On the solvability of nonlinear equations of Schrödinger type in the class of rapidly oscillating functions (Q1905241)

In this paper solvability is established and expansions of the solutions at infinity are constructed for the nonlinear Schrödinger equation \[ i\partial_t u+ a(t) \partial^2_x u+ b(u, u^*; x, t)= 0, \] with the initial data \(u(x, t= 0)= v(x)\) which rapidly oscillates at infinity, i.e. the initial data possesses the exponential factor \(\exp(i\alpha x^2)\) for \(x\to \pm \infty\). The problems considered include: construction of formal asymptotic solutions, choice of the functional space, solvability of the problem with decreasing data.