Author
Guilong Gui
Other affiliations: Chinese Academy of Sciences, The Chinese University of Hong Kong, Northwest University (China)
Bio: Guilong Gui is an academic researcher from Jiangsu University. The author has contributed to research in topics: Cauchy problem & Initial value problem. The author has an hindex of 10, co-authored 11 publications receiving 927 citations. Previous affiliations of Guilong Gui include Chinese Academy of Sciences & The Chinese University of Hong Kong.
Papers
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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.
244 citations
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.
Abstract: In this paper, we investigate the formation of singularities and the existence of peaked traveling-wave solutions for a modified Camassa-Holm equation with cubic nonlinearity. The equation is known to be integrable, and is shown to admit a single peaked soliton and multi-peakon solutions, of a different character than those of the Camassa-Holm equation. Singularities of the solutions can occur only in the form of wave-breaking, and a new wave-breaking mechanism for solutions with certain initial profiles is described in detail.
237 citations
TL;DR: In this article, the authors established the local well-posedness of the two-component Camassa-Holm system in a range of the Besov spaces and derived a wave-breaking mechanism for strong solutions.
Abstract: In this paper we establish the local well-posedness for the two-component Camassa–Holm system in a range of the Besov spaces. We also derive a wave-breaking mechanism for strong solutions. In addition, we determine the exact blow-up rate of such solutions to the system.
155 citations
TL;DR: In this article, the authors studied the well-posedness of the Cauchy problem and the scattering problem for a new nonlinear dispersive shallow water wave equation (the so-called DGH equation) which was derived by Dullin, Gottwald and Holm.
Abstract: In this paper, we study the well-posedness of the Cauchy problem and the scattering problem for a new nonlinear dispersive shallow water wave equation (the so-called DGH equation) which was derived by Dullin, Gottwald and Holm. The issue of passing to the limit as the dispersive parameter tends to zero for the solution of the DGH equation is investigated, and the convergence of solutions to the DGH equation as α2→0 is studied, and the scattering data of the scattering problem for the equation can be explicitly expressed; the new exact peaked solitary wave solutions are obtained in the DGH equation. After giving the condition of existing peakon in the DGH equation, it turns out to be nonlinearly stable for the peakon in the DGH equation.
125 citations
TL;DR: In this article, the authors considered the modified Camassa-Holm equation with cubic nonlinearity, which is integrable and admits the single peaked solitons and multi-peakon solutions.
94 citations
Cited by
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Journal Article•
28,685 citations
01 Jan 2012
TL;DR: In particular, the linear dispersive term in the Korteweg-de Vries equation prevents this from ever happening in its solution as discussed by the authors, and the instability and subsequent modulation of an initially uniform wave profile can be prevented by including dispersive effects in the shallow water theory.
Abstract: Dispersion and nonlinearity play a fundamental role in wave motions in nature. The nonlinear shallow water equations that neglect dispersion altogether lead to breaking phenomena of the typical hyperbolic kind with the development of a vertical profile. In particular, the linear dispersive term in the Korteweg–de Vries equation prevents this from ever happening in its solution. In general, breaking can be prevented by including dispersive effects in the shallow water theory. The nonlinear theory provides some insight into the question of how nonlinearity affects dispersive wave motions. Another interesting feature is the instability and subsequent modulation of an initially uniform wave profile.
864 citations
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.
244 citations
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.
Abstract: In this paper, we investigate the formation of singularities and the existence of peaked traveling-wave solutions for a modified Camassa-Holm equation with cubic nonlinearity. The equation is known to be integrable, and is shown to admit a single peaked soliton and multi-peakon solutions, of a different character than those of the Camassa-Holm equation. Singularities of the solutions can occur only in the form of wave-breaking, and a new wave-breaking mechanism for solutions with certain initial profiles is described in detail.
237 citations
TL;DR: In this paper, the Cauchy problem for the integrable Novikov equation is shown to be locally well-posed in the Besov space with 1 max{1 + 1 + 1/p, 3/2} for all t is an element of [0, T] and for all u(t) is a vertices.
98 citations