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

Abstract

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A self-adaptive moving mesh method for the short pulse equation via its hodograph link to the sine-Gordon equation

“…It was then shown that breather solition solutions of the SP equation can be generated from those of the SG equation [48]. Another interesting outcome of this SG formulation is that wide range of multi-valued, exotic solutions such as loop solutions can be regarded as the solutions of the SP equation; see, for example, [34,42,32,18,22]. From the theoretical aspects as the partial differential equation (PDE), the global well-posedness of the SP equation (and the SG equation) was discussed by Pelinovsky-Sakovich [43], whereas the local well-posedness was already proved by Schäfer-Wayne [50].…”

Solitary-wave solutions of the Degasperis–Procesi equation by means of the homotopy analysis method

“…In [26] we attempted to formulate the HAM in order to find an analytic approximation to this exact solution. The formulation involved the introduction of a new independent variable as was done in [27] for the short-pulse equation; the resulting equation, corresponding to (4.2), involved higher-order nonlinearities. We found that the approximate solution did not agree well with the exact solution.…”