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

Authors: Xin Chen; Hongwei Yang; Min Guo; Baoshu Yin
Journal: Mathematical Problems in Engineering
 (2017)

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## Abstract

Using the method of multiple scales and perturbation method, a set of coupled models describing the envelope Rossby solitary waves in (2+1)-dimensional condition are obtained, also can be called coupled NLS (CNLS) equations. Following this, based on trial function method, the solutions of the NLS equation are deduced. Moreover, the modulation instability of coupled envelope Rossby waves is studied. We can find that the stable feature of coupled envelope Rossby waves is decided by the value of S. Finally, learning from 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.


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## Body

## 1. 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 β along with the changes of latitude were discussed by Luo [6] and β 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 [22–24], Hirota method [25, 26], homogeneous balance method [27], Jacobi elliptic function method [28], Symmetry method [29, 30], Rational solutions [31–33]; meanwhile the features of equations are also discussed [34–36]. In this paper, we plan to adopt the trial function method and derive the exact solution of model. The difference between the two-dimensional 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.


## 2. 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].(1)∂∂t+U¯y∂∂x∇2ψ+β-U¯′′∂ψ∂x=-ϵJψ,∇2ψ,and the boundary condition is(2)∂ψ∂x=0,asy=0,Ly,where x and y are the local Cartesian coordinates pointing east and north. In this β=β∗L2/U, β∗ is Rossby parameter. U is characteristic velocity, L is characteristic length, and Ly is the width of the beta-channel. ϵ is small parameter, on behalf of nonlinear strength. J is the Jacobian of (A,B).(3)JA,B=∂A∂x∂B∂y-∂A∂y∂B∂x,and ∇2 is Laplace operator(4)∇2=∂2∂x2+∂2∂y2.In general, it is difficult to get analytic solution of (1). But, because of the nonlinear term with a small parameter ε, 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(5)T1=ϵt,T2=ϵ2t,X1=ϵx,X2=ϵ2x,Y=ϵy,so long time and space scales are defined as(6)∂∂x⟶∂∂x+ϵ∂∂X1+ϵ2∂∂X2,∂∂y⟶∂∂y+ϵ∂∂Y,∂∂t⟶∂∂t+ϵ∂∂T1+ϵ2∂∂T2.Substituting (6) into (1), we obtain(7)∂∂t+ϵ∂∂T1+ϵ2∂∂T2+U¯y∂∂x+ϵ∂∂X1+ϵ2∂∂X2∂2ψ∂x2+∂2ψ∂y2+2ϵ∂2∂x∂X1+∂2ψ∂y∂Y+ϵ2∂2ψ∂X1+2∂2ψ∂x∂X2+∂2ψ∂Y2+2ϵ3∂2ψ∂X1∂X2+ϵ4∂2ψ∂X22+β-U¯′′∂ψ∂x+ϵ∂ψ∂X1+ϵ2∂ψ∂X2+ϵ∂ψ∂x∂∇2ψ∂y-∂ψ∂y∂∇2ψ∂x=0,∂ψ∂x=0,y=0,Ly.Further(8)∂∂t+U¯∂∂x∂2ψ∂x2+∂2ψ∂y2+β-U¯′′∂ψ∂x+ϵ∂∂T1+U¯∂∂X1∂2ψ∂x2+∂2ψ∂y2+∂∂t+U¯∂∂x2∂2ψ∂x∂X1+2∂2ψ∂y∂Y+β-U¯′′∂ψ∂X1+∂ψ∂x∂3ψ∂y∂x2+∂3ψ∂y3-∂ψ∂y∂3ψ∂x3+∂3ψ∂x∂y2+ϵ2∂∂T2+U¯∂∂X2∂2ψ∂x2+∂2ψ∂y2+∂∂T1+U¯∂∂X12∂2ψ∂x∂X1+2∂2ψ∂y∂Y+∂∂t+U¯∂∂x∂2ψ∂X12+∂2ψ∂Y2+β-U¯′′∂ψ∂X2+∂ψ∂x2∂3ψ∂y∂x∂X1+3∂3ψ∂y2∂Y+∂3ψ∂x2∂Y+∂ψ∂X1∂3ψ∂y∂x2+∂3ψ∂y3-∂ψ∂y2∂3ψ∂x2∂X1+2∂3ψ∂y∂x∂Y+∂3ψ∂y2∂X1-∂ψ∂Y∂3ψ∂x3+∂3ψ∂x∂y2+ϵ3∂∂t+U¯∂∂x∂2ψ∂X1∂X2+∂ψ∂x∂3ψ∂y∂X12+∂3ψ∂y∂x∂X2+2∂3ψ∂x∂Y∂X1+3∂3ψ∂y∂Y2+∂ψ∂X12∂3ψ∂y∂x∂X1+2∂3ψ∂y2∂Y+∂3ψ∂Y∂x2+∂3ψ∂Y∂y2+oϵ3=0,and the boundary condition is(9)∂ψ∂x=0,asy=0,Ly.The stream function ψ is expanded according to the small parameter ϵ(10)ψ=ψ0+ϵψ1+ϵ2ψ2+ϵ3ψ3+⋯,and, substituting (9) into (8), we get the multiple order questions of the stream function ψ.First, one introduces an operator(11)L=∂∂t+U¯∂∂x∇2+β-U¯′′∂∂x,so that(12)oε0:Lψ0=∂∂t+U¯∂∂x∇2ψ0+β-U¯′′∂ψ0∂x,oε1:Lψ1=∂∂T1+U¯∂∂X1∂2ψ0∂x2+∂2ψ0∂y2+∂∂t+U¯∂∂x2∂2ψ0∂x∂X1+2∂2ψ0∂y∂Y+β-U¯′′∂ψ0∂X1+∂ψ0∂x∂3ψ0∂y∂x2+∂3ψ0∂y3-∂ψ0∂y,oε2:Lψ2=∂∂T2+U¯∂∂X2∂2ψ0∂x2+∂2ψ0∂y2+∂∂T1+U¯∂∂X12∂2ψ0∂x∂X1+2∂2ψ0∂y∂Y+∂∂t+U¯∂∂x∂2ψ0∂X12+∂2ψ0∂Y2+β-U¯′′∂ψ0∂X2+∂ψ0∂x2∂3ψ0∂y∂x∂X1+3∂3ψ0∂y2∂Y+∂3ψ0∂x2∂Y+∂ψ0∂X1∂3ψ0∂y∂x2+∂3ψ0∂y3-∂ψ0∂y2∂3ψ0∂x2∂X1+2∂3ψ0∂y∂x∂Y+∂3ψ0∂y2∂X1+∂ψ0∂Y∂3ψ0∂x3+∂3ψ0∂x∂y2-∂∂T1+U¯∂∂X1∂2ψ1∂x2-∂2ψ1∂y2-∂∂t+U¯∂∂x2∂2ψ1∂x∂X1+2∂2ψ1∂y∂Y+β-U¯′′∂ψ1∂X1-∂ψ1∂x∂3ψ1∂y∂x2+∂3ψ1∂y3+∂ψ1∂y∂3ψ1∂x3+∂3ψ1∂x∂y2∂ψ∂x=0,y=0,Ly.Assume(13)ψ0=∑j=12AjT1,T2,X1,X2,YϕjyeiKjx-ωjt+c.c.,and, substituting (13) into L(ψ(0)), we have(14)Lψ0=∂∂t+U¯∂∂x-∑j=12AjϕjKj2eiKjx-ωjt+∑j=12Ajd2ϕjdy2eiKjx-ωjt+β-U¯′′∑j=12AjϕjiKjeiKjx-ωjt=∑j=12AjeiKjx-ωjtiωjKj2ϕj-ωjd2ϕjdy2-iU¯Kj3ϕj+iU¯Kjd2ϕjdy2+iKjβ-U¯′′ϕj≡0,and further(15)-iωj-iU¯Kjd2ϕjdy2-iU¯Kj3-iωjKj2-iKjβ-U¯′′ϕj=0,(16)d2ϕjdy2-Kj2-β¯-U¯′′U¯-Cjϕj=0,and the boundary condition is(17)∂ψ∂x=0,asy=0,Ly,where(18)Cj=ωjKj.From (15), we can get the solution of ϕj:(19)ϕjy=2Lyjsin⁡my,m=n1πLyjn1=±1,2,3,….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 U¯(y) takes some specific functions. Under normal circumstance one can only seek numerical solution, so we consider the higher order question(20)oϵ1:Lψ1=-∂∂T1+U¯∂∂X1∂2ψ∂x2+∂2ψ∂y2-∂∂t+U¯∂∂x2∂2ψ∂x∂X1+2∂2ψ∂y∂Y-β-U¯′′∂ψ∂X1-∂ψ∂x∂3ψ∂y∂x2+∂3ψ∂y3+∂ψ∂y∂3ψ∂x3+∂3ψ∂x∂y2,∂ψ∂x=0,y=0,Ly,and, further, we get(21)Lψ1=-∑j=12g1je2iKjx-ωjt-∑j=12g2je2iKjx-ωjtAj2-ig3eiK1+K2x-ω1+ω2tA1A2-ig4eiK1-K2x-ω1-ω2tA1A2,where(22)g1j=β-U¯′′U¯-Cjϕj∂Aj∂T1+Cgj∂Aj∂X1-2iKjU¯-Cjdϕjdy∂Aj∂Y,g2j=Kjϕjddy-dϕjdyd2ϕjdy2,g3=K1ϕ1ddy-K2dϕ1dyd2ϕ2dy2-K22ϕ2+K2ϕ2ddy-K1dϕ2dyd2ϕ1dy2-K12ϕ1,g4=K1ϕ1ddy+K2dϕ1dyd2ϕ2dy2-K22ϕ2-K2ϕ2ddy+K1dϕ2dyd2ϕ1dy2-K12ϕ1.Cgj=Cj+2Kj2U¯-Cj2β-U¯′′,and the special solution corresponding to the second, third, and forth item of (21) right side is(23)ψ2j1=Aj2ϕ2je2iKjx-ωjt+c.c,ψ31=A1A2ϕ3eiK1+K2x-ω1+ω2t+c.c,ψ41=A1A2ϕ4eiK1-K2x-ω1-ω2t+c.c,where ϕ2j, ϕ3, and ϕ4 satisfied (24)d2ϕ2jdy2-4Kj2-β-U¯′′U¯-cjϕ2j=g2j2U¯Kj-ωj,d2ϕ3dy2-K1+K22-β-U¯′′K1+K2U¯K1+K2-ω1+ω2ϕ3=g3U¯K1+K2-ω1+ω2,d2ϕ4dy2-K1-K22-β-U¯′′K1-K2U¯K1-K2-ω1-ω2ϕ4=g4U¯K1-K2-ω1-ω2,and boundary condition(25)ϕ2jy=0=ϕ2jy=Ly=ϕ3y=0=ϕ3y=Ly=ϕ4y=0=ϕ4y=Ly.We assume the special solution corresponding to the first item of the right side of (20) is(26)ψ1j1=ψ¯1j1eiKjx-ωjt+c.c,ψ¯1j1y=0=ψ¯1j1y=Ly=0,where ψ1j(1) satisfied (27)d2ψ1j1dy2-Kj2-β-U¯′′U¯-Cjψ1j1=β-U¯′′U¯-Cjϕj∂Aj∂T1+Cgj∂Aj∂X1-2iKjU¯-Cjdϕjdy∂Aj∂Y.The following operators are introduced:(28)ξ=X1-CgjT1,T2=εT1.Substituting (28) into (27) and because ε is infinitesimal, we can ignore the first item (29)U¯-Cgj∂∂ξ∂ψ¯1j∂y2+β∂ψ¯1j∂ξ=-4Kj2mLyjsin⁡2mjy∂Aj2∂ξ,ψ¯1jy=0=ψ¯1jy=Ly=0.Obviously the solution for ψ1j may be expressed in the following form:(30)ψ¯ij=4Kj2Kj2+m22msin⁡2myAj23m2m2-2Kj2-kj4βLyj.So we obtained the solution of (20):(31)ψ1=ψ1j1+ψ2j1+ψ31+ψ41+ϕy,T1,X1.So as to obtain the solution of A, we continue to consider the question of o(ε2). Substituting (19) and (30) into o(ε2) and collecting the secular-producing items proportional to ei(Kx-ωt), we have(32)Lψ2=-∑j=12∂Aj∂T2+U¯∂Aj∂X2d2ϕjdy2-Kj2ϕj+iϕj3U¯Kj-ωj∂2Aj∂X12+iϕjU¯Kj-ωj∂2Aj∂Y2+2iKjϕj∂2Aj∂T1∂X1+2dϕjdy∂2Aj∂T1∂Y+U¯∂2Aj∂X1∂Y+β-U¯′′∂Aj∂X2ϕj+8iKj3m2Kj2+m22Aj2Ajcos⁡2my3m2m2-2Kj2-Kj4βLyj,∂ψ∂x=0,y=0,Ly.For the sake of obtaining the evolution equation of Aj, we consider another class of nonhomogeneous solutions, assuming(33)Lψ2=∑j=12FjT1,T2,X1,X2,YeiKjx-ωjt+c.c.,introducing (33) into (32), where(34)F1=f1∂A1∂T2+α1∂A1∂X2+β1∂2A1∂Y2+f10∂2A1∂X12+f11A12A1+f12A22A1+f13A1,F2=f2∂A2∂T2+α2∂A2∂X2+β2∂2A2∂Y2+f20∂2A2∂X12+f21A12A2+f22A22A2+f23A2.We assume two special nonhomogeneous solutions of (33) are(35)ψ12=Φ12eiK1x-ω1t+c.c.,ψ22=Φ22eiK2x-ω2t+c.c.,and, substituting (35) into (34), we get(36)d2Φj2dy2-Kj-β-U¯′′U¯-cjΦj2=FjKjU¯-cj,multiplying (35) by φj, and integrating on y from 0 to Ly. The two sides of (35) are equal to zero, so we get the solution conditions:(37)∂∂T2+cg1∂∂X2A1-iα1∂2A1∂X12-iβ1∂2A1∂Y2-iσ1A12+γ12A22+λ1jX1,Y,T1A1=0,∂∂T2+cg2∂∂X2A2-iα2∂2A2∂X12-iβ2∂2A2∂Y2-iσ2A22+γ21A12+λ2jX1,Y,T1A2=0.We get the evolution equations group of wave amplitude, that is, the coupled equations group. To simplify, we introduce the following transformation:(38)X=1εX2-CgT2=X1-CgT1,T=T2,and then (37) can be written as(39)i∂A1∂T+α1∂2A1∂X2+β1∂2A1∂Y2+σ1∗A12+r12∗A22A1=0,i∂A2∂T+Cg2∗∂A2∂X+α2∂2A2∂X2+β2∂2A2∂Y2+σ2∗A22+r21∗A12A2=0,where (40)Cg2∗=1εCg1-Cg2,σ1∗=σ1+λ11,σ2∗=σ2+λ22,γ12∗=γ12+λ12,γ21∗=γ21+λ21.In (38), coefficients α1, α2, β1, and β2 are dispersion coefficients, σ1∗, σ2∗ are Landau coefficients, γ12∗, γ21∗ are interaction coefficients, and from their expressions it can be seen that their values are related to the base flow function U¯(y). The (38) is called CNLS equations group.


## 3. 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:(41)iAj,T+αjAj,XX+βjAj,YY+γjAj2Aj=0,j=1,2,where αj, βj represent the coefficients of dispersion term. γj 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:(42)AjT,X,Y=A¯jT,X,YexpiφT,X,Y,where A¯j(T,X,Y) is the amplitude portion of the soliton solutions and φ(T,X,Y) is the phase portion of the soliton solution, which is given as(43)φT,X,Y=ωT+KX+MY.Substituting (42) into (41), let real part and imaginary part be zero, and we can get the following coupled equations:(44)∂Aj¯∂T+αj2∂Aj¯∂X∂φ∂X+Aj¯∂2φ∂X2+βj2∂Aj¯∂Y∂φ∂Y+Aj¯∂2φ∂Y2=0,-Aj¯∂φ∂T+αj∂2Aj¯∂X2-Aj¯∂φ∂X2+βj∂2Aj¯∂Y2-Aj¯∂2φ∂Y2+γjAj¯3=0.Further, replacing φ with (43)(45)∂Aj¯∂T+2αjK∂Aj¯∂X+2βjM∂Aj¯∂Y=0,-ωAj¯+αj∂2Aj¯∂X2-K2Aj¯+βj∂2Aj¯∂Y2-M2Aj¯+γjAj¯3=0.Using the traveling wave transformation, this pair of equations will be analyzed further, let(46)Aj¯T,X,Y=Fjζ,asζ=ω′T+K′X+M′Y,and we have(47)-ω′+2KK′αj+MM′βjdFjdζ=0,αK′2+βM′2d2Fjdζ2-αK2+βM2+ωFj+γFj3=0.From the first term of (47), we get(48)ω′=2KK′α+MM′β.Further, according to the balance principle in trial function method, we will balance Fj′′ with Fj3. Using the solution procedure of the trial function method, we will obtain the system of algebraic equations as follows:(49)F3:2αjK′2+βM′2a4+γj=0,F2:32αjK′2+βjM′2a3=0,F1:αjK′2+βjM′2a2-αjK2+βjM2+ωj=0,F0:12αjK′2+βjM′2a1=0.From the equations above, we have the results of the system(50)a4=-γjαjK′2+βjM′2,a3=0,a2=αjK2+βjM2+ωαjK′2+βjM′2,a1=0.Therefore, we know that F satisfied(51)±ζ-ζ0=∫1a0+αjK2+βjM2+ω/αK′2+βM′2Fj2-γj/αjK′2+βjM′2Fj4dFj.If we set a0=0 in (51) and integrating with respect to Fj, we will obtain the following soliton solutions of (41) as follows:(52)A¯1=±α1K2+β1M2+ωγ1sech⁡α1K2+β1M2+ωα1K′2+β1M′2K′X+M′Y-2KK′α1+MM′β1t-ζ0×exp⁡iKX+MY+ωT,A¯2=±-α2K2+β2M2+ωγ2csch⁡α2K2+βM2+ωα2K′2+β2M′2K′X+M′Y-2KK′α+MM′βt-ζ0×exp⁡iKX+MY+ωT,and these solutions are the soliton solutions for CNLS equations, when KK′αj+MM′βj+ω>0.


## 4. 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(53)i∂A1∂T+α∂2A1∂X2+β∂2A1∂Y2+σ1∗A12+r12∗A22A1=0,i∂A2∂T+α∂2A2∂X2+β∂2A2∂Y2+σ1∗A22+r21∗A12A1=0.Further, introducing(54)Z=Xcos⁡θ+Ysin⁡θ,(53) reduces to(55)i∂A1∂T+γ1∂2A1∂Z2+σ1∗A12+r12∗A22A1=0,i∂A2∂T+γ2∂2A2∂Z2+σ1∗A22+r21∗A12A2=0,where γ1=γ2=α2cos2⁡θ+β2sin2⁡θ.From Section3, we know the exact periodic wave solutions, taking the simple form as follows:(56)A1=aexp⁡im1Z-Ω1T,A2=bexp⁡im2Z-Ω2T,where m1, Ω1, m2, and Ω2 satisfy(57)Ω1=γ1m12-σ1a2+γ12b2,Ω2=γ2m22-σ2b2+γ21a2.We assume that a, b are real numbers and m1, m2 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:(58)A1=exp⁡im1Z-Ω1Ta+ϵϕ1Z,T,A2=exp⁡im2Z-Ω2Tb+ϵϕ2Z,T.Substituting (58) into (53), we can get the linear equations as follows:(59)i∂ϕ1∂T+γ12m1i∂ϕ1∂Z+∂2ϕ1∂Z2+σ1a2ϕ1+ϕ1∗+γ12abϕ2+ϕ2∗=0,i∂ϕ2∂T+γ22m1i∂ϕ2∂Z+∂2ϕ2∂Z2+σ2b2ϕ2+ϕ2∗+γ21abϕ1+ϕ1∗=0.Further, assume(60)ϕ1=p1exp⁡iλZ+σT+q1exp⁡-iλZ+σ∗T,ϕ2=p2exp⁡iλZ+σT+q2exp⁡-iλZ+σ∗T,and, substituting (60) into (59), we can get(61)σ1a2-γ1λ2+iσ-2γ1m1λp1+γ12abp2+σ1a2q1∗+γ12abq2∗=0,σ1a2p1∗+γ12abp2∗+σ1a2-γ1λ2-iσ∗+2γ1m1λq1+γ12abq2=0,γ21abp1+σ2b2-γ2λ2+iσ-2γ2m2λp2+γ21abq1∗+σ2b2q2∗=0,γ21abp1∗-σ2b2p2∗+γ21abq1+σ2b2-γ2λ2+iσ-2γ2m2λq2=0.The above equations are linear homogeneous equations for p1, p2, q1, and q2. If there are nonzero solutions, the coefficients determinant must be zero. So we can get the next type:(62)γ12λ4-2γ1σ1λ2a2-iσ-2γm1λ2γ22λ4-2γ2σ2λ2b2-iσ-2γ2m2λ2=4γ1γ2γ12γ21a2b2λ4,and these are the four algebraic equations of iσ. When the parameters are certain, the value of λ makes Reσ>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(63)γ1m1=γ2m2,and, moreover, set(64)Δ1=γ12λ2λ2-2σ1γ1a2,Δ2=γ22λ2λ2-2σ2γ2b2,S=4γ1γ2γ12γ21a2b2λ4.Clearly, when Δ1≥0 the first wave is stable and there is no interaction and the contrary occurs when Δ1<0 is instable. Similarly, when Δ2≥0 the second wave is stable and there is no interaction, and the contrary occurs when Δ2<0 is instable. From (62), we get(65)σ=Im⁡γ1m1λ±Δ1+Δ2±Δ1-Δ22+4S21/2,where σ represents the gain for the frequency shift, which has been described in Figure 1. Equation (65) shows that when Δ1<0 and Δ2<0, that is, Δ1+Δ2<0, no matter what value S takes, at least there is one which satisfies Reσ>0; therefore, the waves have instability. This conclusion shows that when γ1m1=γ2m2, two modulated unstable waves, the interaction of the two waves is still unstable.Figure 1
Gain spectra for frequency shift.WhenΔ1>0, Δ2>0, that is, Δ1+Δ2>0, corresponding to no interaction, the two waves are stable. If S>0, when S≤Δ1Δ2, two waves are stable after interaction. When S>Δ1Δ2, two waves are unstable after interaction. If S<0, when -1/4(Δ1-Δ2)2≤S≤Δ1Δ2, the two waves are stable. When S<-1/4(Δ1-Δ2)2 or S>Δ1Δ2 two waves are unstable after interaction. From the above analysis, we can find that when Δ1>0 and Δ2>0, even with two stable nonlinear waves through interaction, the stable feature is decided by the value of S.WhenΔ1<0 and Δ2>0, corresponding to no interaction, the first wave is stable, but the second wave is unstable, while when Δ1>0, Δ2<0, the condition is opposite.


## 5. 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.whenα=β, the NLS equation (41) for describing the characteristics of Rossby wave propagation transforms to(66)i∂A∂T+α∂2A∂X2+∂2A∂Y2+γA2A=0,where α is the coefficient of dispersion and γ 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 K=M=K′=M′=1:(67)A=2α+ωγsech⁡2α+ω2αX+Y.


### 5.1. Chirp Effect Caused by Dispersion

Let us consider the dispersion effect of chirp, and (66) becomes(68)AT=-iαAXX+AYY.Reviewing the time T from 0→ΔT, where ΔT is an infinitesimal variable, and introducing (67) into (68), we can get the approximate solution of (68) as follows:(69)AΔT,X,Y=2α+ωγsech⁡2α+ω2αX+Yexp⁡-i2α+ωΔT×sech2⁡2α+ω2αX+Y,so that the phase of the wave meets(70)φD=-i2α+ωΔTsech2⁡2α+ω2αX+Y,and, based on (70), we can obtain the chirp effect caused by the dispersion(71)ΔνD=-∇φD=22α+ω3/2ΔTsech2⁡2α+ω2αX+Ytanh⁡2α+ω2αX+Y.


### 5.2. Chirp Effect Caused by Nonlinearity

Separately considering the nonlinear effect of chirp, the NLS equation (66) becomes(72)AT=-iγA2A,and, investigating the condition of time T from 0→ΔT, where ΔT is an infinitesimal variable, and introducing (67) into (72), we can get the approximate solution of (72) as follows:(73)AΔT,X,Y=2α+ωγsech⁡2α+ω2αX+Yexp⁡-i2α+ωΔT×sech2⁡2α+ω2αX+Y,and the phase of the wave meets(74)φN=-2α+ωΔT×sech2⁡2α+ω2αX+Y,and, based on (74), we can get the chirp effect caused by the nonlinear(75)ΔνN=-∇φN=-42α+ω3/22αΔTsech2⁡2α+ω2αX+Ytanh⁡2α+ω2αX+Y.


### 5.3. Discussion of Total Chirp

According to (71) and (75), the total chirp is(76)Δνs=ΔνD+ΔνN=2-42α2α+ω3/2ΔTsech2⁡2α+ω2αX+Ytanh⁡2α+ω2αX+Y.(1) When the dispersion and nonlinear effects cancel each other, as follows:(77)Δνs=ΔνD+ΔνN=0,we can get α=1/16.(2) When the dispersion effect is greater than the nonlinear effect, as follows:(78)ΔνD>ΔνN,we get α<1/16.(3) When the dispersion is less than the nonlinear effect, as follows:(79)ΔνD<ΔνN,we get α>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 α. Combining Figure 2, Figure 3, and calculation, we can get the following conclusions.Figure 2
Three-dimensional waveform with dispersion and nonlinear effects.Figure 3
The variation of the chirp effect under the dispersion and nonlinearity.(1) Whenα=1/16, the dispersion and nonlinear effects cancel each other, and solitary waves propagate over long distances and keep waveforms constant.(2) Whenα>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) Whenα>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.


## 5.1. Chirp Effect Caused by Dispersion

Let us consider the dispersion effect of chirp, and (66) becomes(68)AT=-iαAXX+AYY.Reviewing the time T from 0→ΔT, where ΔT is an infinitesimal variable, and introducing (67) into (68), we can get the approximate solution of (68) as follows:(69)AΔT,X,Y=2α+ωγsech⁡2α+ω2αX+Yexp⁡-i2α+ωΔT×sech2⁡2α+ω2αX+Y,so that the phase of the wave meets(70)φD=-i2α+ωΔTsech2⁡2α+ω2αX+Y,and, based on (70), we can obtain the chirp effect caused by the dispersion(71)ΔνD=-∇φD=22α+ω3/2ΔTsech2⁡2α+ω2αX+Ytanh⁡2α+ω2αX+Y.


## 5.2. Chirp Effect Caused by Nonlinearity

Separately considering the nonlinear effect of chirp, the NLS equation (66) becomes(72)AT=-iγA2A,and, investigating the condition of time T from 0→ΔT, where ΔT is an infinitesimal variable, and introducing (67) into (72), we can get the approximate solution of (72) as follows:(73)AΔT,X,Y=2α+ωγsech⁡2α+ω2αX+Yexp⁡-i2α+ωΔT×sech2⁡2α+ω2αX+Y,and the phase of the wave meets(74)φN=-2α+ωΔT×sech2⁡2α+ω2αX+Y,and, based on (74), we can get the chirp effect caused by the nonlinear(75)ΔνN=-∇φN=-42α+ω3/22αΔTsech2⁡2α+ω2αX+Ytanh⁡2α+ω2αX+Y.


## 5.3. Discussion of Total Chirp

According to (71) and (75), the total chirp is(76)Δνs=ΔνD+ΔνN=2-42α2α+ω3/2ΔTsech2⁡2α+ω2αX+Ytanh⁡2α+ω2αX+Y.(1) When the dispersion and nonlinear effects cancel each other, as follows:(77)Δνs=ΔνD+ΔνN=0,we can get α=1/16.(2) When the dispersion effect is greater than the nonlinear effect, as follows:(78)ΔνD>ΔνN,we get α<1/16.(3) When the dispersion is less than the nonlinear effect, as follows:(79)ΔνD<ΔνN,we get α>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 α. Combining Figure 2, Figure 3, and calculation, we can get the following conclusions.Figure 2
Three-dimensional waveform with dispersion and nonlinear effects.Figure 3
The variation of the chirp effect under the dispersion and nonlinearity.(1) Whenα=1/16, the dispersion and nonlinear effects cancel each other, and solitary waves propagate over long distances and keep waveforms constant.(2) Whenα>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) Whenα>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.


## 6. 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 S, which implies that the instability condition depends not only on the prescribed perturbation wave numbers p, q, 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.



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