Shou-Fu Tian

Rogue waves, homoclinic breather waves and soliton waves for the (2+1)-dimensional B-type Kadomtsev–Petviashvili equation

Abstract 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.

On integrability and quasi-periodic wave solutions to a (3+1)-dimensional generalized KdV-like model equation

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 Backlund 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.

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 the solitary waves, breather waves and rogue waves to a generalized (3+1)-dimensional Kadomtsev–Petviashvili equation☆

Abstract Under investigation in this work is a generalized ( 3 + 1 )-dimensional Kadomtsev–Petviashvili (GKP) equation, which can describe many nonlinear phenomena in fluid dynamics. By virtue of the Bell’s polynomials, an effective and straightforward way is presented to explicitly construct its bilinear form and soliton solution. Furthermore, based on the bilinear formalism and the extended homoclinic test method, the kinky breather wave solutions and rational breather wave solutions of the equation are well constructed. It is hoped that our results can be used to enrich the dynamical behavior of the ( 3 + 1 )-dimensional nonlinear wave fields.

Characteristics of the breathers, rogue waves and solitary waves in a generalized (2+1)-dimensional Boussinesq equation

Under investigation in this work is a generalized (2+1)-dimensional Boussinesq equation, which can be used to describe the propagation of small-amplitude, long wave in shallow water. By virtue of Bells polynomials, an effective way is presented to succinctly construct its bilinear form. Furthermore, based on the bilinear formalism and the extended homoclinic test method, the breather wave solution, rogue-wave solution and solitary-wave solution of the equation are well constructed. Our results can be used to enrich the dynamical behavior of the generalized (2+1)-dimensional nonlinear wave fields.

Lie symmetry analysis, conservation laws and exact solutions of the generalized time fractional Burgers equation

Under investigation in this work are the invariance properties of the generalized time fractional Burgers equation, which can be used to describe the physical processes of unidirectional propagation of weakly nonlinear acoustic waves through a gas-filled pipe. The Lie group analysis method is applied to consider its vector fields and symmetry reductions. Furthermore, based on the sub-equation method, a new type of explicit solutions for the equation is well constructed with a detailed analysis. By means of the power series theory, exact power series solutions of the equation are also constructed. Finally, by using the new conservation theorem, conservation laws of the equation are well constructed with a detailed derivation.

On quasi-periodic waves and rogue waves to the (4+1)-dimensional nonlinear Fokas equation

Under investigation in this paper is the (4+1)-dimensional nonlinear Fokas equation, which is an important physics model. With the aid of Bell’s polynomials, an effective and straightforward method is presented to succinctly construct the bilinear representation of the equation. By using the resulting bilinear formalism, the soliton solutions and Riemann theta function periodic wave solutions of the equation are well constructed. Furthermore, the extended homoclinic test method is employed to construct the breather wave solutions and rogue wave solutions of the equation. Finally, a connection between periodic wave solutions and soliton solutions is systematically established. The results show that the periodic waves tend to solitary waves under a limiting procedure.Under investigation in this paper is the (4+1)-dimensional nonlinear Fokas equation, which is an important physics model. With the aid of Bell’s polynomials, an effective and straightforward method is presented to succinctly construct the bilinear representation of the equation. By using the resulting bilinear formalism, the soliton solutions and Riemann theta function periodic wave solutions of the equation are well constructed. Furthermore, the extended homoclinic test method is employed to construct the breather wave solutions and rogue wave solutions of the equation. Finally, a connection between periodic wave solutions and soliton solutions is systematically established. The results show that the periodic waves tend to solitary waves under a limiting procedure.

Lie symmetry analysis, conservation laws and analytical solutions for a generalized time-fractional modified KdV equation

ABSTRACT In this paper, a generalized time fractional modified KdV equation is investigated, which is used for representing physical models in various physical phenomena. By Lie group analysis method, the invariance properties and the vector fields of the equation are presented. Then the symmetry reductions are provided. Moreover, we construct the explicit solutions of the equation by using sub-equation method. Based on the power series theory, the approximate analytical solution for the equation are also constructed. Finally, the new conservation theorem is applied to constructed conservation laws for the equation.