselfadjoint type. The authors study the long time dynamics and construct the approximate inertial manifold by using the wild solution of the induced form to the weakly damped forced KdV equation. In order to get the final result the study of the perturbed periodic KdV equation given by \[ \begin{aligned} & u_t+ \varepsilon u_{xxxx}+ u_{xxx}-\eta u_{xx}+ \gamma u+uu_x=f\;(\eta,\gamma>0)\\ & u(x,t)= u(x+1,t),\\ & u(x,0) =u_0(x), \end{aligned} \] where \(u_0\), \(f\in H^3(\Omega)\) (time independent), plays an essential role. The perturbed periodic KdV equation was studied by the authors in a previous paper by using spline wavelets.