The Method of Hirota Bilinearization

Bilinearization of a given nonlinear partial differential equation is very important not only to find soliton solutions but also to obtain other solutions such as the complexitons, positons, negatons, and lump solutions. In this work we study the bilinearization of nonlinear partial differential equations in $(2+1)$-dimensions. We write the most general sixth order Hirota bilinear form in $(2+1)$-dimensions and give the associated nonlinear partial differential equations for each monomial of the product of the Hirota operators $D_{x}$, $D_{y}$, and $D_{t}$. The nonlinear partial differential equations corresponding to the sixth order Hirota bilinear equations are in general nonlocal. Among all these we give the most general sixth order Hirota bilinear equation whose nonlinear partial differential equation is local which contains 12 arbitrary constants. Some special cases of this equation are the KdV, KP, KP-fifth order KdV, and Ma-Hua equations. We also obtain a nonlocal nonlinear partial differential equation whose Hirota form contains all possible triple products of $D_{x}$, $D_{y}$, and $D_{t}$. We give one- and two-soliton solutions, lump solutions with one, two, and three functions, and hybrid solutions of local and nonlocal $(2+1)$-dimensional equations. We proposed also solutions of these equations depending on dynamical variables.