An Approach for Solving Short-Wave Models for Camassa-Holm Equation
and Degasperis-Procesi Equation |
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Authors: | YANG Pei CHEN Yong LI Zhi-Bin |
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Institution: | 1. Department of Computer Science, East China Normal University,
Shanghai 200062, China
;2. Institute of Theoretical Computing, East China Normal University, Shanghai 200062, China |
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Abstract: | In this paper, to construct exact solution of nonlinear partial
differential equation, an easy-to-use approach is proposed. By means of the transformation of the independent variables and the travelling wave transformation, the partial differential equation is reduced to an ordinary differential equation. To solve the ordinary differential equation, we assume the soliton solution in the explicit expression and obtain the travelling wave solution. By
the transformation back to the original independent variables, the soliton solution of the original partial differential equation is derived.
We investigate the short wave model for the Camassa-Holm equation
and the Degasperis-Procesi equation respectively. One-cusp soliton
solution of the Camassa-Holm equation is obtained. One-loop soliton solution of the Degasperis-Procesi equation is also obtained, the approximation of which in a closed form can be
obtained firstly by the Adomian decomposition method. The obtained
results in a parametric form coincide perfectly with those given
in the present reference. This illustrates the efficiency and
reliability of our approach. |
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Keywords: | Camassa-Holm equation Degasperis-Procesi equation one-cusp soliton one-loop soliton Adomian decomposition method |
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