The chiral nonlinear Schrödinger equation (CNSE) [1] [2] we are going to study is given by

i ∂ ψ ∂ t + 1 2 ∂ 2 ψ ∂ x 2 + i λ ( ψ ∗ ∂ ψ ∂ x − ψ ∂ ψ ∂ x ∗ ) ψ = 0 ,     − ∞ < x < ∞ (1)

where t is time, x is spatial coordinate, ψ ( x , t ) is a complex-valued function, and λ is a nonlinear coupling constant appears through derivative coupling. This kind of nonlinearity is also known as the current density, unlike the case of cubic nonlinearity which is also known as the Kerr nonlinearity. The Chiral nonlinear Schrödinger is a non-integrable equation by the classical method of Inverse Scattering method. The single bright soliton solution of Equation (1) is given by [1] [2]

ψ ( x , t ) = A sech ( β ( x − v t ) ) exp ( i ( v x + ω t ) ) , (2)

where

β = A 2 λ v     and     ω = v ( 2 λ A 2 − v ) 2 , (3)

A is the amplitude of the soliton, β is the inverse width of the soliton, v is the soliton velocity, and ω is the wave number. The dark soliton solution for (1) exists for λ v < 0 , and has the form

ψ ( x , t ) = A tanh ( β ( x − v t ) ) exp ( i ( v x + ω t ) ) (4)

where

β = A − 2 λ v     and     ω = v ( 4 λ A 2 − v ) 2 . (5)

Thus Equation (1) has bright or dark solitons that are given by (2) and (4), respectively, depending on the sign of λ v . This phenomenon makes the solitons chiral. Equation (1) has at least four conserved quantities [1] [2], namely

I 1 ( t ) = ∫ − ∞ ∞ | ψ ( x , t ) | 2 d x , (6)

I 2 ( t ) = i ∫ − ∞ ∞ ( ψ ¯ ( x , t ) ψ x ( x , t ) − ψ ( x , t ) ψ ¯ x ( x , t ) ) d x , (7)

I 3 ( t ) = ∫ − ∞ ∞ [ λ | ψ ( x , t ) | 4 + 2 i ( ψ ¯ ( x , t ) ψ x ( x , t ) − ψ ( x , t ) ψ ¯ x ( x , t ) ) ] d x , (8)

and

I 4 = ∫ − ∞ ∞ [ λ | ψ x ( x , t ) | 2 + 2 i ( ψ ¯ ( x , t ) ψ x ( x , t ) − ψ ( x , t ) ψ ¯ x ( x , t ) ) + λ | ψ ( x , t ) | 6 ] d x (9)

Due to the exponential decay of the bright soliton solution (2) when | x | → ∞ , the conserved quantities (6)-(9) are well defined. By using (2), the exact values of the conserved quantities are

I 1 = β λ v , I 2 = 2 β λ v , I 3 = 4 A 2 3 β ( λ A 2 + 6 v ) , I 4 = 4 A 2 3 b ( 15 v 2 + 5 β 2 + 40 λ v A 2 + 8 λ 2 A 4 ) . (10)

The conserved quantities (6)-(9) using dark soliton solution (4) are not well defined due to the nonzero boundary condition as | x | → ∞ .

By assuming ψ ( x , t ) = u ( x , t ) + i v ( x , t ) , where u ( x , t ) , v ( x , t ) are real functions, the CNSE (1) can be written as the nonlinear coupled system [3] [4] [5] as

∂ u ∂ t + 1 2 ∂ 2 v ∂ x 2 + 2 λ ( u ∂ v ∂ x − v ∂ u ∂ x ) v = 0, (11)

∂ v ∂ t − 1 2 ∂ 2 u ∂ x 2 − 2 λ ( u ∂ v ∂ x − v ∂ u ∂ x ) u = 0. (12)

The resulting system obtained (11)-(12) can be displayed in a matrix vector form as

∂ w ∂ t + 1 2 A ∂ 2 w ∂ x 2 + G ( w ) A w = 0 , (13)

where

w = [ u v ] ,   A = ( 0 1 − 1 0 ) ,   G ( w ) = 2 λ ( u ∂ v ∂ x − v ∂ u ∂ x ) . (14)

There are many theoretical and numerical studies in the literature about the Nonlinear Schrödinger Equations (NLS). Most of these works are motivated to single NLS and the coupled NLS (see [3] [4] [5] and reference therein). However, up to the authors knowledge, only a few numerical studies for the chiral nonlinear Schrodinger NLS. In [6], a conservative finite difference schemes for the chiral nonlinear schrodinger equation using finite difference method, three different schemes of second order accuracy are derived, a nonlinear implicit finite difference where we have to solve a nonlinear block tridigonal systems at each time step which is expensive to some extent, a linearization approach is used to overcome this difficulty and to obtain, two linearly implicit finite difference schemes. In this work, we are going to present numerical schemes of fourth order accuracy in both direction using method of lines. The method of lines has been used by several authors. A Method of lines is used to find the numerical solution of the Kortweg-de Vries Equation [7]. The method of lines solution of the regularized long-wave equation using Runge-Kutta time discretization method presented in [8]. The method of lines for solution of the one-dimensional wave equation subject to an integral conservation condition presented in [9]. Application of the method of lines for solving the KdV-Burger equation, two methods are used to solve this equation MOL and the Adomian decomposition method the results reveal that, both methods are comparable [10]. A generalization of the method of lines for the numerical solution of the coupled forced vibration of beams presented in [11]. In this work, we are going to solve the chiral nonlinear Schrodinger equation using method of lines, which can be described in two major steps.

Step 1: We convert the partial differential equation into a system of first order ordinary differential equation by approximating the space derivatives in the CNSE using fourth order finite difference approximation order.

Step 2: We solve the ordinary differential system obtained in Step 1 using Runge-Kutta method of fourth order and Predictor Corrector method of fourth order.

Several Numerical examples single and interaction of two solitons will be investigated to confirm the efficiency of the derived schemes.

The paper is organized as follows: In Section 2, two methods of lines are derived to solve the chiral NLS (1). In Section 3, some numerical results single soliton and interaction of two solitons to check the efficiency of the proposed schemes. Finally, the conclusion is given in Section 4.

We will consider the numerical solution of the nonlinear system (10)-(11) in a finite interval [ x L , x R ] . We assume x m = x L + m h , where m = 1,2, ⋯ , M − 1 , and h is called the space grid size, also we assume t n = n k , k is the time step size. We denote the exact and numerical solutions at the grid point ( x m , t n ) by ( w m n , v m n ) and ( U m n , V m n ) , respectively. Also we approximate the space derivatives using the fourth order approximation [12].

∂ u ( x m ) ∂ x = 1 12 h [ U m − 2 − 8 U m − 1 + 8 U m + 1 − U m + 2 ] (15)

∂ 2 u ( x m ) ∂ x 2 = 1 12 h 2 [ − U m − 2 + 16 U m − 1 − 30 U m + 16 U m + 1 − U m + 2 ] . (16)

By using these approximations into (11)-(12), we will get the first order ordinary differential coupled system

U ˙ m + 1 24 h 2 [ − U m − 2 + 16 U m − 1 − 30 U m + 16 U m + 1 − U m + 1 ] + λ 6 h ( U m [ V m − 2 − 8 V m − 1 + 8 V m + 1 − V m + 2 ] − V m [ U m − 2 − 8 U m − 1 + 8 U m + 1 − U m + 2 ] ) V m = 0, (17)

U − 1 = U 0 = 0 ,   U M = U M + 1 = 0 ,     for   m = 1 , 2 , ⋯ , M − 1

V ˙ m − 1 24 h 2 [ − U m − 2 + 16 U m − 1 − 30 U m + 16 U m + 1 − U m + 1 ] − λ 6 h ( U m [ V m − 2 − 8 V m − 1 + 8 V m + 1 − V m + 2 ] − V m [ U m − 2 − 8 U m − 1 + 8 U m + 1 − U m + 2 ] ) U m = 0, (18)

V − 1 = V 0 = 0 ,   V M = V M + 1 = 0 ,     for   m = 1 , 2 , ⋯ , M − 1 ,

The numerical solution of the coupled system (17)-(18) can be obtained by using the Runge-Kutta method of fourth order or by using a linear multistep method in the following manner. First we write the previous system (17)-(18) in the coupled form

U ˙ m = F ( U m , V m ) , m = 1 , 2 , ⋯ , M − 1 (19)

V ˙ m = F ( U m , V m ) , m = 1 , 2 , ⋯ , M − 1 (20)

The basic idea of the method of lines is to replace the spatial derivatives in the partial differential equation with algebraic approximations. Once this is carried out, the spatial derivatives are no longer stated explicitly in terms of the spatial independent variables. Thus, in effect we have a coupled system of ordinary differential equations that approximate the original PDE. Two methods for solving the ODEs coupled system (19)-(20) are presented; Runge-Kutta method of fourth order and the fourth order Predictor-Corrector methods. The details of these methods will be given next.

In this section, we will test the efficiency of the numerical schemes we derived, by considering different numerical tests. Trapezoidal rule is used to calculate the conserved quantities. Define the error norms

L ∞ ( u ) = max 1 ≤ m ≤ M − 1 | u m n − U m n | (44)

L 2 ( u ) = [ h ∑ m = 1 M − 1 | u m n − U m n | 2 ] . (45)

The method of lines is used to solve the Chiral nonlinear Schrodinger equation, using two methods of fourth order accuracy in both directions space and time. The numerical results using single soliton, and the interaction of two solitons of the CNSE confirmed the efficiency, reliability, and accuracy of the present methods. Concerning the cpu time required for execution, the merit goes for the Predictor-Corrector method. We can conclude that, the methods presented in this paper can be applied to similar type of equations, like nonlinear Schrodinger and coupled nonlinear Schrodinger equations.

The authors declare no conflicts of interest regarding the publication of this paper.

AL-Basyouni, K.S. and Ismail, M.S. (2020) Method of Lines for the Chiral Nonlinear Schrödinger Equation. Applied Mathematics, 11, 447-459. https://doi.org/10.4236/am.2020.116032