**3. Field equation for electromagnetics wave propagation: the NLS equation**

The well-known NLS equation involves *u*, the slowly varying envelope of the axial electric field,

$$u\_{\rm x} - \frac{i}{2}D(\mathbf{x})u\_{tt} - i\gamma |u|^2 u = 0 \tag{1}$$

where *D x*ð Þ, *γ*, *x*, and *t* are the dispersion coefficient, self-phase modulation parameters, the spatial propagation distance, and temporal local time, respectively. Using scaling factors, *xo* and *to*, so that the solution may be generally applicable to various physical systems:

$$\mathbf{x}^\* = \frac{\mathbf{x}}{\mathbf{x}\_o}, \mathbf{t}^\* = \frac{\mathbf{t}}{\mathbf{t}\_o} \tag{2}$$

Then, with

$$D^\* = \frac{D\mathbf{x}\_o}{t\_o^2}, u^\* = (\wp \mathbf{x}\_o)^{0.5} u \tag{3}$$

Eq. (1) becomes dimensionless with γ = 1 and the superscript \* omitted.

#### **3.1 Numerical solution method**

The numerical approach is based on the reduction of Eq. (1) into a set of simultaneous first-order ordinary differential equations (ODEs) by the Lanczos-Chebyshev pseudospectral (LCP) method [7, 8], and the set of simultaneous ordinary differential equations (ODEs) is solved by a stable forward marching procedure. The temporal local time, *t*, is mapped into a numerical window [�1, 1]. The solution is written as an economized power series [7, 8]:

$$u(t, \mathbf{x}) = \sum\_{n=0}^{N} u\_n(\mathbf{x}) t^n \tag{4}$$

The derivatives can be obtained from Eq. (4) by means of the term-by-term differentiation to give

$$u'(t, \mathbf{x}) = \sum\_{n=1}^{N} n u\_n(\mathbf{x}) t^{n-1}, \\ u''(t, \mathbf{x}) = \sum\_{n=2}^{N} n(n-1) u\_n(\mathbf{x}) t^{n-2}. \tag{5}$$

Chose *N* – 1 collocation points, *x*i, within the interval [�1,1], that are the roots of the Chebyshev polynomial *TN -2* (*x*) [7, 8]:

$$t\_n = -\cos\left(\frac{(2n+1)\pi}{2(N-2)}\right), n = 0, \dots, N-2\tag{6}$$

Together with the two boundary conditions, a set of *N* + 1 ODEs is obtained.

As *u* is a spike, to give the needed accuracy, the computational domain in the *t*-direction needs to be divided into *K* divisions. Use a *N*-order power series for each subdivision, the set of ODE is in the form:

$$iAU\_x(\mathbf{x}) - iLU(\mathbf{x}) = iQ(\mathbf{x}, U) \tag{7}$$

where *U* is a [(*N + 1) x K*] vector consisting of the coefficients of the power series used. For numerical integration in the x-direction, we have used the unconditionally stable and implicit equations. With step size Δ*x* in the propagation distance,

$$A\left(U^{m+1} + U^m\right) - \frac{i\Delta x}{2}\left[L\left(U^{m+1} + U^m\right)\right] = \frac{i\Delta x}{2}\left[Q\left(\mathbf{x}, U^{m+1}\right) + Q\left(\mathbf{x}, U^m\right)\right] \tag{8}$$

where *U<sup>m</sup>* is the value of *U* at step *m*. Because of the term *Q x*, *U<sup>m</sup>*þ<sup>1</sup> in the righthand side (RHS), Eq. (8) is nonlinear; it must be solved by an iterative procedure. Since both operators, *A* and *L,* at the LHS are linear, for the entire case history, the matrix inversion needs not be done at every step.

#### **3.2 The exactly periodic (EP) soliton solution**

The NLS equation is a robust system that provides countless solutions depending on the many variants, the system parameters, and boundary conditions used [9, 10]. As an initial value problem, the initial input also occupies a vital role. For solutions that may be classified as solitons, they are stationary waves that oscillate and repeat themselves over a soliton period along the propagation distance. However, the period could be controlled by specially designed periodic system parameters. One widely used design is the dispersion managed (DM) systems, where a period consists of two halves in length and each with a dispersion coefficient of opposite sign. Making use of this characteristic, the periodic solution could be found by the shooting method that is an iterative algorithm using for a dispersion map [9, 10]:

$$u\_{in}^{i+1} = \mathbf{0}.\mathbf{5}(u\_{in}^{i} + u\_{out}^{i})\tag{9}$$

where *uin*, and *uou*<sup>t</sup> are the input and output pulse to the dispersion map, respectively, and the superscript *i* denote the iteration number.

The DM solitons are used in the design of long-distance optical transmission systems. They are used in this chapter to find out the propagation characteristic in each half of the dispersion map.

#### **3.3 Bright soliton solution**

It is necessary to have the initial input pulse close to a bright EP soliton [9]. By using trial and error, a Gaussian pulse [9] is chosen:

*Wave Propagation Theory Denies the Big Bang DOI: http://dx.doi.org/10.5772/intechopen.103848*

$$u(t,0) = \beta \exp\left[-a(t-0.5L)^2\right] \tag{10}$$

where *L* is the given length for *t*, *α* is an arbitrarily chosen constant, and *β* is an adjusting parameter to give a specified pulse energy, *E*:

$$E(\boldsymbol{x}) = \int\_{-\frac{L}{2}}^{\frac{L}{2}} \left( \left| u(t, \boldsymbol{x}) \right|^2 \right) dt \tag{11}$$

It is important to set the boundary conditions as:

$$u(t, \mathbf{x}) = \mathbf{1000} \frac{\partial u}{\partial t} - u(t, \mathbf{x}), \text{ at } \mathbf{x} = \pm 0.5L \tag{12}$$

The large constant associated with the derivative term will force *u* to assume a near zero value with zero gradient so that the reflection at the boundaries is eliminated.

An example used *L* = 40, *K* = 10, *N* = 20, *Δx* = 0.001, *D* = 0.1, α = 1.5, and *E* = 0.25. The dispersion map has a length of 6. How the solutions converged to periodic and linear wavelength changes can be seen in the plots in **Figure 1**. The distance, x, shown in this plot is the cumulated distance with each iteration, and the pulse traveled through a distance equal to the dispersion map length of 6 (or 12). As the step size is 0.0005, each iteration generates 12,000 pulse histories, but only every 40th history is shown in **Figure 1**.

The pulse histories show evidence of convergence. Moreover, when doubling the period length from 6.0 to 12.0, the pulse has shown the same linear broadening and narrowing characteristics. **Figure 2** shows the changing pulse shapes in traveling through a period.

**Figure 1.** *The convergence of the iterative method.*

**Figure 2.** *Broadening and narrowing of pulse width through a dispersion map.*

### **3.4 Dark soliton solution**

A dark soliton [10] is obtained if the initial input pulse is taken to be

$$u(t,0) = \beta \exp\left(\mathbf{1} - \left[a(t - \mathbf{0.5L})^2\right]\right) \tag{13}$$

with boundary conditions:

$$
u(t, \pm 0.5L) = 0\tag{14}$$

In an example [10], the followings are used: *D* = 0.4, *L* = 40, *E* = 2.0, and α = 0.15. The constant, *β*, is found from the pulse energy. The dispersion map is 6.0 in length with a positive *D* for the first half and a negative one for the second half. The step size along with the propagation distance, Δx = 0.0005. To cater for the special shape of a soliton and the fact pulse width is changing, the size of the numerical window must be carefully chosen. The observed propagating characteristics of the pulse width are expanding in the first half of the dispersion map, where dispersion is positive and contracting in the second half where dispersion is negative. **Figure 3** shows how the iteration has converge to an EP solution even when the period is increased to twice of its length in the last cycle. Although after 40 iteration cycle, there was still an approximately 0.1% decrease in the pulse energy per cycle; however, the pulse histories show clear evidence of convergence (**Figure 3**). **Figure 4** shows the change of pulse shape in the first half of the dispersion map.

#### **3.5 Calibration to redshift-distance relationship**

In astronomy, redshift, z, is defined by wavelength changes:

**Figure 3.** *Iteration leading to periodic dark soliton solution.*

**Figure 4.** *Transmission of a dark EP soliton in the normal dispersion segment.*

$$z = \frac{\lambda\_2 - \lambda\_1}{\lambda\_1} \tag{15}$$

where λ<sup>1</sup> and λ<sup>2</sup> are the starting and ending wavelength of a spectral line. The redshift-distance relationship [10] is known as the Hubble's law:

**Figure 5.** *Linear relationship between* W *and* x.

$$\mathbf{z} = \mathbf{H}\_o \mathbf{d} / \mathbf{c} \tag{16}$$

where *Ho* is the Hubble's constant and is determined experimentally. Instead of *x*, the distance *d* is in unit of Mpc, while *c*, the velocity of light, is in km/s. As the linear relationship found numerically in Sections 3.2 and 3.3 is in different units, calibration must used to convert the findings to the same as in Eq. (16).

If λ<sup>1</sup> and λ<sup>2</sup> are each proportional to full width at half maximum (FWHM) *W1* and *W2*, then the redshift:

$$\mathbf{z} = \frac{\lambda\_2 - \lambda\_1}{\lambda\_1} = \frac{W\_2 - W\_1}{W\_1} \tag{17}$$

Using a larger scale, the last iterative cycle shown in **Figure 3** is replotted to give **Figure 5**. Based on two selected points, A(241.05, 6.4788) and B(245.55, 11.347), the linear relation is found to be

$$W = -254.34 + 1.082x^\* \tag{18}$$

As both Eq. (16) and (18) are linear, we could use A and B as calibration points and covert Eq. (18) to the same form as Eq. (16). If point A (*xA* = 241.05, *WA* = 64,788) is selected as the reference point, using Eq. (17), redshift *zA* for reaching the point B (*xB* = 245.55, *WB* = 11.347) is

$$z\_A = \frac{W\_B - W\_A}{W\_A} = 0.7514\tag{19}$$

With this amount of redshift and using Eq. (16) with *Ho* = 70.0 km/s/Mpc, an unit often used in astronomy, the distance traveled by the light wave would be

$$(d/c)\_A = \frac{z\_A}{H\_\theta} = 0.01073 \text{ Mpc/c} \tag{20}$$

**Figure 6.** *Redshift,* z*, versus distance from the earth.*

The factor to convert *x* into *d/c*:

$$f\_x = \frac{(d/c)\_A}{(x\_B - X\_{A)}} = 0.002384 \text{ Mpc/c} \tag{21}$$

Applying conversions to the data points over the pulse width expanding segment AB in **Figure 5**, the new calibrated plot, **Figure 6**, confirms that our results have the same linear relationship as given by the Hubble's Law. It should be noted that as numerical solutions contain inaccuracies and noises, some data points are not exactly on the linear line.

To apply, as an example, the numerical simulation to a real physical system [10], that is the space, the dark spectral line due to Lyman-alpha hydrogen has a wavelength of 121.6 nm. The corresponding period is 405 ps. For the chosen reference point, A, *WA* = 6.4788 and the temporal time scaling factor:

$$t\_o = \frac{405}{2\pi W\_A} = 31.25\text{ ps} \tag{22}$$

For the distance scaling factor, xo, using Eq. (20):

$$\mathcal{L}f\_{\mathbf{x}}(\mathbf{x}\_{B} - \mathbf{x}\_{R}) = d\_{R} \tag{23}$$

where *dR* is the physical distance measured from the point corresponding to *xR*, and,

$$d\_{\mathbb{R}} = (\mathfrak{x}\_{\mathbb{B}} - \mathfrak{x}\_{\mathbb{R}}) \mathfrak{x}\_{\circ},$$

If *xA* is the reference point, *xo =cfx* = 0.002384 Mpc, or 9.30 � <sup>10</sup><sup>16</sup> km (which is about 10 light years). Now, an estimation of the dispersion coefficient:

$$D = \frac{t\_o^2 D^\*}{\varkappa\_o} = 2.10 \times 10^{-15} \text{ps}^2/\text{km} \tag{24}$$

There is insufficient information available to work out the self-phase modulation parameter, *γ*. If the light source is the same as our sun, the power of light emitted is known to be 3.9 � <sup>10</sup><sup>26</sup> W. Assuming, just as an order of study, (1) absorption to form the dark soliton is taking place at a distance of 1000 km away from the surface and the law of one over distance square law is used, and (2) the most of the power is in the short and utter-short spectrum and only 10�<sup>10</sup> of the power is associated with the hydrogen absorption spectrum, |*u*| <sup>2</sup> = 3.9 � <sup>10</sup><sup>4</sup> W. Then,

$$\gamma = \frac{\left| u^\* \right|^2}{\varkappa\_o \left| u \right|^2} = \mathbf{1.9} \times \mathbf{10}^{-26} / \text{mW} \tag{25}$$
