Journal ArticleDOI
On the Well-Posedness Problem and the Scattering Problem for the Dullin-Gottwald-Holm Equation
Lixin Tian,Guilong Gui,Yue Liu +2 more
TLDR
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.read more
Citations
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Journal ArticleDOI
Dispersive of propagation wave solutions to unidirectional shallow water wave Dullin–Gottwald–Holm system and modulation instability analysis
Journal ArticleDOI
Blow-up of solutions to the DGH equation
TL;DR: In this paper, the best constants for two convolution problems on the unit circle via a variational method were found and applied to a nonlinear integrable shallow water equation (the DGH equation) to give sufficient conditions on the initial data, which guaranteed finite time singularity formation for the corresponding solutions.
Journal ArticleDOI
Global existence and blow-up solutions for a nonlinear shallow water equation
TL;DR: In this paper, the existence of global solutions and the formation of singularities for a new nonlinear shallow water wave equation derived by Dullin, Gottward and Holm are considered.
Journal ArticleDOI
Global existence and blow-up phenomena for the peakon b-family of equations
Guilong Gui,Yue Liu,Lixin Tian +2 more
TL;DR: In this paper, the authors established the local well-posedness for the peakon b-family of equations, including both the Camassa-Holm equation and Degasperis-Procesi equation as special cases.
Journal ArticleDOI
Optimal control of the viscous Camassa–Holm equation
TL;DR: In this article, the authors studied the problem of optimal control of the viscous Camassa-Holm equation and proved the existence and uniqueness of weak solution to weak solution in a short interval.
References
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Book
Solitons, Nonlinear Evolution Equations and Inverse Scattering
M. A. Ablowitz,Peter A. Clarkson +1 more
TL;DR: In this article, the authors bring together several aspects of soliton theory currently only available in research papers, including inverse scattering in multi-dimensions, integrable nonlinear evolution equations in multidimensional space, and the ∂ method.
Journal ArticleDOI
An integrable shallow water equation with peaked solitons
Roberto Camassa,Darryl D. Holm +1 more
TL;DR: A new completely integrable dispersive shallow water equation that is bi-Hamiltonian and thus possesses an infinite number of conservation laws in involution is derived.
Journal ArticleDOI
Commutator estimates and the euler and navier‐stokes equations
Tosio Kato,Gustavo Ponce +1 more
Journal ArticleDOI
Symplectic structures, their Bäcklund transformations and hereditary symmetries
TL;DR: In this paper, it was shown that compatible symplectic structures lead in a natural way to hereditary symmetries, and that a hereditary symmetry is an operator-valued function which immediately yields a hierarchy of evolution equations, each having infinitely many commuting symmetry all generated by this hereditary symmetry.
Journal ArticleDOI
Stability theory of solitary waves in the presence of symmetry, II☆
TL;DR: In this article, the effect of group invariance on the stability of solitary waves was studied and applications were given to bound states and traveling wave solutions of nonlinear wave equations, where the authors considered an abstract Hamiltonian system which is invariant under a group of operators.
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