WebCAMASSA-HOLM CUSPONS, SOLITONS AND THEIR INTERACTIONS VIA THE DRESSING METHOD ROSSEN IVANOV, TONY LYONS, AND NIGEL ORR Abstract. A dressing method is applied to a matrix Lax pair for the Camassa- Holm equation, thereby allowing for the construction of several global solutions of the system. The Camassa–Holm equation can be written as the system of equations: $${\displaystyle {\begin{aligned}u_{t}+uu_{x}+p_{x}&=0,\\p-p_{xx}&=2\kappa u+u^{2}+{\frac {1}{2}}\left(u_{x}\right)^{2},\end{aligned}}}… In fluid dynamics, the Camassa–Holm equation is the integrable, dimensionless and non-linear partial differential equation The equation was introduced by Roberto Camassa and See more Introducing the momentum m as $${\displaystyle m=u-u_{xx}+\kappa ,\,}$$ then two compatible Hamiltonian descriptions of the Camassa–Holm equation are: See more Traveling waves are solutions of the form $${\displaystyle u(t,x)=f(x-ct)\,}$$ representing waves of permanent shape f that propagate at constant speed c. These waves are called … See more In the spatially periodic case, the Camassa–Holm equation can be given the following geometric interpretation. The group $${\displaystyle \mathrm {Diff} (S^{1})}$$ See more The Camassa–Holm equation is an integrable system. Integrability means that there is a change of variables (action-angle variables) such that the evolution equation in the new variables is equivalent to a linear flow at constant speed. This change of variables … See more The Camassa–Holm equation models breaking waves: a smooth initial profile with sufficient decay at infinity develops into either a wave that exists for all times or into a breaking … See more • Degasperis–Procesi equation • Hunter–Saxton equation See more

Camassa–Holm, Korteweg–de Vries and related models …

WebCamassa{Holm, Korteweg{de Vries and related models for water waves 65 description. This approach will enable us to make clear the various shortcomings of these models, … WebThe Camassa-Holm equation, which was derived physically as a shallow water wave equation by Camassa and Holm in [7, 8], takes the form u t +κu x −u xxt +3uu x = 2u xu xx +uu xxx (1.1) where u= u(x,t) is the fluid velocity in the xdirection and the constant κis related to the critical cycling in a park

A blow-up result for the periodic Camassa-Holm equation

WebCamassa-Holm system on the real line. We obtain the global weak solu-tions via a coordinate transformation into the Lagrangian coordinates. The key ingredients in our analysis are the energy density given by the positive Radon measure and the proposed new distance functions as well. 2010 Mathematics Subject Classification: 35G55, 35B35, … Web6 Aug 2009 · Abstract We classify generalized Camassa–Holm-type equations which possess infinite hierarchies of higher symmetries. We show that the obtained equations … Web8 Feb 2024 · In this paper, we consider the rotational Camassa–Holm equation on the circle. Sufficient conditions on the initial data to guarantee wave breaking are established. 1. Introduction Recently, a rotational Camassa–Holm equation was proposed in [ 1 – 4 ], which reads as with where is a parameter related to the Coriolis effect. cheap wooden craft boxes