TY - RPRT
T1 - Solutions to the nonlinear Schroedinger equation carrying momentum along a curve. Part I: study of the limit set and approximate solutions
Y1 - 2007
A1 - Fethi Mahmoudi
A1 - Andrea Malchiodi
A1 - Marcelo Montenegro
AB - We prove existence of a special class of solutions to the (elliptic) Nonlinear Schroeodinger Equation $- \\\\epsilon^2 \\\\Delta \\\\psi + V(x) \\\\psi = |\\\\psi|^{p-1} \\\\psi$, on a manifold or in the Euclidean space. Here V represents the potential, p an exponent greater than 1 and $\\\\epsilon$ a small parameter corresponding to the Planck constant. As $\\\\epsilon$ tends to zero (namely in the semiclassical limit) we prove existence of complex-valued solutions which concentrate along closed curves, and whose phase is highly oscillatory. Physically, these solutions carry quantum-mechanical momentum along the limit curves. In this first part we provide the characterization of the limit set, with natural stationarity and non-degeneracy conditions. We then construct an approximate solution up to order $\\\\epsilon^2$, showing that these conditions appear naturally in a Taylor expansion of the equation in powers of $\\\\epsilon$. Based on these, an existence result will be proved in the second part.
UR - http://hdl.handle.net/1963/2112
U1 - 2577
U2 - Mathematics
U3 - Functional Analysis and Applications
ER -