In this study, we will construct numerical techniques for tackling the logarithmic Schr ödinger’s nonlinear equation utilizing the explicit scheme and the Crank-Nicolson scheme of the finite difference method. These schemes will be subjected to accuracy and stability tests before being used. Efficacy and robustness of the techniques under consideration will be demonstrated using an exact solution, one-Gausson, as well as conserved quantities. Interaction of two-soliton will be conducted. The numerical findings revealed, the interplay behavior is flexible.
One of the most significant fields of nonlinear optical study recently has been the dynamics of solitons in optical communication [
Biswas and Aceves presented a perturbation approach for studying optical solitons defined by a perturbed nonlinear Schrödinger’s equation in the year 2000 [
Biswas et al. in 2011 it was described how Schrödinger’s nonlinear equation works, which governs optical solitons, is explored using non-Kerr law nonlinearity in the existence of perturbation components with complete nonlinearity [
Wazwaz used the variation iteration approach to conduct research on the logarithmic Schrodinger equation, examining it both with and without a detuning term in 2019 [
In this study, we shall provide numerical solutions to Schrödinger’s nonlinear logarithmic equation in limited domains.
Now we’ll consider, the log NLS equation
i ∂ u ∂ t + a ∂ 2 u ∂ x 2 + b log ( | u | 2 ) u = 0 , (1)
in the finite domain x L ≤ x ≤ x R , 0 ≤ t ≤ T , using the following initial and boundary conditions:
u ( x , 0 ) = g ( x ) , x L ≤ x ≤ x R ,
u ( x L , t ) = 0 = u ( x R , t ) , 0 ≤ t ≤ T ,
where a denotes the dispersion coefficient of the group velocity and b is the log law nonlinearity coefficient. Group velocity dispersion is a dispersive medium characteristic used to determine how it affects pulse duration. The wave profile is the dependent variable u ( x , t ) , whereas the spatial and temporal variables are independent variables x and t, respectively [
To sidestep complex calculations, we suppose
u ( x , t ) = v ( x , t ) + i w ( x , t ) , (2)
v ( x , t ) and w ( x , t ) are both real-valued functions. By replacing (2) into (1), we can get the coupled system shown below by segregating the real and imaginary portions
∂ v ∂ t + a ∂ 2 w ∂ x 2 + b z w = 0 , (3)
∂ w ∂ t − a ∂ 2 v ∂ x 2 − b z v = 0 , (4)
where z = log ( v 2 + w 2 ) . The system contained in Equations (3) and (4), may be expressed as a matrix-vector as
∂ u ∂ t + a ψ ∂ 2 u ∂ x 2 + b ψ z u = 0 . (5)
where
u = [ v w ] , ψ = [ 0 1 − 1 0 ] .
The log NLS equation’s exact soliton solution is provided by [
u ( x , t ) = A e − B 2 ( x − s t ) 2 e i ( − k x + ζ t + θ ) , (6)
where s = − 2 a k , B = b / 2 a , ζ = 2 b log ( A ) − a k 2 − b , and a b > 0 .
Here A denotes the amplitude of Gaussons and θ represents the center of Gausson’s phase. The wave number from the phase component is expressed by ζ , s is the Gausson velocity which is related to its frequency and B indicates the inverse width.
The three preserved quantities in the log NLS Equation (1) are, [
I 1 = ∫ − ∞ ∞ | u | 2 d x , (7)
I 2 = i ∫ − ∞ ∞ [ u ∗ u x − u u x ∗ ] d x , (8)
I 3 = ∫ − ∞ ∞ [ a | u x | 2 − b | u | 2 log | u | 2 + b | u | 2 ] d x , (9)
where u ∗ represents u’s complex conjugate.
To derive numerical methods for solving Equation (1), numerically, the problem domain is specified as x L ≤ x ≤ x R , 0 ≤ t ≤ T , these coordinates are covered by a rectangular grid of points, x = x m = x L + m h , m = 0 , 1 , 2 , ⋯ , N , t = t n = n k , n = 0 , 1 , 2 , ⋯ , h = x R − x L N , where h and k denote the spatial and temporal increases.
To construct an explicit technique for resolving the log NLS equation, we assume U m n to be the numerical solutions, and u m n to be accurate solutions, at the point ( x m , t n ) , where U m n = [ V m n W m n ] . To solve Equations (3) and (4), by using the basic explicit finite difference method, which uses forward approximation for time and central difference approximation for space, as follows
V m n + 1 − V m n k + a 1 h 2 δ x 2 W m n + b z m n W m n = 0 (10)
W m n + 1 − W m n k − a 1 h 2 δ x 2 V m n − b z m n V m n = 0 (11)
Equations (10) and (11) can be written as
V m n + 1 = V m n − a r [ W m − 1 n − 2 W m n + W m + 1 n ] − b k z m n W m n (12)
W m n + 1 = W m n + a r [ V m − 1 n − 2 V m n + V m + 1 n ] + b k z m n V m n , (13)
where r = k h 2 , m = 1 , 2 , ⋯ , N , n = 0 , 1 , 2 , ⋯ , the resulting system in Equations
(12) and (13) is two level explicit scheme U m n + 1 , U m n , we can solve this system for V m n + 1 , W m n + 1 directly, where we enforce the boundary conditions U 0 n = 0 = U M n .
To test the effectiveness of the suggested method, we examine the rigor of (10). To achieve this, we substitute the numerical solution V m n , W m n with the accurate solution v m n , w m n , and by applying Taylor’s series expansion of all terms in Equation (10), around the point ( x m , t n ) , we get the following expansions.
v m n + 1 − v m n k = ∂ v ∂ t + k 2 ∂ 2 v ∂ t 2 + O ( k 2 ) ,
1 h 2 δ x 2 w m n = ∂ 2 w ∂ x 2 + h 2 12 ∂ 4 w ∂ x 4 + O ( h 4 ) , (14)
b z m n w m n = b z w ,
Equations (14) are substituted into Equation (10) to get
[ ∂ v ∂ t + a ∂ 2 w ∂ x 2 + b z w ] + k 2 ∂ 2 v ∂ t 2 + h 2 12 ∂ 4 w ∂ x 4 + ⋯
The value of the first bracket will be zero, if differential Equation (3) is used, As a result, the local truncation error will be
L T E = k 2 ∂ 2 v ∂ t 2 + h 2 12 ∂ 4 w ∂ x 4 + ⋯ (15)
The schema is second order in space and first order in time. It is possible to research the accuracy of the scheme (11) the same way.
This section looks at the numerical method’s stability, or how sensitive the numerical solution, to examine scheme (1) stability, using the explicit scheme, we write it as follows
i ( U m n + 1 − U m n ) + a r δ x 2 U m n + b k z U m n = 0, (16)
where r = k h 2 . We note that schema (16) is non-linear and the von-Neumann stability analysis can only be used for linear schemas. We can achieve the linear version by freezing the phrase that makes the scheme nonlinear.
i ( U m n + 1 − U m n ) + ( a r δ x 2 + b k λ ) U m n = 0, (17)
where λ = max ( z ) . Now we’ll do a von-Neumann stability analysis, as shown below, let
U m n = e α n k e i β m h , (18)
where α and β are constant, i = − 1 , then
δ x 2 U m n = − 4 sin 2 ( β h 2 ) U m n . (19)
Substitute Equations (18) and (19) into Equation (17) we will get
i ( e α k − 1 ) − 4 a r sin 2 ( β h 2 ) + b k λ = 0,
it can also be expressed as
| e α k | 2 = 1 + [ 4 a r sin 2 ( β h 2 ) − b k λ ] 2 ,
which shows that e α k > 1 , and hence the scheme is unconditionally unstable.
The explicit scheme developed in the preceding part is second order in space and first order in time, but it has an issue with stability since it is unstable. In this part, we will overcome this problem by designing a Crank-Nicolson-like schema that is unconditionally stable and has second-order precision of space and time. We assume that v ( x m , t n ) , w ( x m , t n ) and V ( x m , t n ) , W ( x m , t n ) are the exact and approximate solutions respectively at the point ( x m , t n ) .
The numerical solution to Equations (3) and (4) are obtained by employing a Crank-Nicolson-like technique,
V m n + 1 − V m n k + a 2 h 2 δ x 2 ( W m n + 1 + W m n ) + b ( z m n + 1 + z m n 2 ) ( W m n + 1 + W m n 2 ) = 0, (20)
W m n + 1 − W m n k − a 2 h 2 δ x 2 ( V m n + 1 + V m n ) − b ( z m n + 1 + z m n 2 ) ( V m n + 1 + V m n 2 ) = 0. (21)
Equations (20) and (21) can be written as
V m n + 1 − V m n + r 1 [ ( W m − 1 n + 1 − 2 W m n + 1 + W m + 1 n + 1 ) + ( W m − 1 n − 2 W m n + W m + 1 n ) ] + r 2 ( z m n + 1 + z m n ) ( W m n + 1 + W m n ) = 0 , (22)
W m n + 1 − W m n − r 1 [ ( V m − 1 n + 1 − 2 V m n + 1 + V m + 1 n + 1 ) + ( V m − 1 n − 2 V m n + V m + 1 n ) ] − r 2 ( z m n + 1 + z m n ) ( V m n + 1 + V m n ) = 0 , (23)
where r 1 = a k 2 h 2 , r 2 = b k 4 .
The resultant scheme is an implicit nonlinear scheme that generates a block nonlinear tridiagonal system. The nonlinear system obtained may be solved using a variety of approaches. We will, in this paper, utilize the fixed point method to solve Equations (22) and (23). As follows
V m n + 1 , ( s + 1 ) + r 1 [ W m − 1 n + 1 , ( s + 1 ) − 2 W m n + 1 , ( s + 1 ) + W m + 1 n + 1 , ( s + 1 ) ] + r 2 [ z m n + 1 , ( s ) + z m n ] W m n + 1 , ( s + 1 ) = V m n − r 1 [ W m − 1 n − 2 W m n + W m + 1 n ] − r 2 [ z m n + 1 , ( s ) + z m n ] W m n , (24)
W m n + 1 , ( s + 1 ) − r 1 [ V m − 1 n + 1 , ( s + 1 ) − 2 V m n + 1 , ( s + 1 ) + V m + 1 n + 1 , ( s + 1 ) ] − r 2 [ z m n + 1 , ( s ) + z m n ] V m n + 1 , ( s + 1 ) = W m n + r 1 [ V m − 1 n − 2 V m n + V m + 1 n ] + r 2 [ z m n + 1 , ( s ) + z m n ] V m n . (25)
where s = 0 , 1 , 2 , etc . , The resulting system in Equations (24) and (25) may be expressed as a matrix-vector,
J U n + 1 , ( s + 1 ) = F ( U n + 1 , ( s ) , U n ) (26)
The block tridiagonal matrix J may be expressed in the following way:
J ( u ) = [ A 1 C 1 0 ⋯ 0 B 2 A 2 C 2 ⋱ ⋮ 0 ⋱ ⋱ ⋱ 0 ⋮ ⋱ B n − 1 A n − 1 C n − 1 0 ⋯ 0 B n A n ] ,
hence the elements of the matrix J may be described in detail as follows:
B m = [ 0 r 1 − r 1 0 ] m , A m = [ 1 a 12 a 21 1 ] m , C m = [ 0 r 1 − r 1 0 ] m ,
where
a 12 = − 2 r 1 + r 2 [ z m n + 1 , ( s ) + z m n ] ,
a 21 = 2 r 1 − r 2 [ z m n + 1 , ( s ) + z m n ] ,
m = 1 , 2 , ⋯ , N .
Crout’s method may be used to solve the system in Equation (26) and we take the initial guess as U _ n + 1, ( 0 ) = U _ n . The iteration process is repeated until the following condition is met:
‖ ( U m n + 1 ) ( s + 1 ) − ( U m n + 1 ) ( s ) ‖ ∞ < t o l ,
where “tol” is a small number used to quantify error.
To investigate the system’s accuracy in Equations (20) and (21), we replace the numerical solution V m n , W m n by the exact solution v m n and w m n to get
v m n + 1 − v m n k + a δ x 2 ( w m n + 1 + w m n 2 h 2 ) + b ( z m n + 1 + z m n 2 ) ( w m n + 1 + w m n 2 ) = 0 , (27)
w m n + 1 − w m n k − a δ x 2 ( v m n + 1 + v m n 2 h 2 ) − b ( z m n + 1 + z m n 2 ) ( v m n + 1 + v m n 2 ) = 0. (28)
By expanding the terms in Equation (27), using Taylor’s series, we get,
v m n + 1 − v m n k = ∂ v ∂ t + k 2 ∂ 2 v ∂ t 2 + k 2 6 ∂ 3 v ∂ t 3 + ⋯ ,
1 2 h 2 δ x 2 ( w m n + 1 + w m n ) = ∂ 2 w ∂ x 2 + k 2 ∂ 3 w ∂ x 2 ∂ t + h 2 12 ∂ 4 w ∂ x 4 + k 2 4 ∂ 4 w ∂ x 2 ∂ t 2 + ⋯ ,
( z m n + 1 + z m n 2 ) ( w m n + 1 + w m n 2 ) = z w + k 2 ∂ ∂ t ( z w ) + k 2 4 [ ∂ ∂ t ( z ∂ w ∂ t ) + ∂ 2 z ∂ t 2 w ] + ⋯ , (29)
Then, if we put Equation (29) into Equation (27) we’ll get
( ∂ v ∂ t + k 2 ∂ 2 v ∂ t 2 + k 2 6 ∂ 3 v ∂ t 3 + ⋯ ) + a ( ∂ 2 w ∂ x 2 + k 2 ∂ 3 w ∂ x 2 ∂ t + h 2 12 ∂ 4 w ∂ x 4 + k 2 4 ∂ 4 w ∂ x 2 ∂ t 2 + ⋯ ) + b { z w + k 2 ∂ ∂ t ( z w ) + k 2 4 [ ∂ ∂ t ( z ∂ w ∂ t ) + ∂ 2 z ∂ t 2 w ] + ⋯ }
and it’s also possible to write it as
[ ∂ v ∂ t + a ∂ 2 w ∂ x 2 + b z w ] + k 2 ∂ ∂ t [ ∂ v ∂ t + a ∂ 2 w ∂ x 2 + b z w ] + k 2 2 { 1 3 ∂ 3 v ∂ t 3 + 1 2 ∂ 4 w ∂ x 2 ∂ t 2 + 1 2 [ ∂ ∂ t ( z ∂ w ∂ t ) + ∂ 2 z ∂ t 2 w ] } + a h 2 12 ∂ 4 W ∂ x 4 + ⋯
When differential Equation (3) is used, all terms within the first and second brackets both have a value of zero, resulting in the local truncation error (LTE) as
L T E = k 2 2 { 1 3 ∂ 3 v ∂ t 3 + 1 2 ∂ 4 w ∂ x 2 ∂ t 2 + 1 2 [ ∂ ∂ t ( z ∂ w ∂ t ) + ∂ 2 z ∂ t 2 w ] } + a h 2 12 ∂ 4 W ∂ x 4 + ⋯ , (30)
this shows that in both time and space, the scheme is second-order, It is possible to research the accuracy of the scheme (21) the same way.
The linear form of schema (5), is obtained by freezing the term that makes the scheme nonlinear,
U m n + 1 − U m n + k 2 [ a h 2 ψ δ x 2 + b η ψ ] ( U m n + 1 + U m n ) = 0, (31)
where η = max { z } . Now we’ll do a von-Neumann stability analysis, as shown below.
We assume
U m n = U 0 n e i β m h , U m n + 1 = G U 0 n e i β m h , (32)
where U 0 n is referred to as the constant vector, while G is referred to as the amplification matrix and
δ x 2 U m n = − 4 sin 2 ( β h 2 ) U m n (33)
Substitute Equations (32) and (33) into Equation (31) we will get
[ I + ψ γ ] G = [ I − ψ γ ] (34)
where γ = k 2 [ b η − 4 a h 2 sin 2 ( β h 2 ) ] , Equation (34) may be expressed in the following way:
G = [ I + ψ γ ] − 1 [ I − ψ γ ] (35)
The G matrix’s eigenvalues are
λ 1 = 1 − γ 2 − 2 i γ 1 + γ 2 and λ 2 = 1 − γ 2 + 2 i γ 1 + γ 2 (36)
The max | λ m | ≤ 1 , m = 1 , 2 criterion is required for stability when utilizing von-Neumann. It is clear from (36) that, | λ m | = 1 , and hence the scheme is unconditionally stable.
We will try to solve the cases listed below to see how well the suggested strategies work.
To use the exact solution in Equation (6), we use the initial condition at t = 0 , u ( x , 0 ) = A e − B 2 x 2 e i ( − k x + θ ) , this test makes use of the following parameters: h = 0.1 , k = 0.0001 , s = 0.4 , A = 0.4 , a = 0.5 , b = 1 , θ = 0.5 . We investigate the suggested method’s accuracy by computing the L ∞ -error norm, as defined by
L ∞ = ‖ U n − u n ‖ ∞ = max | U ( x m , t m ) − u ( x m , t m ) | ,
The error analysis of the explicit technique and the Crank-Nicolson method employing L ∞ norms is presented in
| Time | L ∞ ( V ) | L ∞ ( W ) |
|---|---|---|
| 0.00 | 0.000000 | 0.000000 |
| 1.00 | 0.000911 | 0.001779 |
| 2.00 | 0.001946 | 0.001852 |
| 3.00 | 0.004109 | 0.002869 |
| 4.00 | 0.005308 | 0.004220 |
| 5.00 | 0.007105 | 0.006551 |
| Iteration | Time | L ∞ ( V ) | L ∞ ( W ) |
|---|---|---|---|
| 0 | 0.00 | 0.000000 | 0.000000 |
| 2 | 1.00 | 0.000833 | 0.001875 |
| 2 | 2.00 | 0.001344 | 0.001595 |
| 2 | 3.00 | 0.002503 | 0.003054 |
| 2 | 4.00 | 0.002844 | 0.003667 |
| 2 | 5.00 | 0.004200 | 0.003933 |
| Time | I1 | I2 | I3 |
|---|---|---|---|
| 0.00 | 0.200530 | −0.159582 | 0.783269 |
| 1.00 | 0.200703 | −0.159802 | 0.783770 |
| 2.00 | 0.200877 | −0.160041 | 0.784309 |
| 3.00 | 0.201050 | −0.160266 | 0.784816 |
| 4.00 | 0.201223 | −0.160503 | 0.785346 |
| 5.00 | 0.201397 | −0.160733 | 0.785867 |
| Iteration | Time | I1 | I2 | I3 |
|---|---|---|---|---|
| 0 | 0.00 | 0.200530 | −0.159582 | 0.783269 |
| 2 | 1.00 | 0.200530 | −0.159571 | 0.783248 |
| 2 | 2.00 | 0.200530 | −0.159579 | 0.783266 |
| 2 | 3.00 | 0.200530 | −0.159572 | 0.783251 |
| 2 | 4.00 | 0.200530 | −0.159578 | 0.783260 |
| 2 | 5.00 | 0.200530 | −0.159577 | 0.783257 |
In this exam, we will make a collision between two solitons, by using the following form
u ( x , t ) = A ∑ m = 1 2 e − B 2 ( x − x m − s m t ) 2 e i ( − k m ( x − x m ) + ζ m t + θ ) ,
where
s m = − 2 a k m , B = b / 2 a , ζ m = 2 b log ( A ) − a k m 2 − b , m = 1 , 2 and a b > 0.
In this test, the parameters below are utilized.
h = 0.1 , k = 0.0001 , s 1 = 0.8 , s 2 = − 0.4 , A = 0.4 , a = 0.5 , b = 1 , θ = 0.5 , x 1 = 0 , x 2 = 4.5
Detailed numerical study of the interaction between two solitons is reported in
| Time | I1 | I2 | I3 |
|---|---|---|---|
| 0.00 | 0.401069 | −0.159232 | 1.613564 |
| 1.00 | 0.401454 | −0.159432 | 1.614715 |
| 2.00 | 0.401843 | −0.158895 | 1.615874 |
| 3.00 | 0.402258 | −0.158897 | 1.614267 |
| 4.00 | 0.402680 | −0.160935 | 1.619449 |
| 5.00 | 0.403103 | −0.161372 | 1.619203 |
| Iteration | Time | I1 | I2 | I3 |
|---|---|---|---|---|
| 0 | 0.00 | 0.401069 | −0.159233 | 1.613564 |
| 2 | 1.00 | 0.401069 | −0.159125 | 1.613450 |
| 2 | 2.00 | 0.401069 | −0.158250 | 1.613315 |
| 2 | 3.00 | 0.401068 | −0.157637 | 1.610117 |
| 2 | 4.00 | 0.401068 | −0.158945 | 1.613603 |
| 2 | 5.00 | 0.401068 | −0.158667 | 1.611674 |
In this paper, we numerically investigated the log NLS equation using both the explicit and implicit scheme of finite difference method. In comparison to the Explicit approach, we found that the Crank-Nicolson method yields findings that are almost as accurate as possible while still conserving the conserved values. The explicit scheme’s precision is of the second order in space and the first order in time, and it is unstable. The Crank-Nicolson scheme is a second-order spatial and temporal accuracy scheme that is also stable under all conditions. Two solitons are seen interacting with each other. We have observed that the collision between the two solitons is flexible.
The authors state that they have no conflicts of interest to disclose in relation to the current study.
The article contains the supporting data for the study’s findings.
Al-Harbi, A., Al-Hamdan, W. and Wazzan, L. (2022) Numerical Methods for Solving Logarithmic Nonlinear Schrödinger’s Equation. Journal of Applied Mathematics and Physics, 10, 3635-3648. https://doi.org/10.4236/jamp.2022.1012242