Green Functions for Nonlinear Klein-Gordon Models

In introducing collective coordinates for the center of mass motion for a nonlinear Klein-Gordon kink, one arrives at the following closed-form expression for the "phonon field" generated by the interaction of a moving kink with a perturbation:

where I(x,t) represents the inhomogenity.

Having a closed form is nice, but as the Green functions themselves are given by

G(x,x¢,t) = N å i=1  fb,i*(x)fb,i(x¢) ¥ ó õ -¥    dweiwt 2p(wb,i2-w2) + ¥ ó õ -¥  fk*(x)fk(x¢) ¥ ó õ -¥    dweiwt 2p(wk2-w2)  , (2)

the utility of Eq. is rather diminished. In this paper an analytic form for these Green Functions are found in terms of modified Lommel functions. Given this analytic form, asymptotic expansions for the Green functions are derives which aid in making statements about the long time behavior of the phonon field for example. Fast algorithms for the numerical calculation of the Green functions are also provided.