Integrability of One Bilinear Equation: Singularity Analysis and Dimension

The integrability of a four-dimensional sixth-order bilinear equation associated with the exceptional affine Lie algebra D(1)4 is studied by means of the singularity analysis. This equation is shown to pass the Painlevé test in three distinct cases of its coefficients, exactly when the equation is effectively a three-dimensional one, equivalent to the BKP equation.

Lump-type solutions, interaction solutions, and periodic lump solutions of the generalized (3+1)-dimensional Burgers equation

In this paper, the lump-type solutions, interaction solutions, and periodic lump solutions of the generalized ([Formula: see text])-dimensional Burgers equation were obtained by using the ansatz method. Based on a variable transformation, the generalized ([Formula: see text])-dimensional Burgers equation was transformed into a bilinear equation. And then, lump-type solutions, two kinds of interaction solutions, and periodic lump solutions were obtained via solutions of the bilinear equation. Fission and fusion phenomena are found in the process of interaction between lump-type soliton and one stripe soliton, which can derive the lumpoff wave solution. The dynamic characteristics of these solutions were vividly displayed by graphics.

General description on extended homoclinic orbit solutions of the KdV-type bilinear equations

The Korteweg–de Vries (KdV)-type bilinear equations always allow 2-soliton solutions. In this paper, for a general KdV-type bilinear equation, we interpret how the so-called extended homoclinic orbit solutions arise from a special case of its 2-soliton solution. Two properties of bilinear derivatives are developed to deal with bilinear equation deformations. A non-integrable (3+1)-dimensional bilinear equation is employed as an example.

Bilinear Equation of the Nonlinear Partial Differential Equation and Its Application

The homogeneous balance of undetermined coefficient method is firstly proposed to derive a more general bilinear equation of the nonlinear partial differential equation (NLPDE). By applying perturbation method, subsidiary ordinary differential equation (sub-ODE) method, and compatible condition to bilinear equation, more exact solutions of NLPDE are obtained. The KdV equation, Burgers equation, Boussinesq equation, and Sawada-Kotera equation are chosen to illustrate the validity of our method. We find that the underlying relation among the G′/G-expansion method, Hirota’s method, and HB method is a bilinear equation. The proposed method is also a standard and computable method, which can be generalized to deal with other types of NLPDE.

Study of lump and lump-kink solitons of a coupled reduced Hirota bilinear equation

By symbolic computation and searching for the solutions of the positive quadratic functions of the related bilinear equations, two kinds of lump solutions of the (3[Formula: see text]+[Formula: see text]1)-dimensional weakly coupled Hirota bilinear equation are derived, and the practicability of this method is verified. Then we add an exponential function to the original positive quadratic function, and obtain a new solution of the Hirota bilinear equation. The interaction between the lump solutions and lump-kink solutions is included in the new solution. On this basis, we give the possibility of adding multiple exponential functions. Finally, we give the coupled reduced Hirota bilinear equation lump-kink solitons by combining the above two methods. In order to ensure the analyticity and reasonable localization of the block, two sets of necessary and sufficient conditions are given for the parameters involved in the solution. The local characteristics and energy distribution of bulk solution are analyzed and explained.