Eigenfunction Spectrum Analysis for Self-Organization in Dissipative Solitons

An attractor of dissipative structures in solitons described by the Korteweg-de Vries (K-dV) equation with a viscous dissipation term is investigated, with the use of an eigenfunction spectrum analysis associated with the dissipative dynamical operator [Phys. Rev. E 49 (1994) 5546]. It is shown numerically and quantitatively that the basic procesess for the self-organization of dissipative soliton are spectrum transfer by nonlinear interaction and selective dissipation among the eigenmodes of the dissipative operator. It is quantitatively shown that an interchange between the dominant operators occurs during nonlinear self-organization processes, which leads to a final self-similar coherent structure uniquely determined by the dissipative operator.