title: Variational approach to studying solitary waves in the nonlinear Schrödinger equation with complex potentials.
abstract: We discuss the behavior of solitary wave solutions of the nonlinear Schrödinger equation (NLSE) as they interact with complex potentials, using a four-parameter variational approximation based on a dissipation functional formulation of the dynamics. We concentrate on spatially periodic potentials with the periods of the real and imaginary part being either the same or different. Our results for the time evolution of the collective coordinates of our variational ansatz are in good agreement with direct numerical simulation of the NLSE. We compare our method with a collective coordinate approach of Kominis and give examples where the two methods give qualitatively different answers. In our variational approach, we are able to give analytic results for the small oscillation frequency of the solitary wave oscillating parameters which agree with the numerical solution of the collective coordinate equations. We also verify that instabilities set in when the slope dp(t)/dv(t) becomes negative when plotted parametrically as a function of time, where p(t) is the momentum of the solitary wave and v(t) the velocity.
authors.affiliation: Physikalisches Institut, Universität Bayreuth, D-95440 Bayreuth, Germany.
authors.forename: Franz G
authors.initials: FG
authors.lastname: Mertens
authors.affiliation: Santa Fe Institute, Santa Fe, New Mexico 87501, USA and Center for Nonlinear Studies and Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA.
authors.forename: Fred
authors.initials: F
authors.lastname: Cooper
authors.affiliation: Pontifical Catholic University of Chile, Departamento de Física, Santiago, Región Metropolitana, Chile.
authors.forename: Edward
authors.initials: E
authors.lastname: Arévalo
authors.affiliation: Physics Department, Savitribai Phule Pune University, Pune 411007, India.
authors.forename: Avinash
authors.initials: A
authors.lastname: Khare
authors.affiliation: Center for Nonlinear Studies and Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA.
authors.forename: Avadh
authors.initials: A
authors.lastname: Saxena
authors.affiliation: Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA.
authors.forename: A R
authors.initials: AR
authors.lastname: Bishop