Abstract
Binary Bell polynomials are applied to construct two kinds of bilinear derivative equations of a $(3+1)$ -dimensional nonlinear equation. Based on one of the bilinear forms, we derive a Backlund transformation, the corresponding Lax pair, infinite conservation laws, and explicit solutions with an arbitrary function in y. In the meantime, from the other bilinear form, we get another bilinear Backlund transformation and exact solutions by utilizing the exchange formulas for Hirota’s bilinear operators.
Highlights
Nonlinear evolution equations (NLEEs) have attracted intensive attention in the past few decades, since they can model many important phenomena and dynamic processes in physics, mechanics, chemistry, and biology [, ]
The study of solutions for NLEEs is very important in nonlinear physical phenomena and many effective methods have been used
Once a nonlinear equation is written in bilinear form, its multi-soliton solutions and rational solutions are usually obtained in a systematic way
Summary
Nonlinear evolution equations (NLEEs) have attracted intensive attention in the past few decades, since they can model many important phenomena and dynamic processes in physics, mechanics, chemistry, and biology [ , ]. The Hirota method is a powerful and direct approach to construct exact solutions of NLEEs. Once a nonlinear equation is written in bilinear form, its multi-soliton solutions and rational solutions are usually obtained in a systematic way. Lambert, Gilson et al proposed an effective method based on the use of the Bell polynomials to obtain bilinear Bäcklund transformation and Lax pair for soliton equations in a direct way [ – ]. Fan developed this method to find bilinear Bäcklund transformations, Lax pairs, infinite conservation laws of nonisospectral and variable-coefficient KdV, KP equations [ , ]. Wang applied the binary Bell polynomials to construct bilinear forms, bilinear
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