Structure Preserving Schemes for Coupled Nonlinear Schrödinger Equation

The numerical solution of CNLS equations are studied for periodic wave solutions. We use the first order partitioned average vector field method, the second order partitioned average vector field composition method and plus method. The nonlinear implicit schemes preserve the energy and the momentum. The results show that the methods are successful to get approximation.


CNLS equations are presented by
where u(x, t) and v(x, t) are the complex two dependent variables, x is the space variable and t is the time variable.The disperison coefficients are α 1 and α 2 [13,15].The CNLS system is included in many different area [8,9,12,14,7,3].
As a result of the interaction of wave packets in the system, nonlinear situations arise.In nonintegrable cases, numerical methods should be used to analyze nonlinear situations.Symplectic and multisymplectic method are used to solve CNLS equations and showed that both method preserve mass, energy and momentum in longtime evolution [1].For soliton solution, Dehghan and Taleei examined a Chebyshev pseudospectral multidomain method.In [20], higher-order compact splitting multisymplectic method which is unconditionally stable are developed for solving CNLS equation.In [11], Variational iteration method was studied for solving CNLS equation .Ismail used Galerking method and tested this method for stabilty and accuracy [6].The linearized conservative difference scheme was studied to solve the system [18].In [19], the variational iteration method was applied to get soliton solutions for CNLS.Energy preserving methods for solving PDE and ODE have been paid much attention nowadays.In [5,4],the AVF methods of arbitrarily higher order were studied in which the Hamiltonian is canonical system and non-canonical system.more efficient AVF based method to get numerical solution of CNLS.Up to the author's knowledge, a PAVF method for the CNLS equation (0.1) is never studied before.This work given as follows.In Section 2, we propese four methods.Their application of CNLS equation given in Section 3. In Section 4, some conclusions are performed to give the competency and reliability of the presented methods in long term.Section 5 is included to concluding remarks.

PAVF Method
In this section, we will propese the partitioned average vector field model for the CNLS equation (0.1).Firstly, we give a second-order AVF method.
We consider PDEs with functions y(x, t) ∈ R.
where η is a subset of R × R. System (1.1) can be rewrite using skew-gradient where S is now skew-symmetric matrix.We choose H such that H x is an estimation to H.The variational derivate of H is dedicated by H. AVF method is defined for (1.3) where the point y j n is the numerical value of y(b + nΔx, t0 + jΔ t) for x ∈ [b, a], t0 is the initial time.If S is skew-symmetric matrix such that ıt is approximation to S, then the method exactly preserve the energy.
The AVF method is time-symmetric [5].Higher order linear integral pre-serving AVF methods, constructed using Gaussian quadrature for canonical and non-canonical Hamiltonian systems have even order 2s [4,5].The AVF method is connected to gradient methods [10].
For the partitioned AVF method, we consider the Hamiltonian system [17], Lets choose y = (w, z) T = (y 1 , y 2 , . . ., y m ; y m+1 , y m+2 , . . ., y d ) T , where d is an even number, denoting d = 2m.Therefore, the system (1.5) can be rewritten as The Hamiltonian is conserved, Then the partitioned AVF method is given by where χ is the time step.
In [17],the Hamiltonian of the system is preserved by the PAVF method exactly.
The adjoint of PAVF method (PAVF-ADJ) is defined by If we use the symbol Θ Δt for the PAVF method (1.8), we can denote the PAVF-ADJ method as Θ *

Δt
Fınally, we define composition (PAVF-C) and plus method (PAVF-P) using PAVF method and PAVF-ADJ method.Composition method is defined by and plus method is given by

Discretization of the system
Considering the equation (0.1), we can write the complex functions u, v of (0.1) sum of imaginary and real parts These equations represent an infinite-dimensional Hamiltonian system in the phase space z = (c, s, c, s) where S −1 = −S and the Hamiltonian is Finite difference approximation is applied for the first-order derivatives (2.2) to get Hamiltonian system which is a finite-dimensional.From then, we obtain the discretized Hamiltonian (2.2) The momentum conservation law is given with

Derivation of the Method
We will apply the PAVF method for the CNLS equation (2.4) Notice that this system is nonlinear, we use Newton method to get the values c n+1 , s n+1 , cn+1 , sn+1 .So Thereafter, derivation of the adjoint method, the composition method and the plus method for the CNLS equation be rewrite easly.In the next section, we will get numerical experiments for the CNLS equation.

Numerical Results
In this part, we will apply all methods to the CNLS equation.Using N + 1 uniform grid points, the space interval [b, a] is discretized.Where Δx = h = a−b N is grid spacing.Time interval is 0 ≤ t ≤ T .The system was solved by the four methods The global momentum and energy are defined by where the initial energy is E 0 and the initial momentum is M 0 .The accuracy of all methods checked at their conservation properties.We use Newton method to solve system because for all methods we get the nonlinear equations.

Elliptic polarization
We use symmetric conditions If the conditions are symmetric, the results are symmetric too (see [2]).T = 100 and the space interval is [0, 8π].In the example, the solutions with periodic boundary conditions are considered.
Figure 1 and Figure 2 global errors of momentum and energy are ploted for all the four methods.The global error in energy conservation for PAVF-C method is better than others.For the energy, the global errors in are nearly likewise for three methods.We see that the energy, momentum do not expand with time for all methods.
Figure 3 and Figure 4 give the wave profiles of |u| and |v|.It is clear that they are almost the same.
In Table 1, For the four methods, the CPU time is written .The CPU time of the PAVF-ADJ method is less than other methods.