Multi-wave, breather wave, and interaction solutions of the Hirota–Satsuma–Ito equation

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]:…”

“…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. Xing Lü et al [13] extended the (2 + 1)-dimensional HSI equation to the (3 + 1)-dimensional form and deduced the Bäcklund transformation, exact solutions, and interaction behavior of the (3 + 1)-dimensional Hirota-Satsuma-Ito-like equation.…”