Evolution of near-soliton initial conditions in non-linear wave equations through their Bäcklund transforms

A novel analytic technique for determining the evolution of near-soliton initial conditions in non-linear wave equations is introduced. It is based on the Bäcklund transform connecting soliton solutions of successive order. This transformation lowers the order of the initial condition rendering the determination of the evolution easier. The result of the evolution in this order is transformed to the higher order using again the Bäcklund transform. As a demonstration, the proposed technique is applied to the non-linear Schrödinger (NLS) and Korteweg–de Vries (KdV) equations. The results are in very good agreement with those obtained by other approaches based on the inverse scattering method. Finally, numerical simulations verify the validity of the proposed technique.