Type: Article

Asymptotics for the fourth-order nonlinear Schrodinger equation in 2D

Naumkin, Pavel I.;

Abstract:

Our aim is to study the large time asymptotics of solutions to the fourth-order nonlinear Schrodinger equation in two space dimensions {i partial derivative(t)u + 1/4 Delta(2)u lambda vertical bar u vertical bar(2)u, t > 0, x is an element of R-2, Our(0, x) = u(0) (x), x is an element of R-2, where lambda > 0. We show that the nonlinearity has a dissipative character, so the solutions obtain more rapid time decay rate comparing with the corresponding linear case, if we assume the nonzero total mass condition f(R) u(0)(x)dx not equal 0. We continue to develop the factorization techniques. The crucial points of our approach presented here are the L-2 - estimates of the pseudodifferential operators and the application of the Kato-Ponce commutator estimates.