Radiative tail of solitary waves in an extended Korteweg-de Vries equation

We solve the fifth-order Korteweg-de Vries (fKdV) equation which is a modified KdV equation perturbed by a fifth-order derivative term multiplied by a small parameter $\epsilon^2$, with $0< \epsilon \ll 1$. Unlike the KdV equation, the stationary fKdV equation does not exhibit exactly localized 1-soliton solution, instead it allows a solution which has a well defined central core similar to that of the KdV 1-soliton solution, accompanied by extremely small oscillatory standing wave tails on both sides of the core. The amplitude of the standing wave tail oscillations is $\mathcal{O}(\exp(-1/\epsilon))$, i.e. it is beyond all orders small in perturbation theory. The analytical computation of the amplitude of these transcendentally small tail oscillations has been carried out up to $\mathcal{O}(\epsilon^5)$ order corrections by using the complex method of matched asymptotics. Also the long-standing discrepancy between the $\mathcal{O}(\epsilon^2)$ perturbative result of Grimshaw and Joshi (1995) and the numerical results of Boyd (1995) has been resolved. In addition to the stationary symmetric weakly localized solitary wave-like solutions, we analyzed the stationary asymmetric solutions of the fKdV equation which decay exponentially to zero on one side of the (slightly asymmetric) core and blows up to large negative values on other side of the core. The asymmetry is quantified by computing the third derivative of the solution at the origin which also turns out to be beyond all orders small in perturbation theory. The analytical computation of the third derivative of a function at the origin has also been carried out up to $\mathcal{O}(\epsilon^5)$ order corrections. We use the exponentially convergent pseudo-spectral method to solve the fKdV equation numerically. The analytical and the numerical results show remarkable agreement.