Simulation of envelope Rossby solitons in a pair of cubic Schrödinger equations

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Abstract

The symplectic scheme of the coupled nonlinear Schrödinger equations is obtained. We study the collision interactions of the envelope Rossby solitons by the symplectic scheme and get the collision interactions of the envelope Rossby solitons are sensitive to the coupling constants and the different velocities of the two solitary waves.

Introduction

The nonlinear Rossby wave packets have been the subject of considerable study in the past years. The baroclinic Rossby wave packets can be described by the coupled nonlinear Schrödinger(CNLS) equationsiAt+α1Axx+(σ1|A|2+ν12|B|2)A=0,iBt+α2Bxx+(σ2|B|2+ν21|A|2)B=0,where A, B are the complex amplitudes or “envelopes” of the two wave packets respectively. In the language of the method of multiple scales, t is the slow time and x is slow space variable moving at the linear group velocity. The coefficients α1 and α2 are so-called “dispersion coefficients”, σ1 and σ2 are the “Landau constants” describing the self-modulation of the wave packets, and ν12 and ν21 are the “coupling constant” of the cross-modulation between the two wave packets. The coupled nonlinear schrödinger equations also arise in the intense electromagnetic wave propagation in a birefringent fiber [1], [3]. Zakharov and Schulman have shown that in the case α1 = α2, σ1 = σ2 = ν12 = ν21, or α1 =   α2, σ1 = σ2 =   ν12 =   ν21, the system has an infinite set of motion invariant and may be solved by the inverse scattering methods. Except for the above two case, the system can only be solved by the numerical method [2]. To study the colliding solitary waves, a Fourier pseudospectral method combined with fourth-order Runge–Kutta time integration is used to solve the CNLS equations. Many collision interactions have been observed by the method [4], [8], [9]. However, at present when the parameter numbers varied, the collision regular is not clear, the parameter realms are not still discovered completely.

Recently, specification has been paid to the symplectic geometry. A great deal of numerical experiments have shown the superiority of the symplectic schemes over the nonsymplectic ones, especially, in structural, global and long-term tracking capabilities [5], [7]. When ν12 = ν21, the coupled Schrödinger equations have the symplectic structure. We construct the symplectic scheme of the CNLS equations.

The purpose of this paper is to present the symplectic scheme of the CNLS system based on the hamiltonian formulation to study the collision interactions of the solitary waves. We present the symplectic scheme of the CNLS equations in Section 2. In Section 3, numerical experiments are reported to the CNLS equations and the numerical results are analyzed. At last, we get some conclusions.

Section snippets

The symplectic scheme of the CNLS equations

To the following coupled nonlinear Schrödinger equationsiAt+α1Axx+(σ1|A|2+ν|B|2)A=0,iBt+α2Bxx+(σ2|B|2+ν|A|2)B=0,where A(x, t) = p(x, t) + q(x, t)i, B(x, t) = u(x, t) + v(x, t)i. Taking p(x, t) = p, q(x, t) = q, u(x, t) = u, v(x, t) = v. Eqs. (3), (4) are equivalent topt+α1qxx+(σ1(p2+q2)+ν(u2+v2))q=0,qt-α1pxx-(σ1(p2+q2)+ν(u2+v2))p=0,ut+α2vxx+(σ2(u2+v2)+ν(p2+q2))v=0,vt-α2uxx-(σ2(u2+v2)+ν(p2+q2))u=0.So Eqs. (3), (4) can be written as the Hamiltonian formdzdt=JδH(z)δz,z = (p, u, q, v), J=0I-I0, I is 2 × 2 identity matrix, the

Numerical experiments

In this section, we solve the coupled nonlinear Schrödinger equations by the scheme (22), (23). Supposing that outside the space region, the value of the solution is equivalent to zero.

When the initial condition of the CNLS equations is as follows:A(x,0)=2α1σ1η1sech(η1x)exp(iξ1x),B(x,0)=2α2σ2η2sech(η2(x+x0))exp(iξ2(x+x0)).We all take Δt = 0.001, Δx = 0.1 η1 = 1, ξ1 = 1/2, η2 =   1, ξ2 = −1/2, x0 =  −10, −20 < x < 20.

In Fig. 1, the computation is done for 0  t  6. The left of Fig. 1 is the numerical result of the A(x

Conclusions

The symplectic scheme for the coupled nonlinear Schrödinger system is presented. we simulate the collision interaction of the CNLS system by the symplectic scheme. We observed that the collision interaction is sensitive to the coupling constants and the different velocities of the two solitary waves. It is obvious that the new scheme can well simulate the behaviour of the coupled nonlinear Schrödinger system.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Nos. 10401033, 10475082 and 10471145) and Knowledge Innovation Key Project of Chinese Academy of Science (KZCX1-SW-18).

References (9)

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