Variational Method for Studying Solitons in the Korteweg-DeVries Equation

We use a variational method based on the principle of least action to obtain approximate time-dependent single soliton solutions to the KdV equation. A class of trial variational functions of the form u(x, t) = −A(t) exp [ −β(t) |x− q(t)| ] , with n a continuous real variable, is used to parametrize timedependent solutions. We find that this class of trial functions leads to soliton-like solutions for all n, moving with fixed shape and constant velocity, and with energy and mass conserved. Minimizing the energy of the soliton with respect to the 1 parameter n, we obtain a variational solution that gives an extremely accurate approximation to the exact solution.