High-order soliton solutions and their dynamics in the inhomogeneous variable coefficients Hirota equation

A series of new soliton solutions is presented for the inhomogeneous variable coefficient Hirota equation by using the Riemann–Hilbert method and transformation relationship. Firstly, through a standard dressing procedure, the N-soliton matrix associated with the simple zeros in the Riemann–Hilbert problem for the Hirota equation is constructed. Then the N-soliton matrix of the inhomogeneous variable coefficient Hirota equation can be obtained by a special relationship transformation from the N-soliton matrix of the Hirota equation. Next, using the generalized Darboux transformation, the high-order soliton solutions corresponding to the elementary high-order zeros in the Riemann–Hilbert problem for the Hirota equation can be derived. Similarly, employing the relationship transformation mentioned above can lead to the high-order soliton solutions of the inhomogeneous variable coefficient Hirota equation. In addition, the collision dynamics of Hirota and inhomogeneous variable coefficient Hirota equations are analyzed; the asymptotic behaviors for multi-solitons and long-term asymptotic estimates for the high-order one-soliton of the Hirota equation are concretely calculated. Most notably, by analyzing the dynamics of the multi-solitons and high-order solitons of the inhomogeneous variable coefficient Hirota equation, we discover numerous new waveforms such as heart-shaped periodic wave solutions, O-shaped periodic wave solutions etc. that have never been reported before, which are crucial in theory and practice.