×

The wild solutions of the induced form under the spline wavelet basis in weakly damped forced KdV equation. (English) Zbl 0937.35163

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.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
37L25 Inertial manifolds and other invariant attracting sets of infinite-dimensional dissipative dynamical systems
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] O. Goubet, Construction on approximate inertial manifolds using wavelets,SIAM. J. Math. Anal.,9 (1992), 1455–1481. · Zbl 0770.35003 · doi:10.1137/0523083
[2] Tian Lixin and Xu Zhenyuan, The research of longtime dynamics behavior in weakly damped forced KdV equation,Applied Mathematics and Mechanics (English Ed.),18, 10 (1997), 1021–1028. · Zbl 0904.35082
[3] Lu Dianchen, Tian Lixin and Liu Zengrong, Wavelet basis analysis in perturbed periodic KdV equation,Applied Methametics and Mechanics (English Ed.),19, 11 (1998), 1053–1058. · Zbl 0936.35170 · doi:10.1007/BF02459193
[4] J. M. Ghidaglia, Weakly damped forced KdV equation behave as finite dimensional dynamical system in the longtime,J. Differential Equations,74 (1988), 369–390. · Zbl 0668.35084 · doi:10.1016/0022-0396(88)90010-1
[5] A. Debussche and M. Marion, On the constructure of families of approximate inertial manifolds,J. Differential Equations,100 (1992), 173–201. · Zbl 0760.34050 · doi:10.1016/0022-0396(92)90131-6
[6] C. K. Chui,An Introduction to Wavelet, Academic Press Inc., USA (1992). · Zbl 0925.42016
[7] C. Foias, G. Sell and R. Temann, Inertial manifolds for nonlincar evolutionary equation,J. Differential Equations,73 (1988), 309–353. · Zbl 0643.58004 · doi:10.1016/0022-0396(88)90110-6
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.