Peakons and periodic cusp waves in a generalized Camassa-Holm equation

We study the peakons and the periodic cusp wave solutions of the following equation: u(t) + 2ku(x) - u(xxt) + auu(x) = 2u(x)u(xx) + uu(xxx) with a, k is an element of R, which we will call the generalized Camassa-Ho lm equation, or simply the GCH equation, for when a = 3 it was the so-calle d CH equation given by R. Camassa and D.D. Holm [Phys. Rev. Lett. 71 (11) ( 1993) 1661-1664]. They showed that the CH equation has a class of new solit ary wave solutions called "peakons". J.P. Boyd [Appl. Math. Comput. 81 (2-3 ) (1997) 173-187] studied another class of new periodic wave solutions call ed "coshoidal waves". Using the bifurcation method of the phase plane, we f irst construct peakons and show that a = 3 is the peakon bifurcation parame ter value for the GCH equation. Then we construct some smooth periodic wave solutions, periodic cusp wave solutions, and oscillatory solitary wave sol utions, and show their convergence when either the parameter a or the wave speed c varies. We also illustrate how to identify the existence of peakons acid periodic cusp waves from the phase portraits. It seems that the GCH e quation is a good example to understand the relationships among peakons, pe riodic cusp waves, oscillatory solitary waves and smooth periodic wave solu tions. (C) 2001 Elsevier Science Ltd. All rights reserved.