Journal ArticleDOI
Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support.
Peter J. Olver,Philip Rosenau +1 more
Reads0
Chats0
TLDR
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, and it is demonstrated how their recombination leads toIntegrable hierarchies endowed with nonlinear dispersion that supports compactons, or cusped and/or peaked solitons.Abstract:
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.read more
Citations
More filters
Journal ArticleDOI
A new integrable equation with peakon solutions
TL;DR: In this article, a new partial differential equation, of a similar form to the Camassa-Holm shallow water wave equation, was obtained by Degasperis and Procesi using the method of asymptotic integrability.
Journal ArticleDOI
Well-posedness and Blow-up Solutions for an Integrable Nonlinearly Dispersive Model Wave Equation
Yi A. Li,Peter J. Olver +1 more
TL;DR: In this article, the Sobolev space was established for the Camassa-Holm equation, and conditions on the initial data that lead to finite time blow-up of certain solutions were demonstrated.
Journal ArticleDOI
On an integrable two-component Camassa–Holm shallow water system
TL;DR: In this paper, a two-component integrable system of coupled equations was derived in the context of shallow water theory and it was shown that while small initial data develop into global solutions, for some initial data wave breaking occurs.
Journal ArticleDOI
On the global existence and wave-breaking criteria for the two-component Camassa-Holm system
Guilong Gui,Guilong Gui,Yue Liu +2 more
TL;DR: In this paper, a two-component Camassa-Holm system is considered, and a wave-breaking criterion for strong solutions is determined in the lowest Sobolev space Hs, s>32 by using the localization analysis in the transport equation theory.
Journal ArticleDOI
Wave-Breaking and Peakons for a Modified Camassa-Holm Equation
TL;DR: In this article, the authors investigated the formation of singularities and the existence of peaked traveling-wave solutions for a modified Camassa-Holm equation with cubic nonlinearity.
Related Papers (5)
Symplectic structures, their Bäcklund transformations and hereditary symmetries
Wave breaking for nonlinear nonlocal shallow water equations
Adrian Constantin,Joachim Escher +1 more