Simulation of envelope Rossby solitons in a pair of cubic Schrödinger equations
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) equationswhere 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 equationswhere 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 toSo Eqs. (3), (4) can be written as the Hamiltonian formz = (p, u, q, v), , 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: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).
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