Integrability and high-order localized waves of the (4 + 1)-dimensional nonlinear evolution equation

Constructing the exact solutions of high-dimensional nonlinear evolution equations and exploring their dynamics have always been important and open problems in real-world applications. The celebrated Korteweg-de Vries equation [KdV] and Kadomtsev-Petviashvili [KP] equation are typical examples of one-dimensional and two-dimensional integrable equations respectively. A natural idea is to investigate the integrable analogues of these equations in higher dimensional space. In this paper, an integrable extension of the Kadomtsev-Petviashvili equation, the (4 + 1)-dimensional nonlinear evolution equation (NLEE) [Physical Review Letters 96, (2006) 190201], is investigated. By dimensionality reduction we obtain three new nonlinear equations, namely (3 + 1)-dimensional NLEE, (2 + 1)-dimensional NLEE and (1 + 1)-dimensional NLEE. Bäcklund transformations and multi-soliton solutions are important characteristics of integrable equations. Based on the binary Bell polynomials and Hirota bilinear method we derive bilinear Bäcklund transformations and N-soliton solutions of these new equations, which show that these equations are integrable. We also give the multiple rational solutions of these new equations. These new integrable NLEEs enrich the models of integrable systems and help understand the new characteristics of nonlinear dynamics in real-world applications.