Dynamics of Lump Solutions in a 2 + 1 NLS Equation By Javier Villarroel, J. Prada, and P. G. Est´ evez We derive a class of localized solutions of a 2+1 nonlinear Schr¨ odinger (NLS) equation and study their dynamical properties. The ensuing dynamics of these configurations is a superposition of a uniform, “center of mass” motion and a slower, individual motion; as a result, nontrivial scattering between humps may occur. Spectrally, these solutions correspond to the discrete spectrum of a certain associated operator, comprised of higher-order meromorphic eigenfunctions. 1. Introduction In this paper we derive and study the dynamical properties of a class of solutions of a 2+1-dimensional differential equation with boundary conditions, namely, iu t + u xx + 2u ∂ x  y -∞ ( 1 -|u | 2 ) dy ′ = 0, lim r →∞ |u |(x , y , t ) = 1 (1) where u(x, y, t) is a complex function, depending on three real variables x, y, t and r 2 ≡ x 2 + y 2 . Note that the reduction to the manifold x = y yields the nonlinear Schr¨ odinger (NLS) equation and hence Equation (1) generalizes the latter to the plane. Recall that the NLS equation was formulated in 1968 by Zakharov [1] as an equation describing dynamics of waves in deep water with surface tension; a year later Benney and Roskes [2] showed that the derivation Address for correspondence: J. Villarroel, Facultad de Ciencias, Universidad de Salamanca, Plaza Merced, 37008 Salamanca, Spain; e-mail: javier@usal.es STUDIES IN APPLIED MATHEMATICS 122:395–410 395 C  2009 by the Massachusetts Institute of Technology