Periodic Pulses Of Coupled Nonlinear Schrödinger Equations In Optics

A system of coupled nonlinear Schrödinger equations arising in nonlinear optics is considered. The existence of periodic pulses as well as the stability and instability of such solutions are studied. It is shown the existence of a smooth curve of periodic pulses that are of cnoidal type. The Grillakis, Shatah and Strauss theory is set forward to prove the stability results. Regarding instability a general criteria introduced by Grillakis and Jones is used. The well-posedness of the periodic boundary value problem is also studied. Results in the same spirit of the ones obtained for single quadratic semilinear Schrödinger equation by Kenig, Ponce and Vega are established. Indiana University Mathematics Journal ©.562847877ALBERT, J.P., BONA, J.L., HENRY, D.B., Sufficient conditions for stability of solitary-wave solutions of model equations for long waves (1987) Phys. D, 24, pp. 343-366. , http://dx.doi.org/10.1016/0167-2789(87)90084-4. MR 887857, 89a:35166ALBERT, J.P., BONA, J.L., SAUT, J.-C., Model equations for waves in stratified fluids (1997) Proc. Roy. Soc. London Ser. A, 453, pp. 1233-1260. , MR 1455330 99g:76013ANGULO, J., Non-linear stability of periodic travelling-wave solutions to the Schrödinger and the modified Korteweg-de Vries (2007) J. Diff. Equations, 235, pp. 1-34BANG, O., CLAUSSEN, C.B., KIVSHAR, Y.S., Spatial solitons and induced Kerr effects in quase-phase-matched quadratic media (1995) Phys. Rev. Lett, 197, pp. 4749-4752BOURGAIN, J., Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations (1993) Geom. Funct. Anal, 3, pp. 107-156. , http://dx.doi.org/10.1007/BF01896020. MR 1209299, 95d:35160aBOURGAIN, J., Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. II. The KdV-equation (1993) Geom. Funct. Anal, 3, pp. 209-262. , http://dx.doi.org/10.1007/ BF01895688. MR 1215780, 95d:35160bALEXANDER V. BURYAK and YURI S. KIVSHAR, Solitons due to second harmonic generation, Phys. Lett. A 197 (1995), 407-412, http://dx.doi.org/10.1016/0375-9601(94)00989-3. MR 1314169 (95j:35188)BYRD, P.F., FRIEDMAN, M.D., (1971) Handbook of Elliptic Integrals for Engineers and Scientists, 67. , Die Grundlehren der mathematischen Wissenschaften, Band, Springer-Verlag, New York, Second edition, revised. MR 0277773 43 #3506CRASOVAN, L.-C., LEDERER, F., MIHALACHE, D., Multiple-humped bright solitary waves in second-order nonlinear media (1996) Opt. Eng, 35, pp. 1616-1623DESALVO, R., HAGAN, D.J., SHEIK, M., AHAE, B., STEGEMAN, G., VANHERZEELE, H., VAN STRYLAND, E.W., Self-Focusing and Defocusing by Cascaded Second Order Nonlinearity in KTP (1992) Opt. Lett, 17, pp. 28-30FERRO, P., TRILLO, S., Periodical waves, domain walls, and modulations instability in dispersive quadratic nonlinear media (1995) Phys. Rev. E, 51, pp. 4994-4997. , http://dx.doi.org/10.1103/PhysRevE. 51.4994FRANKEN, P.A., HILL, A.E., PETERS, C.W., WEINREICH, G., Generation of optical harmonics (1961) Phys. Rev. Lett, 7, pp. 118-119. , http://dx.doi.org/10.1103/PhysRevLett.7.118MANOUSSOS, G., (1988) Linearized instability for nonlinear Schrödinger and Klein-Gordon equations, 41, pp. 747-774. , Comm. Pure Appl. Math, MR 948770 89m:35192MANOUSSOS, G., Analysis of the linearization around a critical point of an infinite-dimensional Hamiltonian system (1990) Comm. Pure Appl. Math, 43, pp. 299-333. , MR 1040143 91d:58231MANOUSSOS GRILLAKIS, JALAL SHATAH, and WALTER STRAUSS, Stability theory of solitary waves in the presence of symmetry. I, J. Funct. Anal. 74 (1987), 160-197, http://dx.doi.org/10.1016/0022-1236(87)90044-9. MR 901236 (88g:35169)MANOUSSOS, G., Stability theory of solitary waves in the presence of symmetry. II (1990) J. Funct. Anal, 94, pp. 308-348. , http://dx.doi.org/10.1016/ 0022-1236(90)90016-E. MR 1081647, 92a:35135HE, H., WERNER, M.J., DRUMMOND, P.D., Simultaneous solitary-wave solutions in a nonlinear parametric waveguide (1996) Phys. Rev. E, 54, pp. 896-911. , http://dx.doi.org/10.1103/PhysRevE.54. 896INCE, E.L., The periodic Lame functions (1940) Proc. Roy. Soc. Edinburgh, 60, pp. 47-63. , MR 0002399 2,46cJONES, C.K.R.T., (1988) Instability of standing waves for nonlinear Schrödinger-type equations, 8, pp. 119-138. , Ergodic Theory Dynam. Systems * (, MR 967634 90d:35267KARAMZIN, Y.N., SUKHORUKOV, A.P., Nonlinear interaction of diffracted light beams in a medium with quadratic nolinearity: Mutual focusing of beams and limitation on the efficiency of optical frequency conveners (1974) JETP Lett, 20, pp. 339-342TOSIO, K., (1995) Perturbation Theory for Linear Operators, , Classics in Mathematics, Springer-Verlag, Berlin, ISBN 3-540-58661-X, Reprint of the 1980 edition. MR 1335452 96a:47025CARLOS E. KENIG, GUSTAVO PONCE, and LUIS VEGA, The Cauchy problem for the Koneweg-de Vries equation in Sobolev spaces of negative indices, Duke Math. J. 71 (1993), 1-21, http://dx.doi.org/10.1215/S0012-7094-93-07101-3. MR 1230283 (94g:35196)TOSIO, K., Quadratic forms for the 1-D semilinear Schrödinger equation (1996) Trans. Amer. Math. Soc, 348, pp. 3323-3353. , http://dx.doi.org/10.2307/2154688. MR 1357398, 96j:35233WILHELM, M., STANLEY, W., (1966) Interscience Tracts in Pure and Applied Mathematics, Hill's equation, 20. , Interscience Publishers John Wiley & Sons, New York-London-Sydney, MR 0197830 33 #5991MENYUK, C.R., SCHIEK, R., TORNER, L., Solitary waves due to Χ(2): Χ(2) cascading (1994) J. Opt. Soc. Amer. B, 11, pp. 2434-2443MICHAEL, R., BARRY, S., (1978) Methods of Modern Mathematical Physics. IV. Analysis of Operators, , Academic Press [Harcourt Brace Jovanovich Publishers, New York, ISBN 0-12-585004-2. MR 0493421 58 #12429cALICE C. YEW, Multipulses of nonlinearly coupled Schrödinger equations, J. Differential Equations 173 (2001), 92-137, http://dx.doi.org/10.1006/jdeq.2000.3922. MR 1836246 (2002h:35302)_, Stability analysis of multipulses in nonlinearly-coupled Schrödinger equations, Indiana Univ. Math. J. 49 (2000), 1079-1124, http://dx.doi.org/10.1512/iumj.2000.49.1826. MR 1803222 (2001m:35033)ALICE C. YEW, A.R. CHAMPNEYS, and P.J. MCKENNA, Multiple solitary waves due to second-harmonic generation in quadratic media, J. Nonlinear Sci. 9 (1999), 33-52, http://dx.doi.org/10.1007/s003329900063. MR 1656377 (99i:78025

Critical behavior of long straight rigid rods on two-dimensional lattices: Theory and Monte Carlo simulations

The critical behavior of long straight rigid rods of length $k$ ($k$-mers) on square and triangular lattices at intermediate density has been studied. A nematic phase, characterized by a big domain of parallel $k$-mers, was found. This ordered phase is separated from the isotropic state by a continuous transition occurring at a intermediate density $\theta_c$. Two analytical techniques were combined with Monte Carlo simulations to predict the dependence of $\theta_c$ on $k$, being $\theta_c(k) \propto k^{-1}$. The first involves simple geometrical arguments, while the second is based on entropy considerations. Our analysis allowed us also to determine the minimum value of $k$ ($k_{min}=7$), which allows the formation of a nematic phase on a triangular lattice.Comment: 23 pages, 5 figures, to appear in The Journal of Chemical Physic

Convocation

The origins of phenotypic variation within mimetic Heliconius butterflies have long fascinated biologists and naturalists. However, the evolutionary processes that have generated this extraordinary diversity remain puzzling. Here we examine intraspecific variation across Heliconius cydno diversification and compare this variation to that within the closely related H. melpomene and H. timareta radiations. Our data, which consist of both mtDNA and genome scan from nearly 2250 AFLP loci, reveal a complex history of differentiation and admixture at different geographic scales. Both mtDNA and AFLP phylogenies suggest that H. timareta and H. cydno are probably geographic extremes of the same radiation that likely diverged from H. melpomene during the Pliocene-Pleistocene boundary. MtDNA suggest that this radiation originated in Central America or the Northwestern region of South America, with a subsequent colonization of the eastern and western slopes of the Andes. Our genome-scan data indicate significant admixture among sympatric H. cydno/H.timareta and H. melpomene populations across the extensive geographic ranges of the two radiations. Within H. cydno, both mtDNA and AFLP data indicate significant population structure at local scales, with strong genetic differences even among adjacent H. cydno color pattern races. These genetic patterns highlight the importance of past geoclimatic events, intraspecific gene flow, and local population differentiation in the origin and establishment of new adaptive forms

Constraining the properties of neutron star crusts with the transient low-mass X-ray binary Aql X-1

Aql X-1 is a prolific transient neutron star low-mass X-ray binary that exhibits an accretion outburst approximately once every year. Whether the thermal X-rays detected in intervening quiescent episodes are the result of cooling of the neutron star or due to continued low-level accretion remains unclear. In this work we use Swift data obtained after the long and bright 2011 and 2013 outbursts, as well as the short and faint 2015 outburst, to investigate the hypothesis that cooling of the accretion-heated neutron star crust dominates the quiescent thermal emission in Aql X-1. We demonstrate that the X-ray light curves and measured neutron star surface temperatures are consistent with the expectations of the crust cooling paradigm. By using a thermal evolution code, we find that ~1.2-3.2 MeV/nucleon of shallow heat release describes the observational data well, depending on the assumed mass-accretion rate and temperature of the stellar core. We find no evidence for varying strengths of this shallow heating after different outbursts, but this could be due to limitations of the data. We argue that monitoring Aql X-1 for up to ~1 year after future outbursts can be a powerful tool to break model degeneracies and solve open questions about the magnitude, depth and origin of shallow heating in neutron star crusts.Comment: 14 pages, 5 figures, 3 tables, accepted to MNRA