High-order breathers, lumps, and semi-rational solutions to the (2 + 1)-dimensional Hirota–Satsuma–Ito equation

Under investigation in this work is the (2 + 1)-dimensional Hirota–Satsuma–Ito (HSI) equation. By employing Bell’s polynomials, bilinear formalism of the HSI equation is succinctly obtained. With the aid of the obtained bilinear formalism, we first construct general high-order soliton solutions by using Hirota’s bilinear method combined with the perturbation expansion. By taking particular complex conjugate conditions of the high-order soliton solutions, high-order breather solutions and the mixed solutions consisting of breathers and line solitons are succinctly derived. We further generate rational solutions termed high-order lumps by taking a long wave limit. Finally, we investigate two types of semi-rational solutions, which describe interaction between lumps and line solitons, or between lumps and breathers. These collisions are elastic, which do not lead to any changes of amplitudes and velocities, and shapes of the line solitons, breathers and lumps after interaction.

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