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Under investigation in this paper is a (3+1)-dimensional generalized Jimbo–Miwa equation(gJM), which describes many interesting three dimensional nonlinear waves in physics. Based on the properties of the Bell's polynomial, a very classic method is presented to find the Hirota bilinear form of the equation. By using the resulting bilinear form, we systematically construct its lump solutions and lump-like solutions including lump-kink solutions and periodic lump solutions with their corresponding limitations. Moreover, we show that the conditions satisfying by lump solutions are considered to satisfy several important properties, including localization, positive and analyticity. The interaction solutions among a lump and a kink wave are also provided by considering a special function. The corresponding interaction phenomenon does not possess elasticity, but has a constantly removing process between a lump and a kink wave. Finally, periodic lump solutions are also derived by employing the three-wave method. It is hoped that our results can help rich the dynamic behaviors of the (3+1)-dimensional Jimbo–Miwa-like equations. [ABSTRACT FROM AUTHOR]

In this work, we investigate the (2+1)-dimensional generalized Bogoyavlensky–Konopelchenko (gBK) equation. Based on its bilinear form, the N th-order breather solutions of the gBK equation are successful given by taking appropriate parameters. Furthermore, the N th-order lump solutions of the gBK equation are obtained via the long-wave limit method. In addition, the semi-rational solutions are generated to reveal the interaction between lump solutions, soliton solutions, and breather solutions. [ABSTRACT FROM AUTHOR]

By using symbolic computation methods, we systematically study a generalized (3+1)-dimensional Kadomtsev–Petviashvili–Boussinesq (KP-Boussinesq) equation, which can be used to explain some interesting physical phenomena in the fields of fluids. We first obtain bilinear representation of the equation by employing the properties of Bell's polynomials. Based on that, its a series of exact solutions, such as one kink soliton solutions, two kink soliton solutions and lump wave solutions, are constructed in detail. It is worth mentioning that we also study the interaction solutions between lump waves and kink waves. Moreover, in order to describe the dynamic behaviours of those solutions, some graphical analyses are given by selecting different parameter values in such solutions. Those analyses are helpful for enriching the physical phenomena in KP-type equations. [ABSTRACT FROM AUTHOR]

Under investigation in this work is a generalized nonlinear equation, which can be widely applied to describe various phenomena in nonlinear physical science field. The equation can be reduced to Kadomtsev–Petviashvili-type equations and Jimbo–Miwa-type equations as its special cases. By using Bell's polynomial, its bilinear representation is well constructed. Based on the obtained bilinear formalism, we derive the solitary wave solutions of the equation. We also consider its kinky breather wave, rational breather wave and rogue wave solutions by employing the extended homoclinic test method, respectively. Moreover, the dynamic behaviors of these solutions are given by graphical analysis. It is hoped that our works can be helpful to understand dynamical behavior of the nonlinear equation. [ABSTRACT FROM AUTHOR]

In this work, we study a generalized (2 + 1)-dimensional asymmetrical Nizhnik–Novikov–Veselov (NNV) equation. Its Hirota bilinear form is constructed via the Bell polynomial. Based on the obtained bilinear form, the Nth-order breather waves are derived explicitly under certain parameter constraints. Moreover, we generate the nonsingular Nth-order lump waves through applying the long wave limit method. Additionally, we successfully present the semi-rational waves containing the combination of lump waves and single-soliton waves, the combination of lump waves and breather waves. [ABSTRACT FROM AUTHOR]

We consider the (3 + 1)-dimensional generalized shallow water wave (GSWW) equation. By virtue of the binary Bell polynomials theory, we obtain the bilinear form of the equation. Then its lump wave solutions, a kind of rational solution localized in all directions of the space, are derived by employing its bilinear form at the special situation for z = x. Furthermore, it is worth noting that the lump wave solutions can interact with single-stripe soliton waves and double-stripe solution waves to generate lumpoff waves and a kind of predictable rouge waves, respectively. Especially, it is interesting that we can predict when and where the peculiar rouge waves will occur. Moreover, in order to understand the dynamics and propagation of the lump waves and the interaction solution, some graphic analyses are exhibited by selecting special parameters. The results of this work can be used to understand the propagation behavior of these solutions of the GSWW equation, which is of great significance for ocean engineering. [ABSTRACT FROM AUTHOR]

In this paper, an extended Korteweg-de Vries (eKdV) equation is investigated, which can be used to describe many nonlinear phenomena in fluid dynamics and plasma physics. With the aid of the generalized Bell's polynomials, the Hirota's bilinear equation to the eKdV equation is succinctly constructed. Based on that, its solition solutions are directly obtained. By virtue of the Riemann theta function, a straightforward way is presented to explicitly construct Riemann theta function periodic wave solutions of the eKdV equation. Finally, the asymptotic behaviors of the Riemann theta function periodic waves are presented, which yields a relationship between the periodic waves and solition solutions by considering a limiting procedure. [ABSTRACT FROM AUTHOR]

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]