One-parameter family of soliton solutions with compact support in a class of generalized Korteweg-de Vries equations

Abstract

We study the generalized Korteweg-de Vries (KdV) equations derivable from the Lagrangian L(l,p)=F[½ cphi_xcphi;_t-(cphi_x)^l/ /(l-1)+α(cphi_x)^p (cphi_xx)²]dx, where the usual fields u(x,t) of the generalized KdV equation are defined by u(x,t)=cphi_x(x,t). For p an arbitrary continuous parameter 0<p≤2, l=p+2 we find soliton solutions with compact support (compactons) 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.