On multi-lump solutions to the non-linear Schrodinger equation

• Main Author: Robert Magnus
• Format: Article
• Language: English
• Published: Texas State University 1998-11-01
• Series: Electronic Journal of Differential Equations
• Subjects: Non-linear Schrodinger equation; semi-classical bound state; nonlinear-elliptic equation.
• Online Access: http://ejde.math.txstate.edu/Volumes/1998/29/abstr.html

We present a new approach to proving the existence of semi-classical bound states of the non-linear Schrodinger equation which are concentrated near a finite set of non-degenerate critical points of the potential function. The method is based on considering a system of non-linear elliptic equations. The positivity of the solutions is considered. It is shown how the same method yields ``multi-bump'' solutions ``homoclinic'' to an equilibrium point for non-autonomous Hamiltonian equations. The method provides a calculable asymptotic form for the solutions in terms of a small parameter.