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We study the class of generalized Korteweg-DeVries equations derivable from the Lagrangian:, where the usual fields u(x, t) of the generalized KdV equation are defined by u(x, t) = ϕ x (x, t). This class contains compactons, which are solitary waves with compact support, and when l = p + 2, these solutions have the feature that their width is independent of the amplitude. We consider the Hamiltonian structure and integrability 1 properties of this class of KdV equations. We show that many of the properties of the solitary waves and compactons are easily obtained using a variational method based on the principle of least action. Using a class of trial variational functions of the form u(x, t) = A(t) exp −β(t) |x − q(t)| 2n we find soliton-like solutions for all n, moving with fixed shape and constant velocity, c. We show that the velocity, mass, and energy of the variational travelling wave solutions are related by c = 2rEM −1 , where r = (p + l + 2)/(p + 6 − l), independent of n.

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%e consider a system in which a classical oscillator is interacting with a purely quantum mechanical oscillator, described by the Lagrangian L = 2x + 2A -2 (m +e A )x, where A is a classical variable and x is a quantum operator. With (x(t)) = 0, the relevant variable for the quantum oscillator is (x(t)x(t)) = G(t). The classical Hsmiltonian dynamics governing the variables A(t), II&(t), G(t), and IIa(t) is chaotic so that the results of making measurements on the quantum system at later times are sensitive to initial conditions. This system arises as the zero momentum part of the problem of pair production of charged scalar particles by a strong external electric 6eld.