Nonlinear model of Schrödinger type: conservation laws, Hamiltonian structure, and complete integrability (Q1822283)

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.