Xiu-Bin Wang

Dynamics of the breathers, rogue waves and solitary waves in the (2+1)-dimensional Ito equation☆

Abstract In this paper, the homoclinic breather limit method is employed to find the breather wave and the rational rogue wave solutions of the ( 2 + 1 )-dimensional Ito equation. Moreover, based on its bilinear form, the solitary wave solutions of the equation are also presented with a detailed derivation. The dynamic behaviors of breather waves, rogue waves and solitary waves are analyzed with some graphics, respectively. The results imply that the extreme behavior of the breather solitary wave yields the rogue wave for the ( 2 + 1 )-dimensional Ito equation.

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.

Characteristics of the solitary waves and rogue waves with interaction phenomena in a generalized (3+1)-dimensional Kadomtsev–Petviashvili equation☆

Abstract Under investigation in this work is a generalized ( 3 + 1 )-dimensional Kadomtsev–Petviashvili equation, which can describe many nonlinear phenomena in fluid dynamics. By virtue of Bell’s polynomials, an effective and straightforward way is presented to explicitly construct its bilinear form and soliton solutions. Furthermore, based on the bilinear formalism, a direct method is employed to explicitly construct its rogue wave solutions with an ansatz function. Finally, the interaction phenomena between rogue waves and solitary waves are presented with a detailed derivation. The results can be used to enrich the dynamical behavior of higher dimensional nonlinear wave fields.

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.

Rogue waves, bright–dark solitons and traveling wave solutions of the (3+1)-dimensional generalized Kadomtsev–Petviashvili equation

Abstract In this paper, a ( 3 + 1 ) -dimensional generalized Kadomtsev–Petviashvili (gKP) equation is investigated, which describes the dynamics of nonlinear waves in plasma physics and fluid dynamics. By employing the extended homoclinic test method, we construct a new family of two wave solutions, rational breather wave and rogue wave solutions of the equation. Moreover, by virtue of some ansatz functions and the Riccati equation method, its analytical bright soliton, dark soliton and traveling wave solutions are derived. Finally, we obtain its exact power series solution with the convergence analysis. In order to further understand the dynamics, we provide some graphical analysis of these solutions.

Lie symmetry analysis, conservation laws and explicit solutions for the time fractional Rosenau-Haynam equation

Under investigation in this paper is the invariance properties of the time-fractional Rosenau–Haynam equation, which can be used to describe the formation of patterns in liquid drops. Using the Lie group analysis method, the vector fields and symmetry reductions of the equation are derived, respectively. Moreover, based on the power series theory, a kind of explicit power series solutions for the equation is well constructed with a detailed derivation. The wave propagation pattern of these solutions are presented along the x axis with different t. Finally, using the new conservation theorem, two kinds of conservation laws of the equation are well constructed with a detailed derivation.

Dynamics of the breathers and rogue waves in the higher-order nonlinear Schrödinger equation

Abstract In this paper, the higher-order nonlinear Schrodinger equation, which can be widely used to describe the dynamics of the ultrashort pulses in optical fibers, is under investigation. By means of the modified Darboux transformation, the hierarchies of breather wave and rogue wave solutions are generated from the trivial solution. Furthermore, the main characteristics of the breather and rogue waves are graphically discussed. The results show that the extreme behavior of the breather wave yields the rogue wave for the higher-order nonlinear Schrodinger equation.