# Soliton Content of Fiber-Optic Light Pulses
## 2 The Nonlinear Schrodinger Equation and Some of Its Solutions
### Soliton Solutions
#### 2.1.2 Radiation and Higher-Order Solitons  
To create a soliton, one can launch a sech-shaped pulse as in Equation (3) for \(z=0\) into a fiber with parameters conforming to Equation (4). If, however, the parameters are only approximately right, we can do as follows: In the spirit of [6] and as also detailed in [9] (but they call \(A\) what we call \(N\) as in [1; 2; 10]), we write the initial condition as:  
\[A(0,t)=N\sqrt{P_{0}}\text{ sech}\left(\frac{t}{T_{0}}\right)\quad. \tag{6}\]  
where the pulse is scaled by a factor \(N\geq 0\), \(N\in\mathbb{R}\) called the soliton order. The energy is then:  
\[E=N^{2}E_{\text{sol}}\quad. \tag{7}\]  
Of course, at \(N=1\), there is the fundamental soliton, but at non-integer \(N\), there is also a part of the pulse energy that is not part of the soliton energy. This balance is a linear wave; in the soliton context, it is called radiation. Linear waves are subject to dispersive broadening, so that after a sufficiently long propagation, radiation will be dispersed away from the soliton, and the soliton itself emerges. Until that happens, the different phase evolution of soliton and radiation creates a beat note, visible as a slowly-decaying beat pattern. This is shown for \(N=1.2\) in Figure 2 (left), where lengths are scaled to \(z_{0}=(\pi/2)\,L_{\text{D}}\). Evaluation of the beat pattern can yield soliton parameters, see Section 4.2.  
There is a threshold value of \(N\) to generate a soliton. As long as \(N<\
icefrac{{1}}{{2}}\), all energy goes into radiation. At \(N=1\), the radiative part dips to zero. At \(N=\
icefrac{{3}}{{2}}\), it reaches the same energy as at \(N=\
icefrac{{1}}{{2}}\) again, and that suffices to generate a second soliton. At \(N=2\), there is a combination of two solitons without any radiative part. This structure, for which an explicit expression was already given in [6], is called an \(N=2\) soliton (Figure 2 right). More generally speaking, for any \(N\in\mathbb{N}\), \(N>1\), one has a so-called higher-order soliton of order \(N\) in the pure form, i.e., without radiation. An explicit expression for the \(N=3\) case solitons was found in [11]. The power profile of all higher-order solitons oscillates with period \(z_{0}\). For all non-integer values of \(N\), there is a radiative contribution.  
Figure 3 gives an overview of this situation. In its upper part, the pulse energy according to Equation (7), shown in units of \(E_{\text{sol}}\), is represented by the dashed parabola. Beginning at \(N=\
icefrac{{1}}{{2}}\), the first soliton appears. Its energy then rises linearly and passes through unity at \(N=1\). Similarly, at other half-integer values of \(N\), more solitons begin, each of them contributing energy to a cumulative value shown in blue. The latter is tangent to the parabola at integer \(N\) values; here, the individual soliton energies are also integer in \(E_{\text{sol}}\) units. They take the first \(N\) values of the sequence \(1,3,5,7,\dots\), which add up to \(N^{2}\). For all non-integer \(N\), there is a gap between the cumulative solitonic and the total energy, which represents the radiative part of the energy. Radiation energy is shown in the figure's lower part on an inverted, magnified scale (green curve). Whenever, at some half-integer \(N\), it reaches \(E_{\text{rad}}=\
icefrac{{1}}{{4}}\,E_{\text{sol}}\), the next soliton is created. Always the integer number closest to \(N\) determines  
Figure 2: (Left) If a pulse with soliton order \(N=1.2\) is launched, a slowly decaying beat with radiation lets the soliton emerge gradually; (right) at \(N=2\), one generates a higher-order soliton, a structure with periodic shape oscillation.  
the number of solitons involved, while the fractional part determines the amount of radiation created along with them.