**2. 5G and nonlinear Schrodinger equation and covid-19**

#### **2.1 Theoretical frameworks**

The envelope evolution of nonlinear systems is extracted in the form of nonlinear Schrodinger equation (NLSE) from different methods such as Fourier-Mode coupling e.g. [24], or Multiple-Scale analysis [25]. The NLS equation in nonlinear optics was first derived by Kelley in 1965 using a nonlinear electromagnetic wave Maxwell's equation introduced by Chiao et al. one year earlier [26, 27]. Furthermore, Karpman and Krushkal in 1969 derived the NLS equation using Whitham-Lighthill adiabatic approximation [28].

Consider a system described by the normalized nonlinear Schrodinger equation [(1 + 1)D NLSE]:

$$i\frac{\partial\rho}{\partial t} + \frac{1}{2}\frac{\partial^2\rho}{\partial x^2} + f\left(|\rho|^2\right)\rho = 0\tag{1}$$

*Deep Saturation Nonlinearity of 5G Media and Potential Link to Covid-19 DOI: http://dx.doi.org/10.5772/intechopen.98826*

Eq. (1) describes many physical systems, primarily those in which nonlinear waves propagate in isotropic media [29]. The parameter *φ* describes the slowly varying envelope which modulates a fast underlying carrier wave and the nonlinear term *f* j j *φ* <sup>2</sup> � � is specific to the physical system.

On the (1 + 1)D NLSE, two particular forms of nonlinearity are Kerr-type and saturable type that are common in optics [30]. The Kerr-type, where *f* j j *φ* <sup>2</sup> � � <sup>¼</sup> j j *<sup>φ</sup>* 2 and the saturable type, where *f* j j *φ* <sup>2</sup> � � <sup>¼</sup> j j *<sup>φ</sup>* 2 *=* 1 þ j j *φ* <sup>2</sup> � �.

Of course where j j *φ* 2 *=* 1 þ j j *φ* <sup>2</sup> � �≈<sup>1</sup> � <sup>1</sup>*=*j j *<sup>φ</sup>* <sup>2</sup> � �, the nonlinearity is called the "deep saturation nonlinearity" [31] and most of the interesting properties and energy of the solitons is in the regions where j j *φ* <sup>2</sup> > > 1 [31].

If *φ ξ*ð Þ , *τ* is a solution of the (1 + 1)D NLSE in deep saturation nonlinearity, then a whole family of the solutions is obtained [31] by re-scaling via real parameter *ε* as

$$\rho(\xi,\tau) \to e^{it\left(1-e^2\right)}e^{-1}\rho\left(e\xi,e^2\tau\right) \tag{2}$$

Then all solutions of the same order of (1 + 1)D deep saturable NLSE are related each other by rescaling and the solutions are self-similar to one another in their physical properties, such as intensity, shape, etc. "This is because the natural scale in the saturable NLSE is visible only in the margins of the intensity profile of the soliton, and its effect on the shape is tiny" [31].

On the Nonlinear Schrodinger equation, themodulation instability of the background plane wave motivates to create localized breathers [32]. Of course there are various types of the numerical and analytical solutions describing breathers relevant to the definition of the nonlinear term *f* j j *φ* <sup>2</sup> � �, but qualitatively breathers can be modeled in a specific solution [33, 34] as the localized pulses on the continuum background plane wave that:

$$\varphi = \left[ \frac{2(1 - 2a)\cosh\left(\sqrt{8a(1 - 2a)}\tau\right) + i\sqrt{8a(1 - 2a)}\sinh\left(\sqrt{8a(1 - 2a)}\tau\right)}{\sqrt{2a}\cos\left(2\sqrt{1 - 2a}\xi\right) - \cosh\left(\sqrt{8a(1 - 2a)}\tau\right)} \right] \tag{3}$$

Where a < 1/2, the solution is Akhmediev breather and for a = 1/2 the solution is peregrine soliton and for a>1/2 it is Kuznetsov-Ma breather. Of course amplitude of the background wave has been deleted here for reality that the background plane wave is without modulation and thus, the envelopes do work as the independent wave packets.

First solution of the NLSE was the Kuznetsov-Ma (KM) breather [35, 36] and we consider also a Kuznetsov-Ma breather type solution for perturbation of the carrier wave in deep saturation nonlinearity. This solution does show the localized pulsation of the background wave as the periodic noise.

The experimental results in the optical fibers e.g. [5] verify that the Kuznetsov-Ma breather is matched with real term Re f g*φ* and in fact the imaginary term Imf g*φ* can be imaginary and then the physical answer is real term of the Eq. (3), that is,

$$\varphi = \left[ \frac{2(1 - 2a)\cos\left(\sqrt{8a(2a - 1)}\tau\right)}{\sqrt{2a}\cos\left(2\sqrt{1 - 2a}\xi\right) - \cos\left(\sqrt{8a(2a - 1)}\tau\right)} \right] \tag{4}$$

Notice that cosh ð Þ¼ *iτ* cosð Þ*τ* and then in the Kuznetsov-Ma breather solution in which a > 1/2, we have cosh ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>8</sup>*a*ð Þ <sup>1</sup> � <sup>2</sup>*<sup>a</sup>* <sup>p</sup> *<sup>τ</sup>* � � <sup>¼</sup> cos ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>8</sup>*a*ð Þ <sup>2</sup>*<sup>a</sup>* � <sup>1</sup> <sup>p</sup> *<sup>τ</sup>* � �.

However if we use the *φ* included to both of the imaginary and real terms, the solution numerically is still near to the Eq. (4) and difference is neglect-able.
