Extension on peakons and periodic cusp waves for the generalization of the Camassa-Holm equation

2014 ◽  
Vol 38 (11) ◽  
pp. 2363-2375 ◽  
Author(s):  
Zhenshu Wen
Keyword(s):  
Holm Equation ◽  
Periodic Cusp Waves

PERIODIC WAVES AND THEIR LIMITS FOR THE CAMASSA–HOLM EQUATION

2006 ◽  
Vol 16 (08) ◽  
pp. 2261-2274 ◽  
Author(s):  
ZHENGRONG LIU ◽  
ALI MOHAMMED KAYED ◽  
CAN CHEN
Keyword(s):  
Periodic Wave ◽  
Periodic Waves ◽  
Implicit Functions ◽  
Bifurcation Method ◽  
Periodic Wave Solutions ◽  
Wave Solutions ◽  
Holm Equation ◽  
Implicit Expressions ◽  
Periodic Cusp Waves ◽  
Theoretical Results

In this paper, the bifurcation method of dynamical systems is employed to study the Camassa–Holm equation [Formula: see text] We investigate the periodic wave solutions of form u = φ(ξ) which satisfy φ(ξ + T) = φ(ξ), here ξ = x - ct and c, T are constants. Their six implicit expressions and two explicit expressions are obtained. We point out that when the initial values are changed, the periodic waves may become three waves, periodic cusp waves, smooth solitary waves and peakons. Our results give an explanation to the appearance of periodic cusp waves and peakons. Moreover, three sets of graphs of the implicit functions are drawn, and three sets of numerical simulations are displayed. The identity of these graphs and simulations imply the correctness of our theoretical results.

scholarly journals Conservative finite difference schemes for the modified Camassa-Holm equation

JSIAM Letters ◽  
2011 ◽  
Vol 3 (0) ◽  
pp. 37-40 ◽  
Author(s):  
Yuto Miyatake ◽  
Takayasu Matsuo ◽  
Daisuke Furihata
Keyword(s):  
Finite Difference ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Holm Equation

The Camassa–Holm equation on the half-line with linearizable boundary condition

2010 ◽  
Vol 348 (13-14) ◽  
pp. 775-780 ◽  
Author(s):  
Anne Boutet de Monvel ◽  
Dmitry Shepelsky
Keyword(s):  
Boundary Condition ◽  
Half Line ◽  
Holm Equation

scholarly journals The Cauchy problem for the generalized Camassa–Holm equation in Besov space

2014 ◽  
Vol 256 (8) ◽  
pp. 2876-2901 ◽  
Author(s):  
Wei Yan ◽  
Yongsheng Li ◽  
Yimin Zhang
Keyword(s):  
Cauchy Problem ◽  
Besov Space ◽  
Holm Equation ◽  
The Cauchy Problem
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