With symbolic computation, two classes of lump solutions to the dimensionally reduced equations in (2+1)-dimensions are derived, respectively, by searching for positive quadratic function solutions to the associated bilinear equations. To guarantee analyticity and rational localization of the lumps, two sets of sufficient and necessary conditions are presented on the parameters involved in the solutions. Localized characteristics and energy distribution of the lump solutions are also analyzed and illustrated.

Study of lump dynamics based on a dimensionally reduced Hirota bilinear equation

Bäcklund transformation, exact solutions and interaction behaviour of the (3+1)-dimensional Hirota-Satsuma-Ito-like equation

In this paper, a (3+1)-dimensional nonlinear evolution equation and its reduction is studied by use of the Hirota bilinear method and the test function method. With symbolic computation, diversity of exact solutions is obtained by solving the under-determined nonlinear system of algebraic equations for the associated parameters. Finally, analysis and graphical simulation are given to reveal the propagation and dynamical behavior of the solutions.

Diversity of exact solutions to a (3+1)-dimensional nonlinear evolution equation and its reduction

In this paper, a $$(3+1)$$
 -dimensional nonlinear evolution equation is cast into Hirota bilinear form with a dependent variable transformation. A bilinear Backlund transformation is then presented, which consists of six bilinear equations and involves nine arbitrary parameters. With multiple exponential function method and symbolic computation, nonresonant-typed one-, two-, and three-wave solutions are obtained. Furthermore, two classes of lump solutions to the dimensionally reduced cases with $$y=x$$
 and $$y=z$$
 are both derived. Finally, some figures are given to reveal the propagation of multiple wave solutions and lump solutions.

Bäcklund transformation, multiple wave solutions and lump solutions to a (3 + 1)-dimensional nonlinear evolution equation

Interaction solutions between lump and soliton are analytical exact solutions to nonlinear partial differential equations. The explicit expressions of the interaction solutions are of value for analysis of the interacting mechanism. We analyze the one-lump-multi-stripe and one-lump-multi-soliton solutions to nonlinear partial differential equations via Hirota bilinear forms. The one-lump-multi-stripe solutions are generated from the combined solution of quadratic functions and N exponential functions, while the one-lump-multi-soliton solutions from the combined solution of quadratic functions and N hyperbolic cosine functions. Within the context of the derivation of the lump solution and soliton solution, necessary and sufficient conditions are presented for the two types of interaction solutions, respectively, based on the combined solutions to the associated bilinear equations. Applications are made for a (2+1)-dimensional generalized KdV equation, the (2+1)-dimensional NNV system and the (2+1)-dimensional Ito equation.

Interaction solutions to nonlinear partial differential equations via Hirota bilinear forms: one-lump-multi-stripe and one-lump-multi-soliton types

For the exponential traveling wave solutions to the Hirota bilinear equations, a sufficient and necessary criterion for the existence of linear superposition principle has been given. Motivated by this criterion, we propose a new Hirota bilinear equation via using a multivariate polynomial. Applying the linear superposition principle to this new Hirota bilinear equation, we finally find two types of resonant multiple wave solutions, among which, the resonant three-wave solutions are illustrated with three-dimensional plots.

Yao-Yao Zi

Papers

Bäcklund transformation, multiple wave solutions and lump solutions to a (3 + 1)-dimensional nonlinear evolution equation

Resonant behavior of multiple wave solutions to a Hirota bilinear equation