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In this work, the (2+1)-dimensional Korteweg-de Vries equation is investigated, which can be used to represent the amplitude of the shallow–water waves in fluids or electrostatic wave potential in plasmas. By employing the properties of Bell's polynomial, we obtain bilinear representation of the equation with the aid of an appropriate transformation. Based on the obtained Hirota bilinear form, its lump solutions with localized characteristics are constructed in detail. We then derive the lumpoff solutions of the equation by studying a soliton solution generated by lump solutions. Furthermore, special rogue wave solutions with predictability are well presented, and the time and place of appearance are also derived. Finally, some graphic analysis is represented to better understand the propagation characteristics of the obtained solutions. It is hoped that our results provided in this work can be used to enrich the dynamic behaviors of the equation. [ABSTRACT FROM AUTHOR]

Abstract We consider the (3+1)-dimensional Boiti–Leon–Manna–Pempinelli equation, which can be used to describe the incompressible fluid. We derive the Hirota bilinear form of the equation via the Bell polynomial. By using Hirota's direct method, we obtain the N -soliton wave solutions of the equation. Based on the resulting Hirota's bilinear equation, we directly derive the breather wave solutions and rational solutions of the equation. Moreover, we employ the long wave limit method to derive the first-order and second-order rational solutions. In order to better understand the dynamical behaviors of the equation, we provide some graphical analyses to discuss the properties for the dynamic behaviors of these solutions. [ABSTRACT FROM AUTHOR]

In this paper, the ( 2 + 1 ) -dimensional B-type Kadomtsev–Petviashvili (BKP) equation is investigated, which can be used to describe the stability of soliton in a nonlinear media with weak dispersion. With the aid of the binary Bell polynomial, its bilinear formalism is succinctly constructed, based on which, the soliton wave solution is also obtained. Furthermore, by means of homoclinic breather limit method, its rogue waves and homoclinic breather waves are derived, respectively. Our results show that rogue wave can come from the extreme behavior of the breather solitary wave for ( 2 + 1 ) -dimensional nonlinear wave fields. [ABSTRACT FROM AUTHOR]

Under investigation in this paper is the integrability of a (3+1)-dimensional generalized KdV-like model equation, which can be reduced to several integrable equations. With help of Bell polynomials, an effective method is presented to succinctly derive the bilinear formalism of the equation, based on which, the soliton solutions and periodic wave solutions are also constructed by using Riemann theta function. Furthermore, the Bäcklund transformation, Lax pairs, and infinite conservation laws of the equation can easily be derived, respectively. Finally, the relationship between periodic wave solutions and soliton solutions are systematically established. It is straightforward to verify that these periodic waves tend to soliton solutions under a small amplitude limit. [ABSTRACT FROM AUTHOR]