Europhys. Lett.
Volume 73, Number 5, March 2006
Page(s) 657 - 663
Section General
Published online 25 January 2006
Europhys. Lett., 73 (5), pp. 657-663 (2006)
DOI: 10.1209/epl/i2005-10453-y

On peaked and smooth solitons for the Camassa-Holm equation

Z. J. Qiao1 and G. P. Zhang2

1  Department of Mathematics, The University of Texas-Pan American 1201 W University Drive, Edinburg, TX 78541, USA
2  Applied Mathematics Research Center, Delaware State University 1200 North Dupont Highway, Dover, DE 19901, USA

received 23 October 2005; accepted in final form 6 January 2006
published online 25 January 2006

This letter presents all possible explicit single soliton solutions for the Camassa-Holm (CH) equation $m_t+m_xu+2mu_x=0, \ m=u-u_{xx}$. This equation is studied under the boundary condition $u\rightarrow A$ (A is a constant) as $x\rightarrow\pm\infty $. Regular peakon solutions correspond to the case of A=0. For the case of $A\not=0$, both new peaked solitons and new type of smooth solitons, which are expressed in terms of trigonometric and hyperbolic functions, are tremendously given through investigating a Newton equation with a new potential. Mathematical analysis and numeric graphs are provided for those smooth soliton and new peaked soliton solutions.

02.30.Ik - Integrable systems.
05.45.Yv - Solitons.
03.75.Lm - Tunneling, Josephson effect, Bose-Einstein condensates in periodic potentials, solitons, vortices, and topological excitations.

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