Localized wave solutions of a higher-order short pulse equation

Constructing suitable Lax pair and making use of zero-curvature equation, a higher-order short pulse equation is researched and proved to be Lax integrable. Modulational instability of the above equation is investigated, which is an important possible generation mechanism of localized wave solutions. Then higher-order semi-rational soliton, multi-soliton and breather solutions of the higher-order short pulse equation are derived by generalized Darboux transformation and classical Darboux transformation of a new Lax pair obtained by hodograph transformation, which are represented by the ratio of the corresponding two determinants. Dynamics of these localized wave solutions are represented in several figures and certain important physical quantities for one-loop soliton solutions are analyzed, such as amplitude, wave number, wave width, velocity and initial phase. In particular, Darboux matrices under classical Darboux transformation of negative expansion, positive expansion and positive and negative expansion are given.