3,418 research outputs found

    Periodic Pulses Of Coupled Nonlinear Schrödinger Equations In Optics

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    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

    Approximation of small-amplitude weakly coupled oscillators with discrete nonlinear Schrodinger equations

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    Small-amplitude weakly coupled oscillators of the Klein-Gordon lattices are approximated by equations of the discrete nonlinear Schrodinger type. We show how to justify this approximation by two methods, which have been very popular in the recent literature. The first method relies on a priori energy estimates and multi-scale decompositions. The second method is based on a resonant normal form theorem. We show that although the two methods are different in the implementation, they produce equivalent results as the end product. We also discuss applications of the discrete nonlinear Schrodinger equation in the context of existence and stability of breathers of the Klein--Gordon lattice
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