We consider an integrable nonlinear wave system (anisotropic chiral field model) which exhibits a soliton solution when the Cauchy problem for an infinitely long medium is posed. Whenever the boundary value problem is formulated for the same system but for a medium of finite extension, we reveal that the soliton becomes unstable and the true attractor is a different structure which is called polarization attractor. In contrast to the localized nature of solitons, the polarization attractor occupies the entire length of the medium. By demonstrating the qualitative difference between nonlinear wave propagation in an infinite medium and in a medium of finite extension (with simultaneous change of the initial value problem to the boundary value problem), we would like to point out that solitons may loose their property of being stable attractors. Additionally, our findings show the interest of developing methods of integration for boundary value problems.
37K40 (Soliton theory, asymptotic behavior of solutions) 37C70 (Attractors and repellers, topological structure)