Finding the one-loop soliton solution of the short-pulse equation by means of the homotopy analysis method.

*(English)*Zbl 1159.65348Summary: The homotopy analysis method is applied to the short-pulse equation in order to find an analytic approximation to the known exact solitary upright-loop solution. It is demonstrated that the approximate solution agrees well with the exact solution. This provides further evidence that the homotopy analysis method is a powerful tool for finding excellent approximations to nonlinear solitary waves.

##### MSC:

65M70 | Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs |

35L70 | Second-order nonlinear hyperbolic equations |

##### Keywords:

short-pulse equation; homotopy analysis method; solitary-wave solution; series solution; numerical examples
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\textit{E. J. Parkes} and \textit{S. Abbasbandy}, Numer. Methods Partial Differ. Equations 25, No. 2, 401--408 (2009; Zbl 1159.65348)

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