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

Abstract

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.

Rational Localized Waves and Their Absorb-Emit Interactions in the (2 + 1)-Dimensional Hirota–Satsuma–Ito Equation

“…Obviously, if = 0, the Hirota bilinear form (10) degenerates to (5). Hence, the bilinear form (10) is a generalized form of (5) and has not been discussed in the previous literature [7][8][9][10][11][12][13][14][15]. Now, we use the extended bilinear form (10) and the Hermitian quadratic form to construct the rational localized wave solution of the (2+1)-dimensional HSI equation.…”

“…Periodic wave solutions and their asymptotic analysis were presented in References [5,6]. The Hirota-Satsuma shallow water wave model has a higher dimensional generalized form [7][8][9][10][11][12]:…”

“…u is a function of the variables x, y, t and represents the amplitude of the shallow water wave, and the subscript variables represent the partial derivatives with respect to the spatial coordinate variables x, y and the time variable t, respectively. Usually, the (2+1)-dimensional HSI equation is also written as the following two forms [7][8][9][10][11][12]:…”

“…where u, v, and w are the functions of x, y, and t. Generally, the function u is the physical field, while the functions v and w are the potentials of physical field derivatives. As a (2 + 1)-dimensional extension of the Hirota-Satsuma shallow water wave model, many researchers [7][8][9][10][11][12] investigated this model. By using the logarithm transformation u = 2(ln f ) xx , the (2+1)-dimensional HSI equation can be converted into the following bilinear form [7][8][9][10][11][12]:…”

“…Yaqing Liu et al [10] investigated the N-soliton solution and localized wave interaction solutions by using Hirota's N-soliton method. By employing the perturbation expansion method, Wei Liu et al [11] investigated the high-order breathers, lumps, and semi-rational solutions. Jian-Guo Liu et al [12] studied multi-wave, breather wave, and interaction solutions by using the three wave method.…”