On the traveling waves governed by the Camassa-Holm equation

The Camassa-Holm equation admits undistorted traveling waves that are either smooth or exhibit peaks or cusps. All three wave types can be periodic or solitary. Also waves of different types may be combined. In the present paper it is shown that, apart from peaks and cusps, the traveling waves governed by the Camassa-Holm equation can be found from some simpler equation. In the case of peaked solutions, this reduced equation is even linear. The governing equation of traveling waves in its original form can be interpreted as a nonlinear combination of the reduced equation and its first integral. For a small range of the integration constant, the reduced equation admits bounded solutions, which then are directly inherited by the Camassa-Holm equation. In general, the solutions of the reduced equation are unbounded and cannot be considered to represent traveling waves. The full equation, however, has a nonlinearity in the highest derivative, which is characteristic for the Camassa-Holm and some other equations. This nonlinear term offers the possibility of constructing bounded traveling waves from the unbounded solutions of the reduced equation. These waves necessarily have discontinuities in the slope and are, therefore, solutions only in a generalized sense.