Type: Article

On the new critical exponent for the nonlinear Schrodinger equations

Hayashi, Nakao; Naumkin, Pavel I.;

Abstract:

We consider the Cauchy problem for the pth order nonlinear Schrodinger equation in one space dimension {iu(t) + 1/2 u(xx) = vertical bar u vertical bar(p), x is an element of R, iota > 0, u(0,x) = u(0) (x), x is an element of R, where p > ps = 3+root 17/2. We reveal that p = 4 is a new critical exponent with respect to the large time asymptotic behavior of solutions. We prove that if p (s) < p < 4, then the large time asymptotics of solutions essentially differs from that for the linear case, whereas it has a quasilinear character for the case of p > 4.