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Solitary Waves of a PT-Symmetric Nonlinear Dirac Equation Jesús Cuevas–Maraver University of Seville

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Solitary Waves of a PT-Symmetric Nonlinear Dirac Equation Jesús Cuevas–Maraver University of Seville University of Massachusetts Amherst ScholarWorks@UMass Amherst Mathematics and Statistics Department Faculty Mathematics and Statistics Publication Series 2015 Solitary Waves of a PT-Symmetric Nonlinear Dirac Equation Jesús Cuevas–Maraver University of Seville Panayotis G. Kevrekidis University of Massachusetts Amherst Avadh Saxena Los Alamos National Laboratory Fred Cooper Santa Fe Institute Avinash Khare Savitribai Phule Pune University See next page for additional authors Follow this and additional works at: https://scholarworks.umass.edu/math_faculty_pubs Part of the Mathematics Commons Recommended Citation Cuevas–Maraver, Jesús; Kevrekidis, Panayotis G.; Saxena, Avadh; Cooper, Fred; Khare, Avinash; Comech, Andrew; and Bender, Carl M., "Solitary Waves of a PT-Symmetric Nonlinear Dirac Equation" (2015). Mathematics and Statistics Department Faculty Publication Series. 1244. Retrieved from https://scholarworks.umass.edu/math_faculty_pubs/1244 This Article is brought to you for free and open access by the Mathematics and Statistics at ScholarWorks@UMass Amherst. It has been accepted for inclusion in Mathematics and Statistics Department Faculty Publication Series by an authorized administrator of ScholarWorks@UMass Amherst. For more information, please contact [email protected]. Authors Jesús Cuevas–Maraver, Panayotis G. Kevrekidis, Avadh Saxena, Fred Cooper, Avinash Khare, Andrew Comech, and Carl M. Bender This article is available at ScholarWorks@UMass Amherst: https://scholarworks.umass.edu/math_faculty_pubs/1244 1 Solitary waves of a PT -symmetric Nonlinear Dirac equation Jesus´ Cuevas–Maraver, Panayotis G. Kevrekidis, Avadh Saxena, Fred Cooper, Avinash Khare, Andrew Comech and Carl M. Bender (Invited Paper) Abstract—In the present work, we consider a prototypical analogy of the paraxial approximation of Maxwell’s equations example of a PT -symmetric Dirac model. We discuss the and of the Schrodinger¨ equation formed the basis on which underlying linear limit of the model and identify the threshold the possibility of PT -symmetric realizations initially in optical of the PT -phase transition in an analytical form. We then focus on the examination of the nonlinear model. We consider the waveguide experiments was proposed and then experimentally continuation in the PT -symmetric model of the solutions of the implemented [6]. The success of this program motivated corresponding Hamiltonian model and find that the solutions can further additional initiatives in other directions of experimental be continued robustly as stable ones all the way up to the PT - interest, including, but not limited to, PT -symmetric elec- transition threshold. In the latter, they degenerate into linear tronic circuits [7], [8], mechanical systems [9] and whispering- waves. We also examine the dynamics of the model. Given the stability of the waveforms in the PT -exact phase we consider gallery microcavities [10]. them as initial conditions for parameters outside of that phase. Another theme of research that has been receiving in- We find that both oscillatory dynamics and exponential growth creasing attention recently, both in the physics and in the may arise, depending on the size of the corresponding “quench”. mathematics community is that of the nonlinear Dirac equa- The former can be characterized by an interesting form of bi- tions (NLDEs). While such models were proposed in the frequency solutions that have been predicted on the basis of the SU(1; 1) symmetry. Finally, we explore some special, analytically context of high-energy physics over 50 years ago [11], [12], tractable, but not PT -symmetric solutions in the massless limit they have, arguably, been far less widespread than their non- of the model. relativistic counterpart [13], the nonlinear Schrodinger¨ (NLS) Index Terms—Nonlinear dynamical systems, nonlinear differ- equation [14], [15]. In recent years, however, there has been a ential equations, bifurcation. surge of activity around NLDE models fueled to some extent by analytical solutions and computational issues arising in associated numerical simulations [16]–[19], as well as by the I. INTRODUCTION considerable progress achieved by rigorous techniques towards The study of open systems bearing gain and loss (especially aspects of the spectral, orbital and asymptotic stability of so in a balanced form) is a topic that has emerged over the solitary wave solutions of such models [20]–[25] and towards past two decades as a significant theme of study [1]–[3]. criteria for their spectral stability [26], [27]. Although our em- While the realm of PT -symmetry introduced by Bender and phasis herein will be on the so-called Gross-Neveu model [28] collaborators was originally intended as an alternative to the (sometimes also referred to as the Soler model [29]), we standard Hermitian quantum mechanics, its most canonical also mention in passing that another main stream of activity realizations (beyond the considerable mathematical analysis in this direction has been towards the derivation of NLDEs of the theme in its own right at the level of operators and in the context e.g. of bosonic evolution [30], [31] (or light spectral theory in mathematical physics) emerged elsewhere propagation [32], [33]) in honeycomb optical lattices. In the arXiv:1508.00852v2 [nlin.PS] 1 Oct 2015 in physics. More specifically, in optical systems [4], [5] the latter context, contrary to what we will be focusing on below, there is no nonlinear interaction between the fields (of the J. Cuevas–Maraver is with the Nonlinear Physics Group at Departamento de F´ısica Aplicada I, Universidad de Sevilla. Escuela Politecnica´ Superior, C/ two-component spinor). Virgen de Africa,´ 7, 41011-Sevilla, Spain and the Instituto de Matematicas´ Our aim in the present work is to connect these two de la Universidad de Sevilla (IMUS). Edificio Celestino Mutis. Avda. Reina budding areas of research, namely to propose a prototypical Mercedes s/n, 41012-Sevilla, Spain (email:[email protected]). P.G. Kevrekidis is with the Department of Mathematics and Statis- PT -symmetric nonlinear Dirac equation model (PT-NLDE). tics, University of Massachusetts, Amherst, MA 01003-4515, USA (email: Motivated by the considerable volume of activity, as well [email protected]). as analytical availability of solutions within the Hamiltonian F. Cooper is with the Santa Fe Institute, Santa Fe, NM 87501, USA. A. Khare is with the Department of Physics, Savitribai Phule Pune Univer- limit, we will focus on the Gross-Neveu (or Soler) model. In sity, Pune 411007, India. the next section, we will present the mathematical formulation A. Comech is with the Department of Mathematics, Texas A&M University, of the model. We will analyze its linear limit and discuss the College Station, TX 77843-3368 and the Institute for Information Transmis- sion Problems, Moscow 127994, Russia. existence of a PT -symmetry breaking critical point, i.e., the C.M. Bender is with the Department of Physics, Washington University, St. point of a PT -phase transition. Then, we will turn to the Louis, MO 63130, USA. nonlinear variant of the model exploring the conditions for the P.G. Kevrekidis, A. Saxena and F. Cooper are with the Center for Nonlinear Studies and Theoretical Division, Los Alamos National Laboratory, Los existence of a standing wave solution, as well as discussing Alamos, New Mexico 87545, USA. the linear (spectral) stability setup for such a solution. Finally, 2 we will briefly touch upon the fate of the conservation laws, equivalence of the linear PT -Dirac equation with effective such as the power (squared L2 norm) and the energy. As a mass m~ = pm2 − γ2, as per the above discussion. remarkable feature, we find that the energy remains invariant Having examined the linear states (plane waves) of the within the nonlinear PT-NLDE model, a feature that certainly model, let us now turn to the nonlinear ones, and more distinguishes the model from its NLS counterpart. In Section specifically to standing waves. The relevant coherent structures III we will examine the numerical properties of the standing will be of the form: wave solutions. Remarkably, we will find that these standing wave solutions are stable throughout their interval of existence tending to a linear limit of vanishing amplitude as the linear threshold (i.e., the threshold of the underlying linear model) of the PT -transition is approached. Since no instability is U(x; t) = exp(−iΛt)u(x);V (x; t) = exp(−iΛt)v(x) : encountered within the exact PT -phase, we consider the (2) propagation beyond the relevant critical point (i.e., under a Once such standing wave solutions are calculated (e.g., by quench in the gain-loss parameter γ) finding (a) the possibility fixed point methods as will be discussed in the next section), of oscillatory motion that we identify with a bi-frequency their linear stability is considered by means of a Bogoliubov- state and connect to the invariances of the model and (b) de Gennes linearized stability analysis. We note here, in the possibility of exponential growth. Lastly, a special case passing, that unfortunately, contrary to what is the case for of vanishing mass solutions is analytically identified and also the Hamiltonian limit of the model with γ = 0, for which numerically examined. The remarkable feature in this case explicit solutions exist as: is that these solutions do not respect the PT -symmetry. In Section IV, we summarize our findings and present some interesting directions for future study. s 1=k (m + Λ) cosh2(kβx) (k + 1)β2 u(x)= ; (3a) m + Λ cosh(2kβx) g2(m + Λ cosh(2kβx)) II. MODEL EQUATIONS s (m − Λ) sinh2(kβx) (k + 1)β2 1=k The system of choice in the present context will be the v(x)=sgn(x) (3b); Gross-Neveu model in its generalized PT -symmetric (PT- m + Λ cosh(2kβx) g2(m + Λ cosh(2kβx)) NLDE) form: 2 2 k p i@tU = @xV − g(jUj − jV j ) U + mU + iγV; (1a) (where β = m2 − Λ2) in the present PT-NLDE we have 2 2 k i@tV = −@xU + g(jUj − jV j ) V − mV + iγU: (1b) been unable to identify such explicit solutions (with a notable exception for m = 0 discussed separately below).
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