1996/02/01 by Peter J. Olver, Philip Rosenau · 821 citations
Physics and Astronomy · Mathematics · #Nonlinear Waves and Solitons #Nonlinear Photonic Systems #Advanced Fiber Laser Technologies #Integrable system #Hamiltonian (control theory) #Mathematical physics #Duality (order theory) #Nonlinear system #Dispersionless equation #Physics #Scaling #Hamiltonian system #Mathematical analysis #Mathematics #Pure mathematics #Quantum mechanics #Kadomtsev–Petviashvili equation #Geometry #Burgers' equation
paper · doi:10.1103/physreve.53.1900
published in Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics 53(2), 1900-1906 (American Physical Society)
openalex publication_date 1996/02/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/08
A simple scaling argument shows that most integrable evolutionary systems, which are known to admit a bi-Hamiltonian structure, are, in fact, governed by a compatible trio of Hamiltonian structures. We demonstrate how their recombination leads to integrable hierarchies endowed with nonlinear dispersion that supports compactons (solitary-wave solutions having compact support), or cusped and/or peaked solitons. A general algorithm for effecting this duality between classical solitons and their nonsmooth counterparts is illustrated by the construction of dual versions of the modified Korteweg--de Vries equation, the nonlinear Schr"odinger equation, the integrable Boussinesq system used to model the two-way propagation of shallow water waves, and the Ito system of coupled nonlinear wave equations. These hierarchies include a remarkable variety of interesting integrable nonlinear differential equations. \textcopyright 1996 The American Physical Society.