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

Integrability characteristics of a novel (2+1)-dimensional nonlinear model: Painlevé analysis, soliton solutions, Bäcklund transformation, Lax pair and infinitely many conservation laws

New general interaction solutions to the KPI equation via an optional decoupling condition approach

The N-rational solutions to two (2+1)-dimensional nonlinear evolution equations are constructed by utilizing the long wave limit method. M-lump solutions to the two equations are derived by making some parameters conjugate to each other. We present and discuss the 1-, 2- and 3-lump solutions to the two equations. The amplitude and shape of the one lump wave remain unchanged during the propagation. The dynamic properties of the collisions among multiple lump waves are analyzed, which indicate that the fusion and fission of multiple lump waves might occur. The multiple lump waves might merge into one lump wave, then split into multiple lump waves. The lines which multiple lump waves follow are various if we choose different parameters. These results are helpful to describe some nonlinear phenomena in the areas of optics, fluid dynamics and plasma.

https://iopscience.iop.org/article/10.1088/1402-4896/abf307/pdf

Derivation and simulation of the M-lump solutions to two (2+1)-dimensional nonlinear equations

Under investigation in this paper is a (3+1)-dimensional nonlinear evolution equation, which was proposed and analyzed to study features and properties of nonlinear dynamics in higher dimensions. Using the Hirota bilinear method, we construct a bilinear Backlund transformation, which consists of four equations and involves six free parameters. With test function method and symbolic computation, three sets of lump–kink solutions and new types of interaction solutions are derived, and figures are presented to reveal the interaction behaviors. Setting constraints to the new interaction solution via the test function expressed by “polynomial-cos-cosh,” we simulate the periodic interaction phenomenon. Pfaffian solutions to the (3+1)-dimensional nonlinear evolution equation are obtained based on a set of linear partial differential conditions. According to our results, the diversity of solutions to the (3+1)-dimensional nonlinear evolution equation is revealed.

Bäcklund transformation, exact solutions and diverse interaction phenomena to a (3+1)-dimensional nonlinear evolution equation

With the Hirota bilinear method and symbolic computation, we investigate the $$(3+1)$$
 -dimensional generalized Kadomtsev–Petviashvili equation. Based on its bilinear form, the bilinear Backlund transformation is constructed, which consists of four equations and five free parameters. The Pfaffian, Wronskian and Grammian form solutions are derived by using the properties of determinant. As an example, the one-, two- and three-soliton solutions are constructed in the context of the Pfaffian, Wronskian and Grammian forms. Moreover, the triangle function solutions are given based on the Pfaffian form solution. A few particular solutions are plotted by choosing the appropriate parameters.

Meng-Gang Li

Papers

Bäcklund transformation, Pfaffian, Wronskian and Grammian solutions to the (3 +1 ) -dimensional generalized Kadomtsev-Petviashvili equation