Mei-Juan Xu

On periodic wave solutions with asymptotic behaviors to a (3+1)-dimensional generalized B-type Kadomtsev–Petviashvili equation in fluid dynamics

Abstract In this paper, a ( 3 + 1 ) -dimensional generalized B-type Kadomtsev–Petviashvili equation is investigated, which can be used to describe weakly dispersive waves propagating in a quasi media and fluid mechanics. Based on the Bell polynomials, its multiple-soliton solutions and the bilinear form with some reductions are derived, respectively. Furthermore, by using Riemann theta function, we construct one- and two-periodic wave solutions for the equation. Finally, we study the asymptotic behavior of the periodic wave solutions, which implies that the periodic wave solutions can be degenerated to the soliton solutions under a small amplitude limit.

On Lie symmetries, optimal systems and explicit solutions to the Kudryashov-Sinelshchikov equation

Under investigation in this paper is the Kudryashov-Sinelshchikov equation, which describes influence of viscosity and heat transfer on propagation of the pressure waves. The Lie symmetry method is used to study its vector fields and optimal systems, respectively. Furthermore, the symmetry reductions and exact solutions of the equation are obtained on the basic of the optimal systems. Finally, based on the power series theory, a kind of explicit power series solutions for the equation is well constructed with a detailed derivation.

On Bell polynomials approach to the integrability of a (3+1)-dimensional generalized Kadomtsev–Petviashvili equation

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 Bells polynomials, we succinctly obtain its bilinear representations, bilinear Backlund 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.

Quasi-periodic wave solutions and asymptotic properties to an extended Korteweg–de Vries equation from fluid dynamics

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.

On quasi-periodic wave solutions and asymptotic behaviors to a (2 + 1)-dimensional generalized variable-coefficient Sawada–Kotera equation

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.