An analytical solution for a nonlinear differential equation with logarithmic decay

Abstract

The problem is the ordinary differential equation du/dt = - p exp(- q/u) with u(0) = [alpha]. We prove that the general three-parameter problem may be reduced through a group invariance to computing a single, parameter-free function w(t). This solution may be most compactly expressed in the form w = 1/ln (t*ln2(t*)B([tau])), where [tau] [equiv] ln(ln(t*)) and t* = t + exp(1). We compute a Chebyshev series for B([tau]) for [tau][epsilon] [0,6] and show that B([tau]) ~ 1 + (4[tau] -2) exp(-[tau]) + 4[tau]2exp(-2[tau]) to within 1 part in 104 for [tau] > 6. The problem was motivated by the similar logarithmic decay of certain classes of quasi-solitary waves, such as the "breather" of the [phi]4 field theory.