Narrow Results

In this paper, a (3+1)-dimensional generalized variable-coefficients Kadomtsev–Petviashvili (gvcKP) equation is proposed, which describes many nonlinear phenomena in fluid dynamics and plasma physics. By a very natural way, the integrable constraint conditions on the variable coefficients are presented to investigate the integrabilities of the gvcKP equation. Based on the generalized Bell's polynomials, we succinctly obtain its bilinear representations, bilinear Bäcklund transformation and Lax pair, respectively. Furthermore, by virtue of the binary Bell polynomial form, the infinite conservation laws of the equation are found with explicit recursion formulas as well by using its Lax equations via algebraic and differential manipulation. In addition, by using the Hirota bilinear method, its N-soliton solutions are also obtained.

In this paper, a $(2+1)$-dimensional generalized variable-coefficient Sawada–Kotera (gvcSK) equation is investigated, which describes many nonlinear phenomena in fluid dynamics and plasma physics. Based on the properties of binary Bell polynomials, we present a Hirota’s bilinear equation to the gvcSK equation. By virtue of the Hirota’s bilinear equation, we obtain the N-soliton solutions and the quasi-periodic wave solutions of the gvcSK equation, which can be reduced to the ones of several integrable equations such as Sawada–Kotera, modified Caudrey–Dodd–Gibbon–Sawada–Kotera, isospectral BKP equations and etc. Furthermore, we obtain the relationship between the soliton solutions and periodic solutions by considering the asymptotic properties of the periodic solutions.

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.

Under investigation in this paper is a fifth-order Korteweg–de Vries type (fKdV-type) equation with time-dependent coefficients, which can be used to describe many nonlinear phenomena in fluid mechanics, ocean dynamics and plasma physics. The binary Bell polynomials are employed to find its Hirota’s bilinear formalism with an extra auxiliary variable, based on which its $N$-soliton solutions can be also directly derived. Furthermore, by considering multi-dimensional Riemann theta function, a lucid and straightforward generalization of the Hirota–Riemann method is presented to explicitly construct the multiperiodic wave solutions of the equation. Finally, the asymptotic properties of these periodic wave solutions are strictly analyzed to reveal the relationships between periodic wave solutions and soliton solutions.

In this paper, we consider the (2+1)-dimensional Ito equation, which was introduced by Ito. By considering the Hirota’s bilinear method, and using the positive quadratic function, we obtain some lump solutions of the Ito equation. In order to ensure rational localization and analyticity of these lump solutions, some sufficient and necessary conditions are provided on the parameters that appeared in the solutions. Furthermore, the interaction solutions between lump solutions and the stripe solitons are discussed by combining positive quadratic function with exponential function. Finally, the dynamic properties of these solutions are shown via the way of graphical analysis by selecting appropriate values of the parameters.

In this work, we consider the generalized (3+1)-dimensional Boussinesq equation, which can describe the propagation of gravity wave on the surface of water. Based on the Bell polynomial theory, a powerful technique is employed to explicitly construct its bilinear formalism and two-soliton solutions, based on which the new rational solution is well-constructed. Moreover, the extended homoclinic test approach is presented to succinctly construct the breather wave and rogue wave solutions of the Boussinesq equation. Then the main characteristics of these solutions are graphically discussed. More importantly, they reveal that the extreme behavior of the breather wave can give rise to the rogue wave.

In this paper, with the aid of symbolic computation system Maple, and based on the simplified Hirota method and ansatz technique, we discussed the (3+1)-dimensional generalized Kadomtsev–Petviashvili equation with $p=3$ to obtain lump solutions, lump–kink solutions and three classes of interaction solutions. Comparing our new results with other researchers’ results shows that using this method gives the more opportunity to solve the nonlinear partial differential equations that appear in mathematics, physics, biological engineering and other fields. We also presented profiles of new lump solution, lump–kink solutions and interaction solutions as illustrative examples.