Dynamics and exact traveling wave solutions of the \((2+1)\)-dimensional Davey-Stewartson equation (Q2858058)

Consider the system of partial differential equations NEWLINE\[NEWLINE\begin{gathered} iu_t+ c_0 u_{xx}+ u_{yy}- c_1|u|^2u- c_2 uv_x= 0,\\ v_{xx}+ c_3 v_{yy}- (|u|^2)_x= 0,\end{gathered}\tag{\(*\)}NEWLINE\]NEWLINE where \(u: \mathbb{R}^+\times \mathbb{R}^2\to\mathbb{C}\), \(v: \mathbb{R}^+\times \mathbb{R}^2\to \mathbb{R},\;c_0,\dots, c_4\) are real parameters. The author looks for solutions to \((*)\) of the form NEWLINE\[NEWLINEu(t,x,y)= e^{i\theta}U(\xi),\quad v(t,x,y)= V(\xi),\tag{\(**\)}NEWLINE\]NEWLINE with \(\theta= \delta x+\eta y+\omega t\), \(\xi= k(x+\ell y-\lambda t)\), where \(\delta\), \(\eta\), \(\omega\), \(\ell\), and \(\lambda\) are real parameters, \(\lambda= 2(c_0\delta+ \eta\ell)\). Substituting \((**)\) into \((*)\) one get an ODE system having a first integral. Using this property one finally arrives at the autonomous system NEWLINE\[NEWLINE{d\phi\over d\xi}= y,\quad {dy\over d\xi}= {\alpha\over\gamma} \phi+ {\beta\over\gamma} \phi^3.\tag{\(***\)}NEWLINE\]NEWLINE The author applies methods from the qualitative theory of ODEs and exploits the fact that \((***)\) has a first integral to derive explicit expressions for three types of traveling waves to \((*)\).