Asymptotic Solutions of the Non-Linear Wave Equation of Van-Der-Pol Type on the Infinite Line

Using multiple time and spatial scales it is shown that for the wave equation with a small Van-der-Pol nonlinearity on the infinite line, initially oscillatory waves (with or without slowly-varying amplitudes) leading to saw-tooth waves. If the initial conditions are localised, nonoscillatory, and decay fast enough to zero at infinity then the leading asymptotically valid solution becomes unbounded at large times. But if the initial disturbance vanishes outside a finite interval the leading approximation approaches finite saw-tooth waves at large times.