Europhys.Lett., 73 (5), pp. 657–663 (2006) DOI: 10.1209/epl/i2005-10453-y EUROPHYSICS LETTERS 1 March 2006 On peaked and smooth solitons for the Camassa-Holm equation Z. J. Qiao 1 ( ∗ )and G. P. Zhang 2 1 DepartmentofMathematics,TheUniversityofTexas-PanAmerican 1201WUniversityDrive,Edinburg,TX78541,USA 2 AppliedMathematicsResearchCenter,DelawareStateUniversity 1200NorthDupontHighway,Dover,DE19901,USA received 23 October 2005; accepted in final form 6 January 2006 published online 25 January 2006 PACS. 02.30.Ik – Integrable systems. PACS. 05.45.Yv – Solitons. PACS. 03.75.Lm – Tunneling, Josephson effect, Bose-Einstein condensates in periodic poten- tials, solitons, vortices, and topological excitations. Abstract. – This letter presents all possible explicit single soliton solutions for the Camassa- Holm (CH) equation mt + mxu +2mux =0,m = u − uxx. This equation is studied under the boundary condition u → A (A isaconstant)as x → ±∞. Regular peakon solutions correspond to the case of A = 0. For the case of A = 0, both new peaked solitons and new type of smooth solitons, which are expressed in terms of trigonometric and hyperbolic functions, are tremen- douslygiventhroughinvestigatingaNewtonequationwithanewpotential. Mathematicalanal- ysisandnumericgraphsareprovidedforthosesmoothsolitonandnewpeakedsolitonsolutions. Both the Camassa-Holm (CH) equation [1] m t + m x u +2mu x =0, m = u − α 2 u xx , x ∈ R (1) and the unidirectional shallow-water wave equation [2] ˜ m t +˜ m x ˜ u +2˜ m˜ u x = −c 0 ˜ u x + γ ˜ u xxx , ˜ m =˜ u − α 2 ˜ u xx , x ∈ R (2) have excited much attraction in recent years. The CH equation was implied by the work of Fokas and Fuchssteiner (1981) on hereditary symmetries [3]. It came to be remarkable in the work of Camassa and Holm (1993) where the peakon was described [1]. A peakon is a weak solution with non-smooth property at some points. A discussion of mathematical details is given in several literatures: by Beals etal. (1998) [4]; Contantin, Escher, and McKean (1998, 1999) [5–7], Alber et al. (2001) [8], Johnson (2002) [9], Qiao (2003) [10], and Gesztesy and Holden (2003) [11]. Bothequationsareintegrableandhavepeakedsolitonsandinfinitenumberofconservation laws[1,2]. Moreover,thetwowaveequationsaretransformableoneintotheotherbyasimple transformation, namely ˜ m(x, t)= m  x − t 2 (3γ − c 0 ),t  + 1 2 (γ − c 0 ), ˜ u(x, t)= u  x − t 2 (3γ − c 0 ),t  + 1 2 (γ − c 0 ). ( * ) E-mail: qiao@utpa.edu c  EDP Sciences Article published by EDP Sciences and available at http://www.edpsciences.org/epl or http://dx.doi.org/10.1209/epl/i2005-10453-y