Higher-order rogue waves with controllable fission and asymmetry localized in a (3 + 1)-dimensional generalized Boussinesq equation

The purpose of this paper is to report the feasibility of constructing high-order rogue waves with controllable fission and asymmetry for high-dimensional nonlinear evolution equations.Such a nonlinear model considered in this paper as the concrete example is the(3+1)-dimensional generalized Boussinesq(gB) equation,and the corresponding method is Zhaqilao’s symbolic computation approach containing two embedded parameters.It is indicated by the(3+1)-dimensional gB equation that the embedded param...     

A(2+1)-dimensional nonlinear model for Rossby waves in stratified fluids and its solitary solution

In this paper,we investigate a(2+1)-dimensional nonlinear equation model for Rossby waves in stratified fluids.We derive a forced Zakharov–Kuznetsov(ZK)–Burgers equation from the quasigeostrophic potential vorticity equation with dissipation and topography under the generalized beta effect,and by utilizing temporal and spatial multiple scale transform and the perturbation expansion method.Through the analysis of this model,it is found that the generalized beta effect and basic topography can induce nonlinear waves,and slowly varying topography is an external impact factor for Rossby waves.Additionally,the conservation laws for the mass and energy of solitary waves are analyzed.Eventually,the solitary wave solutions of the forced ZK–Burgers equation are obtained by the simplest equation method as well as the new modified ansatz method.Based on the solitary wave solutions obtained,we discuss the effects of dissipation and slowly varying topography on Rossby solitary waves.     

Propagation and modulational instability of Rossby waves in stratified fluids

Perturbation analysis and scale expansion are used to derive the (2+1)-dimensional coupled nonlinear Schrödinger (CNLS) equations that can describe interactions of two Rossby waves propagating in stratified fluids. The (2+1)-dimensional equations can reflect and describe the wave propagation more intuitively and accurately. The properties of the two waves in the process of propagation can be analyzed by the solution obtained from the equations using the Hirota bilinear method, and the influence factors of modulational instability are analyzed. The results suggest that, when two Rossby waves with slightly different wave numbers propagate in the stratified fluids, the intensity of bright soliton decreases with the increases of dark soliton coefficients. In addition, the size of modulational instable area is related to the amplitude and wave number in y direction.     

Multiple-order rogue wave solutions to a (2+1)-dimensional Boussinesq type equation

In this paper, based on the Hirota bilinear method and symbolic computation approach, multiple-order rogue waves of (2+1)-dimensional Boussinesq type equation are constructed. The reduced bilinear form of the equation is deduced by the transformation of variables. Three kinds of rogue wave solutions are derived by means of bilinear equation. The maximum and minimum values of the first-order rogue wave solution are given at a specific moment. Furthermore, the second-order and third-order rogue waves are explicitly derived. The dynamic characteristics of three kinds of rogue wave solutions are shown by three-dimensional plot.     

Resonance Y-type soliton solutions and some new types of hybrid solutions in the(2+1)-dimensional Sawada–Kotera equation

In this paper, based on N-soliton solutions, we introduce a new constraint among parameters to find the resonance Y-type soliton solutions in (2+1)-dimensional integrable systems. Then, we take the (2+1)-dimensional Sawada–Kotera equation as an example to illustrate how to generate these resonance Y-type soliton solutions with this new constraint. Next, by the long wave limit method, velocity resonance and module resonance, we can obtain some new types of hybrid solutions of resonance Y-type solitons with line waves, breather waves, high-order lump waves respectively. Finally, we also study the dynamics of these interaction solutions and indicate mathematically that these interactions are elastic.     

Novel interaction phenomena of the(3+1)-dimensional Jimbo–Miwa equation

In this paper,we give the general interaction solution to the(3+1)-dimensional Jimbo–Miwa equation.The general interaction solution contains the classical interaction solution.As an example,by using the generalized bilinear method and symbolic computation by using Maple software,novel interaction solutions under certain constraints of the(3+1)-dimensional Jimbo–Miwa equation are obtained.Via three-dimensional plots,contour plots and density plots with the help of Maple,the physical characteristics and structures of these waves are described very well.These solutions greatly enrich the exact solutions to the(3+1)-dimensional Jimbo–Miwa equation found in the existing literature.     

(2+1)-dimensional dissipation nonlinear Schrödinger equation for envelope Rossby solitary waves and chirp effect

In the past few decades, the (1+1)-dimensional nonlinear Schrödinger (NLS) equation had been derived for envelope Rossby solitary waves in a line by employing the perturbation expansion method. But, with the development of theory, we note that the (1+1)-dimensional model cannot reflect the evolution of envelope Rossby solitary waves in a plane. In this paper, by constructing a new (2+1)-dimensional multiscale transform, we derive the (2+1)-dimensional dissipation nonlinear Schrödinger equation (DNLS) to describe envelope Rossby solitary waves under the influence of dissipation which propagate in a plane. Especially, the previous researches about envelope Rossby solitary waves were established in the zonal area and could not be applied directly to the spherical earth, while we adopt the plane polar coordinate and overcome the problem. By theoretical analyses, the conservation laws of (2+1)-dimensional envelope Rossby solitary waves as well as their variation under the influence of dissipation are studied. Finally, the one-soliton and two-soliton solutions of the (2+1)-dimensional NLS equation are obtained with the Hirota method. Based on these solutions, by virtue of the chirp concept from fiber soliton communication, the chirp effect of envelope Rossby solitary waves is discussed, and the related impact factors of the chirp effect are given.     

Dynamical analysis of diversity lump solutions to the (2+1)-dimensional dissipative Ablowitz–Kaup–Newell–Segure equation

The lump solution is one of the exact solutions of the nonlinear evolution equation. In this paper, we study the lump solution and lump-type solutions of (2+1)-dimensional dissipative Ablowitz–Kaup–Newell–Segure (AKNS) equation by the Hirota bilinear method and test function method. With the help of Maple, we draw three-dimensional plots of the lump solution and lump-type solutions, and by observing the plots, we analyze the dynamic behavior of the (2+1)-dimensional dissipative AKNS equation. We find that the interaction solutions come in a variety of interesting forms.     

Consistent Burgers equation expansion method and its applications to high- dimensional Burgers-type equations

In this paper, a novel method, named the consistent Burgers equation expansion (CBEE) method, is proposed to solve nonlinear evolution equations (NLEEs) by the celebrated Burgers equation. NLEEs are said to be CBEE solvable if they are satisfied by the CBEE method. In order to verify the effectiveness of the CBEE method, we take (2+1)-dimensional Burgers equation as an example. From the (1+1)-dimensional Burgers equation, many new explicit solutions of the (2+1)-dimensional Burgers equation are derived. The obtained results illustrate that this method can be effectively extended to other NLEEs.     

Optical solitons and modulation instability analysis of the (1+1)-dimensional coupled nonlinear Schrödinger equation

This study successfully reveals the dark, singular solitons, periodic wave and singular periodic wave solutions of the (1+1)-dimensional coupled nonlinear Schrödinger equation by using the extended rational sine-cosine and rational sinh-cosh methods. The modulation instability analysis of the governing model is presented. By using the suitable values of the parameters involved, the 2-, 3-dimensional and the contour graphs of some of the reported solutions are plotted.     

Folded novel accurate analytical and semi-analytical solutions of a generalized Calogero–Bogoyavlenskii–Schiff equation

This paper studies the analytical and semi-analytic solutions of the generalized Calogero–Bogoyavlenskii–Schiff(CBS) equation. This model describes the(2 + 1)–dimensional interaction between Riemann-wave propagation along the y-axis and the x-axis wave. The extended simplest equation(ESE) method is applied to the model, and a variety of novel solitarywave solutions is given. These solitary-wave solutions prove the dynamic behavior of soliton waves in plasma. The accuracy of the obtained solution is verified using a variational iteration(VI) semi-analytical scheme. The analysis and the match between the constructed analytical solution and the semi-analytical solution are sketched using various diagrams to show the accuracy of the solution we obtained. The adopted scheme's performance shows the effectiveness of the method and its ability to be applied to various nonlinear evolution equations.     

Various Kinds Waves and Solitons Interaction Solutions of Boussinesq Equation Describing Ultrashort Pulse in Quadratic Nonlinear Medium

The consistent tanh expansion (CTE) method is applied to the (2+1)-dimensional Boussinesq equation which describes the propagation of ultrashort pulse in quadratic nonlinear medium. The interaction solutions are explicitly given, such as the bright soliton-periodic wave interaction solution, variational amplitude periodic wave solution, and kink-periodic wave interaction solution. We also obtain the bright soliton solution, kind bright soliton solution, double well dark soliton solution and kink-bright soliton interaction solution by using Painlevé truncated expansion method. And we investigate interactive properties of solitons and periodic waves.     

Breather molecules and localized interaction solutions in the (2+1)-dimensional BLMP equation

The (2+1)-dimensional Boiti–Leon–Manna–Pempinelli (BLMP) equation is an important integrable model. In this paper, we obtain the breather molecule, the breather-soliton molecule and some localized interaction solutions to the BLMP equation. In particular, by employing a compound method consisting of the velocity resonance, partial module resonance and degeneration of the breather techniques, we derive some interesting hybrid solutions mixed by a breather-soliton molecule/breather molecule and a lump, as well as a bell-shaped soliton and lump. Due to the lack of the long wave limit, it is the first time using the compound degeneration method to construct the hybrid solutions involving a lump. The dynamical behaviors and mathematical features of the solutions are analyzed theoretically and graphically. The method introduced can be effectively used to study the wave solutions of other nonlinear partial differential equations.     

Dissipative Nonlinear Schrödinger Equation with External Forcing in Rotational Stratified Fluids and Its Solution

The dissipative nonlinear Schrödinger equation with a forcing item is derived by using of multiple scales analysis and perturbation method as a mathematical model of describing envelope solitary Rossby waves with dissipation effect and external forcing in rotational stratified fluids. By analyzing the evolution of amplitude of envelope solitary Rossby waves, it is found that the shear of basic flow, Brunt-Vaisala frequency and β effect are important factors in forming the envelope solitary Rossby waves. By employing Jacobi elliptic function expansion method and Hirota's direct method, the analytic solutions of dissipative nonlinear Schrödinger equation and forced nonlinear Schrödinger equation are derived, respectively. With the help of these solutions, the effects of dissipation and external forcing on the evolution of envelope solitary Rossby wave are also discussed in detail. The results show that dissipation causes slowly decrease of amplitude of envelope solitary Rossby waves and slowly increase of width, while it has no effect on the propagation speed and different types of external forcing can excite the same envelope solitary Rossby waves. It is notable that dissipation and different types of external forcing have certain influence on the carrier frequency of envelope solitary Rossby waves.     

Nonlinear Shock and Kink Waves with Complete Coriolis Force in Earth's Atmosphere

Nonlinear waves in a Boussinesq fluid model which includes both the vertical and horizontal components of Coriolis force are studied by using the semi-geostrophic approximation and the method of travelling-wave solution. Taylor series expansion has been employed to isolate the characteristics of the linear Rossby waves and to identify the nonlinear shock and kink waves. The KdV-Burgers and the compound KdV-Burgers equations are derived, their shock wave and kink wave solution are also obtained.     

Dynamical Analysis and Exact Solutions of a New (2+1)-Dimensional Generalized Boussinesq Model Equation for Nonlinear Rossby Waves*

In this paper, we study the higher dimensional nonlinear Rossby waves under the generalized beta effect. Using methods of the multiple scales and weak nonlinear perturbation expansions [Q. S. Liu, et al., Phys. Lett. A 383 (2019) 514], we derive a new $(2+1)$-dimensional generalized Boussinesq equation from the barotropic potential vorticity equation. Based on bifurcation theory of planar dynamical systems and the qualitative theory of ordinary differential equations, the dynamical analysis and exact traveling wave solutions of the new generalized Boussinesq equation are obtained. Moreover, we provide the numerical simulations of these exact solutions under some conditions of all parameters. The numerical results show that these traveling wave solutions are all the Rossby solitary waves.     

Exploring Solutions of Nonlinear Rossby Waves with Complete Coriolis Force

Nonlinear Rossby waves in a Boussinesq fluid model which includes both the vertical and horizontal components of Coriolis force are studied by using the semi-geostrophic approximation and the method of travelling-wave solution. Taylor series expansion has been employed to isolate the characteristics of the linear Rossby waves and identify the Rossby cnoidal and solitary waves. Qualitative analysis indicates that if the disturbances are independent of latitude, the effect of horizontal components of Coriolis force disappears.     

Bright,periodic, compacton and bell-shape soliton solutions of the extended QZK and (3+1)-dimensional ZK equations

The(3+1)-dimensional Zakharov–Kuznetsov(ZK) and the new extended quantum ZK equations are functional to decipher the dense quantum plasma, ion-acoustic waves, electron thermal energy,ion plasma, quantum acoustic waves, and quantum Langmuir waves. The enhanced modified simple equation(EMSE) method is a substantial approach to determine competent solutions and in this article, we have constructed standard, illustrative, rich structured and further comprehensive soliton solutions via this method. The solutions are ascertained as the integration of exponential, hyperbolic,trigonometric and rational functions and formulate the bright solitons, periodic, compacton, bellshape, parabolic shape, singular periodic, plane shape and some new type of solitons. It is worth noting that the wave profile varies as the physical and subsidiary parameters change. The standard and advanced soliton solutions may be useful to assist in describing the physical phenomena previously mentioned. To open out the inward structure of the tangible incidents, we have portrayed the three-dimensional, contour plot, and two-dimensional graphs for different parametric values. The attained results demonstrate the EMSE technique for extracting soliton solutions to nonlinear evolution equations is efficient, compatible and reliable in nonlinear science and engineering.     

Solitons and periodic waves for a generalized (3+1)-dimensional Kadomtsev-Petviashvili equation in fluid dynamics and plasma physics

Under investigation in this paper is a generalized (3+1)-dimensional Kadomtsev–Petviashvili equation in fluid dynamics and plasma physics. Soliton and one-periodic-wave solutions are obtained via the Hirota bilinear method and Hirota–Riemann method. Magnitude and velocity of the one soliton are derived. Graphs are presented to discuss the solitons and one-periodic waves: the coefficients in the equation can determine the velocity components of the one soliton, but cannot alter the soliton magnitude; the interaction between the two solitons is elastic; the coefficients in the equation can influence the periods and velocities of the periodic waves. Relation between the one-soliton solution and one-periodic wave solution is investigated.