Abstract
An intimate connection between two mathematical concepts—the theory of solitons on the one hand, and the theory of commuting non-self-adjoint operators on the other hand—is established. The link is provided by the Marchenko method for solving nonlinear partial differential equations in general and the Korteweg–de Vries equation in particular. The solution of this equation is cast in a special form in terms of solitonic combinations such that the role played by a pair of commuting non-self-adjoint operators is evident and lends itself to generalization. Solitonic combinations are closely related to collective motions of spatiotemporal systems, whose theory is reviewed here and extended to the physical space of four dimensions (one time and three space). A novel interpretation of the physical concept of quasiparticle is presented.
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