Special Bäcklund transformations and nonlinear superpositions for the non-integrable phi⁴field model

Abstract

Some special Bäcklund transformation (BT) theorems and a particular nonlinear superposition theorem are established to find exact solutions for a non-integrable model, the (N + 1)-dimensional φ 4 scalar field. Some new types of exact solutions such as the conoid periodic-periodic interaction waves and the periodic-solitary wave interaction solutions are explicitly given. The interaction solutions possess abundant structures, thanks to the intrusion of some arbitrary functions in the expressions of the special solutions.

Modulational Instability and Stationary Waves for the Coupled Generalized Schrödinger-Boussinesq System

“…Here, we look for solutions of the SB system through deforming them to the non-integrable φ 4 model, which possesses abundant known solutions. Moreover, some special Bäcklund transformations and nonlinear superpositions have been obtained for the N + 1-dimensional φ 4 model [14], and thus more new solutions might be produced accordingly.…”

Periodic travelling and non-travelling wave solutions of the nonlinear Klein–Gordon equation with imaginary mass

“…However, solutions of Eqs. (8) and (9) might be more abundant. First, we obtain some periodic travelling wave solutions by means of the Jacobi elliptic function expansion method.…”

“…First, we obtain some periodic travelling wave solutions by means of the Jacobi elliptic function expansion method. Second, following the idea of the deformation mapping method [8][9][10], we directly write down some special non-travelling wave solutions where arbitrary functions are introduced.…”

“…(3) in the real scale field is also known as the φ 4 field model. Recently, some special Bäcklund transformations and nonlinear superpositions for the φ 4 field model have been revealed [8].…”

“…Some periodic travelling and non-travelling wave solutions of Eqs. (8) and (9) are presented in Section 3. Last section contains a brief summary and discussion of our investigation.…”