Evolution of near-soliton initial conditions in nonlinear wave equations through their Bäcklund transforms. (English) Zbl 1069.35066

Summary: A novel analytic technique for determining the evolution of near-soliton initial conditions in nonlinear 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 nonlinear Schrödinger and Korteweg-de Vries 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.


35Q51 Soliton equations
37K35 Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems
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