On the reduction and the existence of approximate analytic solutions of some basic nonlinear ODEs in mathematical physics and nonlinear mechanics

Abstract

Using series of admissible functional transformations we reduce the one-dimensional axisymmetric nonlinear Schrödinger (NLS) equation, as well as the forced damped nonlinear Duffing (NLD) equation to equivalent nonlinear first-order integrodifferential equations. The forced undamped (NLD) equation results as a special case. The reduced integrodifferential equations are exact. In the limits of small or large values of the parameters characterizing these nonlinear problems, we prove that further reductions lead to first-order nonlinear ordinary differential equations which, except in case of the (NLS) equation, are of the Abel classes. The approximate reduced (NLS) equation admits exact analytic solutions. On the other hand, taking into account the known exact analytic solutions of the equivalent Abel classes of equations we show that there do not exist analytic solutions of the above two nonlinear Duffing oscillators. However, if further asymptotic approximations take place, new approximate analytic solutions concerning the (NLD) equations are constructed.