Finite-dimensional integrable systems associated with the Davey-Stewartson I equation (Q2730677)

The authors investigate the Davey-Stewartson I (DSI) equation, NEWLINE\[NEWLINE-iu_t=u_{xx}+u_{yy}+2|u|^2u+2(v_1+v_2)u, \quad v_{1,x}-v_{1,y}=v_{2,x}+v_{2,y}=-(|u|^2)_x,NEWLINE\]NEWLINE which is an integrable equation in \(1+2\) dimensions. It turns out that there exists the Lax pair in \((1+1)\)-dimensional form by nonlinear constraints. It is shown that the \((1+1)\)-dimensional system obtained by nonlinearizing the Lax pair of DSI equation can also be nonlinearized to three \((1+0)\)-dimensional Hamiltonian systems with a constraint of Neumann type. A full set of involutive conserved integrals is obtained and their functional independence is proved. These systems are completely integrable in Liouville sense. In case the number of eigenvalues is two, then the systems can be solved directly. A periodic solution of the DSI equation is obtained as an example.