Invariant tori of nonlinear Schrödinger equation
The authors consider an one-dimensional equation of the Schrödinger type: \[ i\;u_t-u_{xx}+iVx)+f\left (|u|^2\right )=0 \tag{1} \] on the cylinder \([0,\pi]\), under the boundary conditions: \[ a_1u(t,0)-b_1u_x(t,0)=0, a_2u(t,\pi)+b_2u_x(t,\pi)=0,\tag{2} \] where \(a_1^2+b_1^2\neq 0\) and \(a_2^2+b_2^2\neq 0\). In equation (1) \(V(x)\) is a prescribed and analytic positive potential and the nonlinearity \(f\) is analytic and satisfies the conditions \(f(0)=0\) and \(f'(0)\neq 0\). The main aim of the paper is to prove that the equation (1) under conditions (2) admits small-amplitude quasi-periodic solutions. This result is a simple extension of the result due to \textit{S. B. Kuksin} and \textit{J. Pöschel} [Ann. Math. (2) 143, 149--179 (1996; Zbl 0847.35130)] who have solved the same problem in the case \(V(x)=m\), where \(m\in \mathbb R\) is a given constant. The author proved that there are many elliptic invariant tori of the problem (1), (2), for a given analytic potential \(V(x)\) which is not necessary to be constant.
scientific article; zbMATH DE number 5551138