1.4. Application of complex transformation to the KMN equation
In order to get optical solutions of Eq. (1.1), the following transformation is selected [33–38]
\(Q(x,\ y,\ t)\ =\ U(\xi)e^{i\eta(x,y,t)}\), (1.2)
where \(U(\xi)\) represents the amplitude component with\(\xi=\ l_{1}x\ +\ l_{2}y\ \ vt,\) the phase component\(\eta\left(x,\ y,\ t\right)\ \)of the soliton is defined as\(\eta\left(x,\ y,\ t\right)=\ -h_{1}x\ -h_{2}y\ +\ \omega t\ +\ \theta_{0}\)with \(i=\sqrt{-1}\). Here\(,\ h_{1}\) and \(h_{2}\) refer to the frequencies of the soliton in the \(x\)- and \(y\)-directions, respectively, while \(\omega\) and \(\theta_{0}\) , respectively, correspond to the wave number and phase of the soliton. Also, the parameters \(l_{1}\) and \(l_{2}\) in Eq. (1.2) represent the inverse width of the soliton along the \(x\) and y directions, respectively, while represents the velocity of the soliton. Plugging the above transformation into KMN equation specified by Eq. (1.1) and decomposing into real and imaginary parts, the following pair of equations are attained:
\(\alpha l_{1}l_{2}U^{\prime\prime}\ -\ (\omega+\ \alpha h_{1}h_{2})\ U\ \ 2\beta h_{1}U^{3}\ =\ 0\ \)(1.3)
and \(v\ =\ -p\ (l_{1}h_{2}\ +\ l_{2}h_{1})\), (1.4)
where a prime denotes the derivative with respect to ξ.