On Two-Loop Soliton Solution of the Schäfer–Wayne Short-Pulse Equation Using Hirota's Method and Hodnett–Moloney Approach

A two-loop soliton solution to the Schäfer–Wayne short-pulse equation (SWSPE) is shown. The key step in finding this solution is to transform the independent variables in the equation. This leads to a transformed equation for which it is straightforward to find an explicit two-soliton solution using Hirota's method. The two-loop soliton solution to the SWSPE is then found in implicit form by means of a transformation back to the original independent variables. Following Hodnett and Moloney's approach, some computations of the energy of the one- and two-soliton solutions are made.