Analytic smoothing effect for nonlinear Schrödinger equation in two space dimensions
The unknown function u is a complex-valued function that depends on the time variable and a spatial variable which belongs to a two-dimensional Euclidean space. Nonlinearity is given the function ``right member'' \(f\) which is a complex-valued function satisfying a gauge condition. Also, this function behaves like a cubic power at the origin. The authors state some typical examples of the nonlinearities \(f\). Using a Cauchy condition of the form \(u(0)\) associated to equation (1.1), is obtained a Cauchy problem and the purpose of the present paper is to prove the global existence of solutions for this Cauchy problem. Also, the authors study the analyticity of mentioned solutions. The global existence of solutions and the analyticity of solutions are demonstrated for sufficiently small Cauchy data with exponential decay at infinity. Should be noted that the authors describe the analytic smoothing effects for nonlinear Schrödinger equation by using some Lebesgue spaces of a \(p\)-th integrable function on \(\mathbb R^2\), the usual Sobolev space (or the space of Bessel potentials) and the homogeneous Sobolev space (or the space of Riesz potentials). Also, it is used the Galilei transform.
scientific article; zbMATH DE number 6368440
scientific article