title: Stability of solitary waves in the nonlinear Dirac equation with arbitrary nonlinearity.
abstract: We consider the nonlinear Dirac equation in 1 + 1 dimension with scalar-scalar self interaction g(2)/κ+1(̅ΨΨ)(κ+1) and with mass m. Using the exact analytic form for rest frame solitary waves of the form Ψ(x,t)=ψ(x)e(-iωt) for arbitrary κ, we discuss the validity of various approaches to understanding stability that were successful for the nonlinear Schrödinger equation. In particular we study the validity of a version of Derrick's theorem and the criterion of Bogolubsky as well as the Vakhitov-Kolokolov criterion, and find that these criteria yield inconsistent results. Therefore, we study the stability by numerical simulations using a recently developed fourth-order operator splitting integration method. For different ranges of κ we map out the stability regimes in ω. We find that all stable nonlinear Dirac solitary waves have a one-hump profile, but not all one-hump waves are stable, while all waves with two humps are unstable. We also find that the time t(c), it takes for the instability to set in, is an exponentially increasing function of ω and t(c) decreases monotonically with increasing κ.
authors.affiliation: LMAM and School of Mathematical Sciences, Peking University, Beijing 100871, China.
authors.forename: Sihong
authors.initials: S
authors.lastname: Shao
authors.affiliation: IMUS and Departamento de Física Aplicada I, E.S.P. Universidad de Sevilla, 41011 Sevilla, Spain.
authors.forename: Niurka R
authors.initials: NR
authors.lastname: Quintero
authors.affiliation: Physikalisches Institut, Universität Bayreuth, D-95440 Bayreuth, Germany.
authors.forename: Franz G
authors.initials: FG
authors.lastname: Mertens
authors.affiliation: Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA and The Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, New Mexico 87501, USA.
authors.forename: Fred
authors.initials: F
authors.lastname: Cooper
authors.affiliation: Indian Institute of Science Education and Research, Pune 411008, India.
authors.forename: Avinash
authors.initials: A
authors.lastname: Khare
authors.affiliation: Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA.
authors.forename: Avadh
authors.initials: A
authors.lastname: Saxena
mesh_terms: D017711:Nonlinear Dynamics; D011789:Quantum Theory