Special Session 21: Nonlinear Evolution Equations and Applications

The 2-soliton solutions of the Korteweg-de Vries equation satisfy a fourth-order nonlinear ordinary differential equation which depends on two parameters λ, μ. As is well known, for each fixed choice of (λ, μ) in R, the ODE is actually a completely integrable Hamiltonian system with two degrees of freedom. Here we address the question of whether the soliton and 2-soliton solutions of KdV are the only L solutions of this ODE, as (λ, μ) ranges over R. In ruling out alternate solutions, the difficult case seems to be when the eigenvalues of the linearized equation around 0 are two double eigenvalues on the imaginary axis, so there is a four-dimensional center manifold. We discuss the connection between this question and the stability theory of 2-solitons. (Preliminary report.) −→∞ ∞←−