Nonlinear Schrödinger, Infinite Dimensional Tori and Neighboring Tori

of the periodic nonlinear Schrödinger equation, where u(x, t) is a complex valued function in the class of smooth period one functions. In this work, we explain in what sense the generic level set of the constants of motion for the periodic nonlinear Schrödinger equation is an infinite dimensional torus, why the solution of the Hamiltonian equation is almost periodic in time, and describe how neighboring generic infinite dimensional tori are connected. Bourgain [1] has solved the initial value problem for the periodic nonlinear Schrödinger equation. Ma and Ablowitz [2] have reduced the periodic nonlinear Schrödinger equation to an inverse spectral problem for periodic potentials. They provide explicit formulas for the special class of N -soliton solutions of the periodic nonlinear Schrödinger equation and found an infinite sequence of functionals that are in involution and constant along solutions of (1). For the nonlinear Schrödinger equation, Batig et al [3] and Schmidt [4] used the method of inverse spectral theory and integrated the equation in the class of analytic [4] and smooth periodic functions [3]. They identified the generic invariant set of the constants of motion with an infinite dimensional tori. Their study [2, 3] did not describe how neighboring tori are connected. The nonlinear Schrödinger equation is an example of the Hamiltonian equation