On solitary waves for nonlinear schrödinger equations in higher dimensions

Abstract

We investigate the existence of solitary waves of the nonlinear singular Schrödinger equation

in arbitrary dimensions and prove that the phase φ of any solitary wave type solution u(x,t)= r(x,t).exp(iφ(x,t) is a linear function and can be determined by the velocity of the amplitude r . Thus the problem of finding solitary waves of the Schrödinger equation is equivalent to the investigation of solitary waves of the corresponding nonlinear scalar field equation. Further we prove some necessary conditions in the case χ = 0.

AMS (MOS)::

• 35Q20
• 35F30