## On multi-lump solutions to the non-linear Schrödinger equation

### Summary

Summary: 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.

35J60, 35Q55

### Keywords/Phrases

non-linear Schrödinger equation, semi-classical bound state, nonlinear-elliptic equation