One parameter family of Compacton Solutions in a class of Generalized Korteweg-DeVries Equations

Abstract

We study the generalized Korteweg-DeVries equations derivable from the Lagrangian: $L(l,p)=\int \left(\frac{1}{2}{\phi }_x{\phi }_t-\frac{({\phi }_x{)}^l}{l(l-1)}+\alpha ({\phi }_x{)}^p({\phi }_{xx}{)}^2\right)dx,$ where the usual fields $u(x,t)$ of the generalized KdV equation are defined by $u(x,t)={\phi }_x(x,t)$. For $p$ an arbitrary continuous parameter $0<p\le 2,l=p+2$ we find compacton solutions to these equations which have the feature that their width is independent of the amplitude. This generalizes previous results which considered $p=1,2$. For the exact compactons we find a relation between the energy, mass and velocity of the solitons. We show that this relationship can also be obtained using a variational method based on the principle of least action.

Latex 4 pages and one figure available on request