Lian-Li Feng

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.

Nonlocal Symmetries and Consistent Riccati Expansions of the (2+1)-Dimensional Dispersive Long Wave Equation

Abstract In this article, the (2+1)-dimensional dispersive long wave equation (DLWE) is investigated, which is derived in the context of a water wave propagating in narrow infinitely long channels of finite constant depth. By using of the truncated Painlevé expansion, we construct its nonlocal symmetry and Bäcklund transformation. After implanting the equation into an enlarged one, then the residual symmetry is localised. Meanwhile, the symmetry group transformation can be computed from the prolonged system. Furthermore, the equation is verified to be consistent Riccati expansion (CRE) solvable. Outing from the CRE, the soliton-cnoidal wave interaction solution in terms of Jacobi elliptic functions and the third type of incomplete elliptic integral are studied, respectively.

Nonlocal Symmetries, Consistent Riccati Expansion, and Analytical Solutions of the Variant Boussinesq System

Abstract Under investigation in this paper is the variant Boussinesq system, which describes the propagation of surface long wave towards two directions in a certain deep trough. With the help of the truncated Painlevé expansion, we construct its nonlocal symmetry, Bäcklund transformation, and Schwarzian form, respectively. The nonlocal symmetries can be localised to provide the corresponding nonlocal group, and finite symmetry transformations and similarity reductions are computed. Furthermore, we verify that the variant Boussinesq system is solvable via the consistent Riccati expansion (CRE). By considering the consistent tan-function expansion (CTE), which is a special form of CRE, the interaction solutions between soliton and cnoidal periodic wave are explicitly studied.

Quasi-periodic wave solutions, soliton solutions, and integrability to a (2+1)-dimensional generalized Bogoyavlensky-Konopelchenko equation

In this paper, a (2+1)-dimensional generalized Bogoyavlensky–Konopelchenko (gBK) equation is investigated, which can be used to describe the interaction of a Riemann wave propagating along y-axis and a long wave propagating along x-axis. The complete integrability of the gBK equation is systematically presented. By employing Bell’s polynomials, a lucid and systematic approach is proposed to systematically study its bilinear formalism, bilinear Bäcklund transformations, Lax pairs, respectively. Furthermore, based on multidimensional Riemann theta functions, the periodic wave solutions and soliton solutions of the gBK equation are derived. Finally, an asymptotic relation between the periodic wave solutions and soliton solutions are strictly established under a certain limit condition.

Lie Symmetry Analysis, Conservation Laws and Exact Power Series Solutions for Time-Fractional Fordy–Gibbons Equation*

In this paper, the time fractional Fordy–Gibbons equation is investigated with Riemann–Liouville derivative. The equation can be reduced to the Caudrey–Dodd–Gibbon equation, Savada–Kotera equation and the Kaup–Kupershmidt equation, etc. By means of the Lie group analysis method, the invariance properties and symmetry reductions of the equation are derived. Furthermore, by means of the power series theory, its exact power series solutions of the equation are also constructed. Finally, two kinds of conservation laws of the equation are well obtained with aid of the self-adjoint method.

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.

Nonlocal symmetry and consistent Riccati expansion integrability of the (1+1)-dimensional integrable nonlinear dispersive-wave system

Abstract Under investigation in this paper is nonlocal symmetry, consistent Riccati expansion (CRE) integrability of the (1+1)-dimensional integrable nonlinear dispersive-wave system, which can be used to describes a bidirectional soliton for wave propagation. We construct the Bäcklund transformation and consider the truncated Painlevé expansion of the system. It’s Schwarzian form is derived, whose nonlocal symmetry is localized to provide the corresponding nonlocal group. Furthermore, we verify that the system is solvable via the CRE. Based on the CRE, we further present its soliton-cnoidal wave interaction solution in terms of Jacobi elliptic functions and the third type of incomplete elliptic integral.