An Analytical Solution for a Nonlinear Differential Equation with Logarithmic Decay

The problem is the ordinary differential equation du/dt = -p exp( -q/u) with u(O) = u. We prove that the general three-parameter problem may be reduced through a group invariance to computing a single, parameter-free function w(f). This solution may be most compactly expressed in the form w = l/ln(t*ti(t*)B(r)), where 7 = ln(ln(t*)) and t* = t + exp(1). We compute a Chebyshev series for B(7) for T E [0,6] and show that B(T) 1 + (47 2)exp(-7) + 4r2exp(-27) to within 1 part in lo4 for T > 6. The problem was motivated by the similar logarithmic decay of certain classes of quasi-solitary waves, such as the “breather” of the +4 field theory. o 1988 AC&KC PXSS, IIIC.