## 4 The Nonlinear Equations (6)-(8)
### The Solitons of Equation (6)
Therefore, even though the two soliton solutions ofEquation (6) have wavenumbers \(C_{1}\) and \(C_{2}\) that intersect these branches (as the dashed lines show in Figure 4), and this fact favors a resonance between the solitons and the radiation modes, the soliton spectrum is practically zero at the frequencies where \(k(\omega)=C_{1,2}\), thus implying that the solitons do not have frequency components capable of resonating with the radiation modes satisfying the resonance condition \(k(\omega)=C_{1,2}\). This _qualitative_ argument is helpful to understand why these _embedded solitons_ do not resonate with the radiation modes, in spite of the fact of having wavenumbers contained in the range of the dispersion relation. However, if desired, it is also possible to use the procedure shown in Refs. [18; 20] to construct a rigorous proof (not just a _qualitative_ one) which shows that the absence of resonances between the solitons and the radiation modes is the consequence of a delicate balance between the linear and the nonlinear terms of Equation (6). This procedure consists in taking the FT of Equation (6), using a function of the form (52) to calculate the FT of the nonlinear terms. In this way, an expression for the FT of \(u(z,t)\) is obtained, which contains a quotient of two polynomials in \(\omega\). The polynomial in the numerator depends on the nonlinear coefficients, and that in the denominator depends on the coefficients of the linear terms. Moreover, the roots of the polynomial that appears in the denominator determines the resonant frequencies. The next step is to prove that if the values of the parameters \(A\), \(B\) and \(C\) which appear in (52) have the values given by the Equations (53)-(55), then the two polynomials cancel out, and the denominator disappears. With the disappearance of this denominator, the possibility of a resonance also disappears, thus explaining why the solution given by Equations (52)-(55) does not resonate with the radiation modes. The above paragraph shows that the soliton solutions of Equation (6) (defined by Equations (52)-(55)) are _embedded solitons_ if the inequality (17) holds. On the other hand, when the inequality (18) holds, the linear dispersion relation (21) presents a band of forbidden wavenumbers (defined by (19) and (20)), and in this case it is not evident if the soliton wavenumbers \(C_{1}\) and \(C_{2}\) are located within the forbidden band (in which case the solitons would be standard), or they lie outside this band (and they are contained within the linear spectrum of the system), in which case they would be _embedded solitons_. A careful comparison of the soliton wavenumbers \(C_{1}\) and \(C_{2}\) given by Equation (55), and the boundaries \(k_{1}\) and \(k_{2}\) of the band of forbidden frequencies (given by Equation (20)), show that:  
\[C_{2}>k_{2}\text{ and }C_{1}<k_{1} \tag{58}\]  
These inequalities imply that the soliton wavenumbers lie _outside_ the forbidden band \(k_{1}<k<k_{2}\), and consequently there are contained within the range of wavenumbers permitted by dispersion  
Figure 23: The continuous curve shows the scaled dispersion relation \(k(\omega)/a\) of Equation (6) with \(c_{0}=1/15\), \(c_{2}=1/2\), \(c_{4}=1/256\), \(\gamma_{1}=5\), \(\gamma_{2}=1\), \(k(\omega)\) given by Equation (21) and \(a=8.6287\). The dashed line shows the Fourier transform of the initial form of the soliton solution (52), with \(A\) and \(B\) given by Equations (53) and (54). relation, thus implying that also in this case (when the inequality (18) holds) the solitons defined by Equations (52)-(55) are _embedded solitons_. In Figure 3 we showed the dispersion relation \(k(\omega)\) defined by Equation (15) when \(c_{0}=1/30\), \(c_{2}=1/2\) and \(c_{4}=1/80\), and the dashed horizontal lines show the position of the soliton wavenumbers \(C_{1}\) and \(C_{2}\) (corresponding to \(\gamma_{1}=5\) and \(\gamma_{2}=1\)), evidencing that these wavenumbers lie outside the forbidden band. Now let us consider the second type of soliton solutions of Equation (6).