(2 + 1)-Dimensional Coupled Model for Envelope Rossby Solitary Waves and Its Solutions as well as Chirp Effect

1Shandong University of Science and Technology, Qingdao, Shandong 266590, China 2Key Laboratory of Meteorological Disaster of Ministry of Education, Nanjing University of Information Science and Technology, Nanjing 210044, China 3Key Laboratory of Ocean Circulation and Waves, Institute of Oceanology, Chinese Academy of Sciences, Qingdao 266071, China 4Function Laboratory for Ocean Dynamics and Climate, Qingdao National Laboratory for Marine Science and Technology, Qingdao 266237, China


Introduction
Wave phenomenon exists widely in nature.As a special and important branch of waves, Rossby solitary waves have important theoretical significance and research value.Meanwhile, with the intercross and penetration of different knowledge, the Rossby solitary waves theory has applied to many other fields successfully, such as physical oceanography, atmospheric physics, hydraulic engineering, communication engineering, and thermal power engineering.In the case of application of solitary waves in engineering, the application of optical soliton in communication engineering is the most representative.Meanwhile, the Rossby waves theory was widely used in the study of mesoscale eddies and the interaction of large and medium scale motions.Rossby solitary waves in the westerly shear flow were first found by Long [1].Afterward, Benny [2] amplified this research and found that velocity and amplitude of Rossby waves were proportional and depicted the importance of nonlinearity.In view of the barotropic fluid and stratified fluid model, the KdV and mKdV equation are also generated to describe the generation and evolution of Rossby solitary waves by Redekopp [3].Compared to KdV equation, the mKdV equation is more suitable to express the condition with stronger perturbation.With the development of solitary waves, a variety of equation models for describing the Rossby solitary waves such as ILW-Burgers equation and ZK-Burgers equation were discussed by Yang et al. [4,5].Moreover, the generation and evolution of solitary waves in different topography condition and different fluid depths were discussed.Recently, the Rossby parameters  along with the changes of latitude were discussed by Luo [6] and  plane approximation was obtained.These Rossby solitary waves are called classical solitary waves which related to the KdV-type equations.Later, many researchers have studied the Rossby wave equation in many aspects [7,8], such as integrable system [9,10], the integrable coupling of equations [11], and Hamiltonian structures [12].Unlike the KdV-type equation, the nonlinear Schrödinger equations were used to study the evolution of envelope classical Rossby solitary waves.From the end of the 70's to the 80's, driven by the study of atmospheric blocking dynamics [13], nonlinear Rossby wave theory had been developing rapidly and gradually formed Rossby solitary waves theory and dipole waves theory.In addition, beside the above two theories, envelope Rossby solitary waves also dropped in the research scope of the theme.The envelope Rossby solitons in the barotropic shear and uniform flows were first investigated by Benney [14] and Yamagata [15].Afterward, Luo [16] tried to use this envelope Rossby solitons to explain atmospheric blocking phenomenon.Later, dissipative NLS equation in rotational stratified fluids and its solution were obtained by Shi et al. [17].As we all know, using the nonlinear Schrödinger equation describing the Rossby solitary waves, we can introduce the concept of chirp in optical soliton communication [18], to study the influence of dispersion and nonlinearity on solitary waves propagation process.In the optical soliton communication field, the concept of chirp [19,20] is the phenomenon that the central wavelength shifts when the pulse is transmitted.It is helpful to analyze the propagation characteristics and the formation mechanism of solitary waves.
In the domain of solitary wave models, it is necessary to obtain the exact solutions of solitary wave models by all kinds of solving methods and analyze the feature of solitary waves in propagation process based on the exact solutions.Many methods to solve the equations are proposed, such as traveling wave method [21], Darboux transformation method [22][23][24], Hirota method [25,26], homogeneous balance method [27], Jacobi elliptic function method [28], Symmetry method [29,30], Rational solutions [31][32][33]; meanwhile the features of equations are also discussed [34][35][36].In this paper, we plan to adopt the trial function method and derive the exact solution of model.The difference between the twodimensional and three-dimensional model will be given and some features of three-dimensional NLS equation will be discussed.
We note that the above researches commonly considered two-dimensional model or single (2 + 1)-dimensional model to reflect the evolution of envelope Rossby solitary waves.There are two disadvantages: (1) The two-dimensional model can only be applied to describe the evolution of envelope Rossby solitary waves in a line.
(2) The velocity of the KdV-type soliton is larger than the real observation.
While, as we know, the (2 + 1)-dimensional model can be applied to reflect the evolution of envelope Rossby solitary waves in a plane, which is more suitable for the real ocean and atmosphere, in this paper, by using the method of multiple scales and perturbation method, starting from the barotropic atmospheric vorticity equation, we will derive the coupled (2 + 1)-dimensional nonlinear Schrödinger equations for envelope Rossby solitary waves in Section 2.Not only is the model (2 + 1)-dimensional and more suitable to describe the feature of two envelope Rossby solitary waves in a plane, particularly, but also it is a coupled model and can show the interaction process between two waves.Then, based on trial function method, we will deduce the solution of the CNLS equations group and the envelope solitary waves characteristics in Section 3. Thirdly, we study the modulation instability of Rossby waves trains in (2 + 1)-dimensional condition in Section 4. Finally, the concept of chirp in optical soliton communication is introduced, and the chirp effect caused by dispersion and nonlinearity is also discussed in Section 5.

Derivation of the (2 + 1)-Dimensional CNLS Equations
The tropical atmospheric motion is quasihorizontal and quasiconvergent.The governing equation is the quasigeostrophic barotropic vorticity equation [37].
and the boundary condition is where  and  are the local Cartesian coordinates pointing east and north.In this  =  *  2 /,  * is Rossby parameter. is characteristic velocity,  is characteristic length, and   is the width of the beta-channel. is small parameter, on behalf of nonlinear strength. is the Jacobian of (, ).
and ∇ 2 is Laplace operator In general, it is difficult to get analytic solution of (1).But, because of the nonlinear term with a small parameter , we use the multiple scales method and perturbation method to obtain the nonlinear solution of it.As a preliminary study, we will consider two waves.Let us introduce the slow time and space variables Substituting ( 6) into (1), we obtain Further and the boundary condition is The stream function  is expanded according to the small parameter   =  (0) +  (1) and, substituting ( 9) into (8), we get the multiple order questions of the stream function .
First, one introduces an operator so that Assume and, substituting (13) into ( (0) ), we have and further and the boundary condition is where From (15), we can get the solution of   : Formula ( 15) under the boundary condition poses a standard Sturm-Liouville problem.The effects of zonal flows on linear equation trapped waves were treated in detail by many researchers.The analytic solution of ( 15) can be obtained when () takes some specific functions.Under normal circumstance one can only seek numerical solution, so we consider the higher order question  ( (1) ) :  ( (1) and, further, we get where and the special solution corresponding to the second, third, and forth item of (21) right side is where  2 ,  3 , and  4 satisfied and boundary condition We assume the special solution corresponding to the first item of the right side of ( 20) is 1  (  −  ) + ., =0 = (1) 1 = = 0, where  (1)  1 satisfied The following operators are introduced: Substituting ( 28) into ( 27) and because  is infinitesimal, we can ignore the first item Obviously the solution for  1 may be expressed in the following form: Mathematical Problems in Engineering So we obtained the solution of (20): (1) =  (1)  1 +  (1)  2 +  (1)  3 +  (1)  4 +  (,  1 ,  1 ) .
So as to obtain the solution of , we continue to consider the question of ( 2 ).Substituting ( 19) and (30) into ( 2 ) and collecting the secular-producing items proportional to  (−) , we have For the sake of obtaining the evolution equation of   , we consider another class of nonhomogeneous solutions, assuming introducing (33) into (32), where We assume two special nonhomogeneous solutions of (33) are and, substituting ( 35) into (34), we get multiplying (35) by   , and integrating on  from 0 to   .The two sides of ( 35) are equal to zero, so we get the solution conditions: We get the evolution equations group of wave amplitude, that is, the coupled equations group.To simplify, we introduce the following transformation: and then (37) can be written as where

The Solutions of the (2 + 1)-Dimensional CNLS Equations
In this chapter, we will discuss the solutions of the (2 + 1)dimensional CNLS equation.Based on our experience, we should transform the coupled equations into two independent equations.Inspired by [38,39], (39) can be written with the following form: where   ,   represent the coefficients of dispersion term.  is the nonlinear coupling term coefficient, which can be positive or negative.In order to obtain the traveling wave solutions of the CNLS equations, we define the transformation which is the complex number envelope solution: where   (, , ) is the amplitude portion of the soliton solutions and (, , ) is the phase portion of the soliton solution, which is given as Substituting ( 42) into (41), let real part and imaginary part be zero, and we can get the following coupled equations: Further, replacing  with (43) Using the traveling wave transformation, this pair of equations will be analyzed further, let   (, , ) =   () , as  =    +    +   , (46) and we have From the first term of (47), we get Further, according to the balance principle in trial function method, we will balance    with  3  .Using the solution procedure of the trial function method, we will obtain the system of algebraic equations as follows: From the equations above, we have the results of the system Therefore, we know that  satisfied If we set  0 = 0 in (51) and integrating with respect to   , we will obtain the following soliton solutions of (41) as follows: and these solutions are the soliton solutions for CNLS equations, when     +     +  > 0.

Modulation Instabilities of Coupled Envelope Rossby Waves
For coupled envelope nonlinear Rossby waves, they meet the CNLS equations (39).We set the NLS equation of the (2 + 1)dimension with constant coefficients as Further, introducing (53) reduces to where From Section 3, we know the exact periodic wave solutions, taking the simple form as follows: where  1 , Ω 1 ,  2 , and Ω 2 satisfy We assume that ,  are real numbers and  1 ,  2 represent wave number.Equation (54) shows that each nonlinear Rossby wave dispersion not only contains itself wave number and amplitude, but also contains another wave amplitude, which is characteristic of the interaction between wave and wave.
Next, we will analyze the stability of waves solution below.Assume the solutions for the disturbance as follows: Substituting ( 58) into (53), we can get the linear equations as follows: Further, assume and, substituting (60) into (59), we can get The above equations are linear homogeneous equations for  1 ,  2 ,  1 , and  2 .If there are nonzero solutions, the coefficients determinant must be zero.So we can get the next type: and these are the four algebraic equations of .When the parameters are certain, the value of  makes Re > 0. But, for the influence of the interaction between wave and wave on the stability of wave, we will give a special case for discussion.Assume the number of waves satisfies and, moreover, set Clearly, when Δ 1 ≥ 0 the first wave is stable and there is no interaction and the contrary occurs when Δ 1 < 0 is instable.Similarly, when Δ 2 ≥ 0 the second wave is stable and there is no interaction, and the contrary occurs when Δ 2 < 0 is instable.From (62), we get where  represents the gain for the frequency shift, which has been described in Figure 1.Equation (65) shows that when Δ 1 < 0 and Δ 2 < 0, that is, Δ 1 + Δ 2 < 0, no matter what value  takes, at least there is one which satisfies Re > 0; therefore, the waves have instability.This conclusion shows that when  1  1 =  2  2 , two modulated unstable waves, the interaction of the two waves is still unstable.When Δ 1 > 0, Δ 2 > 0, that is, Δ 1 +Δ 2 > 0, corresponding to no interaction, the two waves are stable.If  > 0, when  ≤ Δ 1 Δ 2 , two waves are stable after interaction.When  > Δ 1 Δ 2 , two waves are unstable after interaction.If  < 0, when −(1/4)(Δ 1 − Δ 2 ) 2 ≤  ≤ Δ 1 Δ 2 , the two waves are stable.When  < −(1/4)(Δ 1 − Δ 2 ) 2 or  > Δ 1 Δ 2 two waves are unstable after interaction.From the above analysis, we can find that when Δ 1 > 0 and Δ 2 > 0, even with two stable nonlinear waves through interaction, the stable feature is decided by the value of .

Chirp Effect
With summary of previous studies on Rossby waves, it is not hard to find that nonlinearity and dispersion are important factors affecting the propagation of Rossby waves.In this section, we use the concept of chirp in the field of optical soliton communication to study the chirp effect caused by nonlinearity and dispersion in the propagation of Rossby waves.
when  = , the NLS equation (41) for describing the characteristics of Rossby wave propagation transforms to where  is the coefficient of dispersion and  is the nonlinear coefficient.Here, based on the soliton solution of the NLS equation, we take the initial wave form of (2+1)-dimensional Rossby solitary waves as follows, setting  =  =   =   = 1: Reviewing the time  from 0 → Δ, where Δ is an infinitesimal variable, and introducing (67) into (68), we can get the approximate solution of (68) as follows: so that the phase of the wave meets and, based on (70), we can obtain the chirp effect caused by the dispersion

Chirp Effect Caused by Nonlinearity.
Separately considering the nonlinear effect of chirp, the NLS equation (66) becomes and, investigating the condition of time  from 0 → Δ, where Δ is an infinitesimal variable, and introducing (67) into (72), we can get the approximate solution of (72) as follows: and the phase of the wave meets and, based on (74), we can get the chirp effect caused by the nonlinear Δ]  = −∇  = −4 (2 + ) (1) When the dispersion and nonlinear effects cancel each other, as follows: we can get  = 1/16.
(3) When the dispersion is less than the nonlinear effect, as follows: Δ]      <     Δ]      , we get  > 1/16.As we know, the total chirp effect is related to the sea conditions in the propagation region and the amplitude of the initial wave.If the propagation area and the sea condition parameter is determined, the total chirp effect is only related to the initial amplitude.The chirp effect caused by dispersion and nonlinearity is described in Figures 2 and 3.The above calculation shows that the magnitude of the initial amplitude is determined by the parameter .
Combining Figure 2, Figure 3, and calculation, we can get the following conclusions.
(1) When  = 1/16, the dispersion and nonlinear effects cancel each other, and solitary waves propagate over long distances and keep waveforms constant.
(2) When  > 1/16, the nonlinear effect is greater than the dispersion effect, so that the total chirp is not zero.In this case, the isolated waves present nonlinear characteristics, and the amplitude of isolated waves is changed periodically.
(3) When  > 1/16, the dispersion effect is greater than the nonlinear effect, so that the total chirp is not zero.In this case, the isolated waves present dispersion property, and the amplitude of isolated waves is changed periodically.

Conclusions
In this paper, we obtained the (2 + 1)-dimensional coupled NLS (CNLS) equations and discussed the solutions of the single nonlinear Schrödinger equation.In addition, we know that, for two nonlinear Rossby waves, they can be described by CNLS equations.The equations show that no matter the two waves' interaction process, their respective energy and momentum are conserved.Furthermore, modulational instability of coupled envelope Rossby waves in (2 + 1)dimensional condition is also discussed.we can find that the stable feature of coupled envelope Rossby waves is decided by the value of , which implies that the instability condition depends not only on the prescribed perturbation wave numbers , , but also on the amplitude of the Rossby waves.Finally, introducing the concept of chirp in the optical soliton communication field we study the chirp effect caused by nonlinearity and dispersion in the propagation of Rossby waves.
* 21 are interaction coefficients, and from their expressions it can be seen that their values are related to the base flow function ().The(38)is called CNLS equations group.