Nonlinear model of Schrödinger type: Conservation laws, Hamiltonian structure, and complete integrability

A method is proposed for finding Lax type representations for nonlinear evolution (one-dimensional) equations of mathematical physics. It is shown that the Schrödinger type nonlinear model $\psi_t-i\psi_{xx}+2|\psi|^2\psi_x=0$ admits a Lax-type representation and is a Hamiltonian completely integrable dynamical system. Exact quasiperiodic (finite-gap, i.e  having only a finite number of stability bands in its spectrum) solutions of this system are obtained in terms of Riemann theta functions.