SOLITARY WAVES IN A CLASS OF GENERALIZED KORTEWEG-DEVRIES EQUATIONS

We study the class of generalized Korteweg-de Vries equations derivabl e from the Lagrangian L(l,p) = integral [1/2phixphit - (phix)l/l(l - 1 ) + alpha(phixx)2] dx, where the usual fields u(x, t) of the generaliz ed KdV equation are defined by u(x, t) = phix(x, t). This class contai ns compactons, which are solitary waves with compact support, and when l = p + 2, these solutions have the feature that their width is indep endent of the amplitude. We consider the Hamiltonian structure and int egrability properties of this class of KdV equations. We show that man y of the properties of the solitary waves and compactons are easily ob tained using a variational method based on the principle of least acti on. Using a class of trial variational functions of the form u(x, t) = A(t) exp [-beta(t) \x - q(t)\2n] We find solitonlike solutions for al l n, moving with fixed shape and constant velocity c. We show that the velocity, mass, and energy of the variational traveling-wave solution s are related by c = 2rEM-1, where r = (p + l + 2)/(p + 6 - l), indepe ndent of n.