Solutions to the nonlinear Schroedinger equation carrying momentum along a curve. Part I: study of the limit set and approximate solutions

2007/08/01 by Mahmoudi, Fethi, Malchiodi, Andrea, Montenegro, Marcelo
#34B18 #35B25 #35B34 #35J20 #35J60 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.0708.0125

Abstract

We prove existence of a special class of solutions to the (elliptic) Nonlinear Schroeodinger Equation - ε² Δψ+ V(x) ψ= |ψ|^p-1 ψ, on a manifold or in the Euclidean space. Here V represents the potential, p an exponent greater than 1 and ε a small parameter corresponding to the Planck constant. As ε 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 ε², showing that these conditions appear naturally in a Taylor expansion of the equation in powers of ε. Based on these, an existence result will be proved in the second part.

Related

• Solutions to the nonlinear Schroedinger equation carrying momentum along a curve. Part II: proof of the existence result
• Schrödinger-type equations with sign-changing nonlinearities: a survey
• Kinematics of the Three Moving Space Curves Associated with the Nonlinear Schrödinger Equation
• Solutions concentrating around the saddle points of the potential for Schrödinger equations with critical exponential growth
• Existence of solution for Schr "odinger equation with discontinuous\n nonlinearity and critical Growth