Algebraic Solitary Waves Research Articles

A new nonlinear integral-differential equation describing Rossby waves and its related properties

This paper focuses on algebraic Rossby solitary waves models in stratified fluids. Starting with the quasi-geostrophic potential vorticity equation, we present the Boussinesq-intermediate long-wave (Boussinesq-ILW) model for the first time using the multi-scale analysis and the perturbation expansion method. Compared with previous models describing algebraic Rossby solitary waves, the Boussinesq-ILW model is more general: the equation transforms into the Boussinesq-BO equation when h1→∞, and when [Ξ(B)]XXX changes to BXXXX, the model reduces to the Boussinesq equation. The conservation law is important for exploring the properties of the model, therefore we propose several common conservation laws: mass, momentum, and energy conservation. Finally, based on the trial function method, we obtain the solution of the Boussinesq-ILW equation and investigate the wave-wave interaction. The results show that the peak value of the Mach stem increases as the parameter γ increases, but the length becomes shorter and changes more rapidly.

On the existence of chirped algebraic solitary waves in optical fibers governed by Kundu–Eckhaus equation

This paper reports the first analytical demonstration of propagation of algebraic solitary waves accompanied with a nonlinear chirp in optical fibers governed by the Kundu–Eckhaus equation. Both chirped dark and bright algebraic solitary waves can exist in the system with higher-order effects such as Raman effect and quintic nonlinearity. The chirp associated with each of these algebraic solitary waves is shown to be dependent on the intensity of the wave. It is found that the Raman process has a profound impact in the chirp of these localized pulses.

Fractional Theoretical Model for Gravity Waves and Squall Line in Complex Atmospheric Motion

The evolution of nonlinear gravity solitary waves in the atmosphere is related to the formation of severe weather. The nonlinear concentration of gravity solitary wave leads to energy accumulation, which further forms the disastrous weather phenomenon such as squall line. This paper theoretically proves that the formation of squall line in baroclinic nonstatic equilibrium atmosphere can be reduced to the fission process of algebraic gravity solitary waves described by the (2 + 1)-dimensional generalized Boussinesq-BO (B-BO) equation. Compared with previous models describing isolated waves, the Boussinesq-BO model can describe the propagation process of waves in two media, which is more suitable for actual atmospheric conditions. In order to explore more structural features of this solitary wave, the derived integer order model is transformed into the more practical time fractional-order model by using the variational method. By obtaining the exact solution and the conservation laws, the fission properties of algebraic gravity solitary waves are discussed. When the disturbance with limited width appears along the low-level jet stream, these solitary waves can be excited. When the disturbance intensity and width reach a certain value, solitary wave formation takes place, which is exactly the squall line or thunderstorm formation observed in the atmosphere.

Abound rogue wave type solutions to the extended (3+1)-dimensional Jimbo–Miwa equation

Under investigation in this article is an extended (3+1)-dimensional Jimbo–Miwa equation (JM), which can be used to describe many nonlinear phenomena in fluid dynamics. By using the Hirota bilinear form of the extended (3+1)-dimensional JM equation, thirty classes of rogue wave type solutions are found with the help of symbolic computations. The rogue wave type solutions contain two important parameters a and b. When a and b get different values, we can present obvious rogue wave type solutions. For example, (i) taking a=b=0, then we have algebraic solitary waves (lump); (ii) if one of a and b is fixed at 0 and the other is nonzero, then we have an interaction solution between an algebraically decayed soliton and exponentially decayed soliton (lumpoff); (iii) let a≠0 and b≠0, then we have an interaction solution between lump and exponentially localized twin soliton wave (instant or rogue wave). The new rogue wave type solutions help us to know different physical worlds.

Two-dimensional fractional algebraic internal solitary waves model and its solution

With the development of theory and advancement of scientific research, fractional calculus has begun to be considered as a method for the study of physical systems. In this work, based on the basic system of equations for internal solitary waves, we have derived two-dimensional Benjamin–Ono equation. Then, the integer-order two-dimensional Benjamin–Ono equation is transformed into the time-fractional Benjamin–Ono equation. To study the properties of the algebraic internal solitary waves, we discuss the conservation laws of the new model. Also, using the Hirota bilinear method, the derived new model is solved. Finally, we explore the characteristics of motion of the algebraic internal solitary waves with the help of the multi-soliton solutions.

Time-fractional Benjamin-Ono equation for algebraic gravity solitary waves in baroclinic atmosphere and exact multi-soliton solution as well as interaction

In this article, we will consider algebraic gravity solitary waves of the baroclinic atmosphere. The Benjamin-Ono (BO) equation will be obtained by employing multi-scale analysis and perturbation method. Based on the semi-inverse method and the Agrawal’s method, the time-fractional BO equation will be derived to describe the algebraic gravity solitary waves. We need to emphasis that the time-fractional differential-integral equation(time-fractional BO equation) is great different from the existing fractional order differential equation. Further, the conservation laws of the time-fractional BO equation will also be discussed. Finally, we will use the Hirota bidglinear method to find the exact multi-soliton solutions to the above equation and research the interaction between the soliton solutions. The results show that the propagation processes and characteristics of algebraic gravity solitary waves change with the change of the fractional order. Therefore, constructing time-fractional BO equation is not only of great theoretical value but also of more practical value for the study of algebraic gravity solitary waves of actual ocean and atmosphere.

A new ZK-ILW equation for algebraic gravity solitary waves in finite depth stratified atmosphere and the research of squall lines formation mechanism

The research of algebraic gravity solitary waves is an advanced field which has important practical and theoretical value in physical, oceanography, aerology and etc. By calculation condition and theoretical method limit, previous researches mainly focused on the (1+1) dimensional models, and (2+1) dimensional models were few considered. In this paper, from the non-static equilibrium equation, a new ZK-ILW equation is derived by using multi-scale analysis and perturbation method in finite depth stratified atmosphere, which is the first time obtained. The model can reduce to ZK-BO model (h→∞), and ZK model (h→0) and is the generalization of the above two models. In order to further understand the nature of algebraic gravity solitary waves, we get the analytical solution of ZK-ILW equation by using the trial function method and discuss the conservation laws. Furthermore, the fission process of algebraic gravity solitary waves is studied, and we can judge that one of the possible formation mechanism of squall lines is the fission of algebraic gravity solitary waves.

A new ZK–BO equation for three-dimensional algebraic Rossby solitary waves and its solution as well as fission property

The BO equation depicts the evolution of algebraic Rossby solitary waves in ocean and atmosphere. But, despite its overt fame, the BO equation is restricted as a two-dimensional model. On the basis of the great success in the soliton theory, a lot of works have recently been directed to three-dimensional models and investigations of solitary waves properties in three-dimensional systems. In this paper, a new ZK–BO equation for three-dimensional algebraic Rossby solitary waves is derived by employing perturbation expansions and stretching transformations of time and space. By virtue of the trial function method, the exact solution of the ZK–BO equation is presented. By comparing the solution with the solution of two-dimensional algebraic Rossby solitary waves, we can find that the wave length of three-dimensional algebraic Rossby solitary waves is shorter while the wave amplitude is higher. Based on the exact solution of ZK–BO equation, the dissipation effect is studied. The results reveal the influence of dissipation effect on the three-dimensional algebraic solitary waves. Further, after theoretical analysis, the conservation laws of three-dimensional algebraic Rossby solitary waves are discussed and the fission property is studied.

The Boussinesq-BO equation for algebraic gravity solitary waves in baroclinic atmosphere and the research of squall lines formation mechanism

The gravity solitary waves are a kind of waves which are caused by the disturbance of static equilibrium. The nonlinearity concentration of the gravity solitary waves makes the energy assemble together and forms disastrous weather phenomena, such as squall lines. By the calculation condition and theoretical method limit, previous studies tried hard to reduce the variable numbers and discussed the gravity solitary waves in barotropic atmosphere, but the baroclinic problem of atmosphere is inevitable topic. In this paper, from the basic kinetic equations in baroclinic non-static equilibrium atmosphere, by using multi-scale analysis and perturbation method, a new model is derived to describe the algebraic gravity solitary waves, we call it Boussinesq-BO equation. Comparing with the former models, the Boussinesq-BO model can describe the propagation process of waves in two directions and is more suitable for the real atmosphere condition. With the help of the trial function method, an exact solution of Boussinesq-BO equation is obtained and the fission property of algebraic gravity solitary waves is discussed. Finally, we can find that the fission of algebraic gravity solitary waves is also a possible formation mechanism of squall lines.

The Rational Solutions and the Interactions of the N-Soliton Solutions for Boiti-Leon-Manna-Pempinelli-Like Equation

These rational solutions which can be described a kind of algebraic solitary waves which have great potential in applied value in atmosphere and ocean. It has attracted more and more attention recently. In this paper, the generalized bilinear method instead of the Hirota bilinear method is used to obtain the rational solutions to the (2 + 1)-dimensional Boiti-Leon-Manna-Pempinelli-like equation (hereinafter referred to as BLMP equation). Meanwhile, the (2 + 1)-dimensional BLMP-like equation is derived on the basis of the generalized bilinear operators D3,x D3,y and D3,t. And the rational solutions to the (2 + 1)-dimensional BLMP-like equation are obtained successively. Finally, with the help of the N-soliton solutions of the (2 + 1)-dimensional BLMP equation, the interactions of the N-soliton solutions can be derived. The results show that the two soliton still maintained the original waveform after happened collision.

ZK-Burgers equation for three-dimensional Rossby solitary waves and its solutions as well as chirp effect

Two-dimensional Rossby solitary waves propagating in a line have attracted much attention in the past decade, whereas there is few research on three-dimensional Rossby solitary waves. But as is well known, three-dimensional Rossby solitary waves are more suitable for real ocean and atmosphere conditions. In this paper, using multiscale and perturbation expansion method, a new Zakharov-Kuznetsov (ZK)-Burgers equation is derived to describe three-dimensional Rossby solitary waves that propagate in a plane. By analyzing the equation we obtain the conservation laws of three-dimensional Rossby solitary waves. Based on the sine-cosine method, we give the classical solitary wave solutions of the ZK equation; on the other hand, by the Hirota method we also obtain the rational solutions, which are similar to the solutions of the Benjamin-Ono (BO) equation, the solutions of which can describe the algebraic solitary waves. The rational solutions of the ZK equations are worth of attention. Finally, with the help of the classical solitary wave solutions, similar to the fiber soliton communication, we discuss the dissipation and chirp effect of three-dimensional Rossby solitary waves.

The Rational Solutions and Quasi-Periodic Wave Solutions as well as Interactions ofN-Soliton Solutions for 3 + 1 Dimensional Jimbo-Miwa Equation

The exact rational solutions, quasi-periodic wave solutions, andN-soliton solutions of 3 + 1 dimensional Jimbo-Miwa equation are acquired, respectively, by using the Hirota method, whereafter the rational solutions are also called algebraic solitary waves solutions and used to describe the squall lines phenomenon and explained possible formation mechanism of the rainstorm formation which occur in the atmosphere, so the study on the rational solutions of soliton equations has potential application value in the atmosphere field; the soliton fission and fusion are described based on the resonant solution which is a special form of theN-soliton solutions. At last, the interactions of the solitons are shown with the aid ofN-soliton solutions.

A new model for algebraic Rossby solitary waves in rotation fluid and its solution**Project supported by the Shandong Provincial Key Laboratory of Marine Ecology and Environment and Disaster Prevention and Mitigation Project, China (Grant No. 2012010), the National Natural Science Foundation of China (Grant Nos. 41205082 and 41476019), the Special Funds for Theoretical Physics of the National Natural Science Foundation

A generalized Boussinesq equation that includes the dissipation effect is derived to describe a kind of algebraic Rossby solitary waves in a rotating fluid by employing perturbation expansions and stretching transformations of time and space. Using this equation, the conservation laws of algebraic Rossby solitary waves are discussed. It is found that the mass, the momentum, the energy, and the velocity of center of gravity of the algebraic solitary waves are conserved in the propagation process. Finally, the analytical solution of the equation is generated. Based on the analytical solution, the properties of the algebraic solitary waves and the dissipation effect are discussed. The results point out that, similar to classic solitary waves, the dissipation can cause the amplitude and the speed of solitary waves to decrease; however, unlike classic solitary waves, the algebraic solitary waves can split during propagation and the decrease of the detuning parameter can accelerate the occurrence of the solitary waves fission phenomenon.

Interaction of algebraic Rossby solitary waves with topography and atmospheric blocking

In this article, a new equation governing the behavior of Rossby solitary waves is derived by employing perturbation expansions and stretching transformations of time and space, which is called mBO-mKdV-Burgers equation. The equation is different from the common BO equation, it is more suitable for describing the Rossby solitary waves when the perturbation is stronger. Based on the analytical solution of mBO-mKdV-Burgers equation, the features of Rossby solitary waves including conserved laws, fission property and dissipation effect are studied. It is found that dissipation causes the amplitude and speed of solitary waves decrease; breakup phenomenon will happen during propagation. Finally, numerical simulation is carried out to investigate the effect of detuning parameter α, topographical altitude and dissipation on the interaction of Rossby solitary waves with topography. It is pointed out that with decreasing of detuning parameter α, the propagation speed of solitary waves decreases and the interaction time increases, which is beneficial to form the large amplitude disturbance; with increasing of topographical altitude, the free solitary waves can not cross over topography; decreasing detuning parameter α and increasing topographical altitude are both mechanisms to generate atmospheric blocking; with the occurence of small dissipation, damping oscillation phenomenon will happen.

A kind of new algebraic Rossby solitary waves generated by periodic external source

In the article, by employing multiple-scale, perturbation method, a new model is derived to describe the algebraic Rossby solitary waves generated by periodic external source in stratified fluid. The local conservation laws and analytic solutions of the model are obtained, and the breakup properties are discussed. By numeric simulation, some problems on the generation and evolution of the algebraic solitary waves under the influence of periodic external source are theoretically investigated. The results show that besides the solitary waves, an additional harmonic wave appears in the region of the external source forcing. Furthermore, the periodic variation of the external source forcing can prevent solitary waves from breaking. Meanwhile, the detuning parameter has an important effect on the breakup of the algebraic Rossby solitary waves.

Benjamin-Ono-Burgers-MKdV Equation for Algebraic Rossby Solitary Waves in Stratified Fluids and Conservation Laws

In the paper, by using multiple-scale method, the Benjamin-Ono-Burgers-MKdV (BO-B-MKdV) equation is obtained which governs algebraic Rossby solitary waves in stratified fluids. This equation is first derived for Rossby waves. By analysis and calculation, some conservation laws are derived from the BO-B-MKdV equation without dissipation. The results show that the mass, momentum, energy, and velocity of the center of gravity of algebraic Rossby waves are conserved and the presence of a small dissipation destroys these conservations.

Algebraic Rossby Solitary Waves Excited by Non-Stationary External Source

The paper deals with the effects of non-stationary external source forcing and dissipation on algebraic Rossby solitary waves. From quasi-geostrophic potential vorticity equation, basing on the multiple-scale method, an inhomogeneous Korteweg-de Vries—Benjamin—Ono—Burgers (KdV-B-O-Burgers) equation is obtained. This equation has not been previously derived for Rossby waves. By analysis and calculation, four conservation laws associated with the above equation are first obtained. With the help of pseudo-spectral method, the waterfall plots are obtained and the evolutional characters of algebraic Rossby solitary waves are studied. The results show that non-stationary external source and dissipation have great effect on the generation and evolution of algebraic solitary Rossby waves.

Sub-ODE method and solitary wave solutions for higher order nonlinear Schrödinger equation

With the aid of an ordinary differential equation (ODE) involving an arbitrary positive power of dependent variable and its various positive solutions, three types of solitary wave solution of the higher order nonlinear Schrodinger equation (NLSE) with non-Kerr terms have been found out, which are the bell type solitary waves, the kink type solitary waves and the algebraic solitary waves, provided that the coefficients of the higher order NLSE satisfy certain constraint conditions.

Solitary Waves for Cubic-Quintic Nonlinear Schrödinger Equation withVariable Coefficients

The cubic-quintic nonlinearSchrödinger equation (CQNLS) plays importantparts in the optical fiber and the nuclear hydrodynamics. By using thehomogeneous balance principle, the bell type, kink type, algebraic solitarywaves, and trigonometric traveling waves for the cubic-quintic nonlinearSchrödinger equation with variable coefficients (vCQNLS) are derivedwith the aid of a set of subsidiary high-order ordinary differentialequations (sub-equations for short). The method used in this paper mighthelp one to derive the exact solutions for the other high-order nonlinearevolution equations, and shows the new application of the homogeneousbalance principle.

Various exact solutions of nonlinear Schrödinger equation with two nonlinear terms

Four kinds of exact solutions to nonlinear Schrödinger equation with two higher order nonlinear terms are obtained by a subsidiary ordinary differential equation method (sub-equation method for short). They are the bell type solitary waves, the kink type solitary waves, the algebraic solitary waves and the sinusoidal waves.