2. Overview of the methods
In this section, we will recount the workflow of the gKM and NAEM to investigate some new optical solutions to a nonlinear evolution equation (NLEE).
Assume a general (2+1)-dimensional NLEE in the following form:
\(P(Q,{Q_{x},Q_{y},Q_{t},Q}_{\text{xx}},Q_{\text{xy}},{Q_{\text{xt}},Q_{\text{yt}},Q_{\text{tt}},Q}_{\text{xtt}},\ldots\ldots)=0\), (2.1)
where \(Q=Q(x,y,t)\) is an unknown function of complex-valued, \(P\)is a polynomial of \(Q(x,y,t)\), and its various partial derivatives, in which the linear and nonlinear partial derivatives are involved.
With the introduction of the transformation\(Q\left(x,y,t\right)=U(\xi)e^{i\eta(x,y,t)},\) where\(\text{\ \ ΞΎ}=\ l_{1}x\ +\ l_{2}y\ \ vt,\) and\(\eta\left(x,\ y,\ t\right)=\ -h_{1}x\ -h_{2}y\ +\ \omega t\ +\ \theta_{0}\), Eq. (2.1) is converted to the following nonlinear ordinary differential equation (ODE):
\(O\left(U,U^{{}^{\prime}},U^{"}\ldots\ldots\ldots.\right)=0\), (2.2)
where \(O\) is a polynomial of \(U\) and its derivatives, and the superscripts indicate the total derivatives with respect to \(\xi\).