Skip to main content Accessibility help
×
Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-26T02:34:12.312Z Has data issue: false hasContentIssue false

References

Published online by Cambridge University Press:  05 June 2012

Mark J. Ablowitz
Affiliation:
University of Colorado, Boulder
Get access

Summary

Image of the first page of this content. For PDF version, please use the ‘Save PDF’ preceeding this image.'
Type
Chapter
Information
Nonlinear Dispersive Waves
Asymptotic Analysis and Solitons
, pp. 334 - 344
Publisher: Cambridge University Press
Print publication year: 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ablowitz, M.J., Bakirtas, I., and Ilan, B. 2005. Wave collapse in nonlocal nonlinear Schrödinger systems. Physica D, 207, 230–253.CrossRefGoogle Scholar
Ablowitz, M.J., and Biondini, G. 1998. Multiscale pulse dynamics in communication systems with strong dispersion-management. Opt. Lett., 23, 1668–1670.CrossRefGoogle ScholarPubMed
Ablowitz, M.J., Biondini, G., Biswas, A., Docherty, A., Hirooka, T., and Chakravarty, S. 2002a. Collision-induced timing shifts in dispersion-managed soliton systems. Opt. Lett., 27, 318–320.CrossRefGoogle Scholar
Ablowitz, M.J., Biondini, G., and Blair, S. 1997. Multi-dimensional pulse propagation in non-resonant ξ(2) materials. Phys. Lett. A, 236, 520–524.CrossRefGoogle Scholar
Ablowitz, M.J., Biondini, G., and Blair, S. 2001a. Nonlinear Schrödinger equations with mean terms in non-resonant multi-dimensional quadratic materials. Phys. Rev. E, 63, 605–620.CrossRefGoogle Scholar
Ablowitz, M.J., Biondini, G., Chakravarty, S., and Horne, R.L. 2003a. Four wave mixing in dispersion-managed return-to-zero systems. J. Opt. Soc. Am. B, 20, 831–845.CrossRefGoogle Scholar
Ablowitz, M.J., Biondini, G., Chakravarty, S., Jenkins, R.B., and Sauer, J.R. 1996. Four-wave mixing in wavelength-division-multiplexed soliton systems: damping and amplification. Opt. Lett., 21, 1646–1648.CrossRefGoogle ScholarPubMed
Ablowitz, M.J., Biondini, G., and Olson, E. 2000b. On the evolution and interaction of dispersion-managed solitons. In: Massive WDM and TDM Soliton Transmission Systems, edited by Akira, Hasegawa. Kyoto, Japan: Kluwer Academic, pp. 362–367.Google Scholar
Ablowitz, M.J., and Clarkson, P.A. 1991. Solitons, Nonlinear Evolution Equations and Inverse Scattering. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Ablowitz, M.J., Docherty, A., and Hirooka, T. 2003b. Incomplete collisions in strongly dispersion-managed return-to-zero communication systems. Opt. Lett., 28, 1191–1193.CrossRefGoogle Scholar
Ablowitz, M.J., and Fokas, A.S. 2003. Complex Variables: Introduction and Applications. Second edition. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Ablowitz, M.J., Fokas, A.S., and Musslimani, Z. 2006. On a new nonlocal formulation of water waves. J. Fluid Mech., 562, 313–344.CrossRefGoogle Scholar
Ablowitz, M.J., and Haberman, R. 1975. Resonantly coupled nonlinear evolution equations. Phys. Rev. Lett., 38, 1185–1188.CrossRefGoogle Scholar
Ablowitz, M.J., Hammack, J., Henderson, D., and Schober, C.M. 2000a. Modulated periodic waves in deep water. Phys. Rev. Lett., 84, 887–890.CrossRefGoogle Scholar
Ablowitz, M.J., Hammack, J., Henderson, D., and Schober, C.M. 2001. Long time dynamics of the modulational instability of deep water waves. Physica D, 152, 416–433.CrossRefGoogle Scholar
Ablowitz, M.J., and Haut, T.S. 2009a. Asymptotic expansions for solitary gravity–capillary waves in two and three dimensions. Proc. R. Soc. Lond. A, 465, 2725–2749.CrossRefGoogle Scholar
Ablowitz, M.J., and Haut, T.S. 2009b. Coupled nonlinear Schrödinger equations from interfacial fluids with a free surface. Theor. Math. Phys., 159, 689–697.CrossRefGoogle Scholar
Ablowitz, M.J., and Haut, T.S. 2010. Asymptotic expansions for solitary gravity-capillary waves in two dimensions. J. Phys. Math Theor., 43, 434005.Google Scholar
Ablowitz, M.J., Hirooka, T., and Biondini, G. 2001b. Quasi-linear optical pulses in strongly dispersion-managed transmission systems. Opt. Lett., 26, 459–461.CrossRefGoogle Scholar
Ablowitz, M.J., and Hirooka, T. 2002. Managing nonlinearity in strongly dispersion-managed optical pulse transmission. J. Opt. Soc. Am. B, 19, 425–439.CrossRefGoogle Scholar
Ablowitz, M.J., Hirooka, T., and Inoue, T. 2002b. Higher order asymptotic analysis of dispersion-managed transmission systems: solitons and their characteristics. J. Opt. Soc. Am. B, 19, 2876–2885.CrossRefGoogle Scholar
Ablowitz, M.J., and Horikis, T.P. 2008. Pulse dynamics and solitons in mode-locked lasers. Phys. Rev. A, 78, 011802.CrossRefGoogle Scholar
Ablowitz, M.J., and Horikis, T.P. 2009a. Solitons and spectral renormalization methods in nonlinear optics. Eur. Phys. J. Special Topics, 173, 147–166.CrossRefGoogle Scholar
Ablowitz, M.J., and Horikis, T.P. 2009b. Solitons in normally dispersive mode-locked lasers. Phys. Rev. A, 79, 063845.CrossRefGoogle Scholar
Ablowitz, M.J., Horikis, T.P., and Ilan, B. 2008. Solitons in dispersion-managed mode-locked lasers. Phys. Rev. A, 77, 033814.CrossRefGoogle Scholar
Ablowitz, M.J., Horikis, T.P., and Nixon, S.D. 2009c. Soliton strings and interactions in mode-locked lasers. Opt. Comm., 282, 4127–4135.CrossRefGoogle Scholar
Ablowitz, M.J., Horikis, T.P., Nixon, S.D., and Zhu, Y. 2009a. Asymptotic analysis of pulse dynamics in mode-locked lasers. Stud. Appl. Math., 122, 411–425.CrossRefGoogle Scholar
Ablowitz, M.J., Ilan, B., and Cundiff, S.T. 2004a. Carrier-envelope phase slip of ultra-short dispersion-managed solitons. Opt. Lett., 29, 1818–1820.CrossRefGoogle Scholar
Ablowitz, M.J., Kaup, D.J., Newell, A.C., and Segur, H. 1973a. Method for solving sine–Gordon equation. Phys. Rev. Lett., 30, 1262–1264.CrossRefGoogle Scholar
Ablowitz, M.J., Kaup, D.J., Newell, A.C., and Segur, H. 1973b. Nonlinear-evolution equations of physical significance. Phys. Rev. Lett., 31, 125–127.CrossRefGoogle Scholar
Ablowitz, M.J., Kaup, D.J., Newell, A.C., and Segur, H. 1974. Inverse scattering transform—Fourier analysis for nonlinear problems. Stud. Appl. Math., 53, 249–315.CrossRefGoogle Scholar
Ablowitz, M.J., Kruskal, M.D., and Segur, H. 1979. A note on Miura's transformation. J. Math. Phys., 20, 999–1003.CrossRefGoogle Scholar
Ablowitz, M.J., Manakov, S.V., and Schultz, C.L. 1990. On the boundary conditions of the Davey–Stewartson equation. Phys. Lett. A, 148, 50–52.CrossRefGoogle Scholar
Ablowitz, M.J., and Musslimani, Z. 2003. Discrete spatial solitions in a diffraction-managed nonlinear waveguide array: a unified approach. Physica D, 184, 276–303.CrossRefGoogle Scholar
Ablowitz, M.J., and Musslimani, Z. 2005. Spectral renormalization method for computing self-localized solutions to nonlinear systems. Opt. Lett., 30, 2140–2142.CrossRefGoogle ScholarPubMed
Ablowitz, M.J., Nixon, S.D., and Zhu, Y. 2009b. Conical diffraction in honeycomb lattices. Phys. Rev. A, 79, 053830.CrossRefGoogle Scholar
Ablowitz, M.J., Prinari, B., and Trubatch, A.D. 2004b. Discrete and Continuous Nonlinear Schrödinger Systems. Cambridge: Cambridge University Press.Google Scholar
Ablowitz, M.J., and Segur, H. 1979. On the evolution of packets of water waves. J. Fluid Mech., 92, 691–715.CrossRefGoogle Scholar
Ablowitz, M.J., and Segur, H. 1981a. Solitons and the Inverse Scattering Transform. Philadelphia: SIAM.CrossRefGoogle Scholar
Ablowitz, M.J., and Villarroel, J. 1991. On the Kadomtsev–Petviashili equation and associated constraints. Stud. Appl. Math., 85, 195–213.CrossRefGoogle Scholar
Ablowitz, M.J., and Wang, X.P. 1997. Initial time layers in Kadomtsev–Petviashili type equations. Stud. Appl. Math., 98, 121–137.CrossRefGoogle Scholar
Abramowitz, M., and Stegun, I.R. 1972. Handbook of Mathematical Functions. Tenth edition. New York: Dover.Google Scholar
Agrawal, G.P. 2001. Nonlinear Fiber Optics. New York: Academic Press.Google Scholar
Agrawal, G.P. 2002. Fiber-Optic Communication Systems. New York: Wiley-Interscience.CrossRefGoogle Scholar
Ahrens, C. 2006. The asymptotic analysis of communications and wave collapse problems in nonlinear optics. Ph.D. thesis, University of Colorado.Google Scholar
Airy, G.B. 1845. Tides and waves. Encyc. Metrop., 192, 241–396.Google Scholar
Akhmediev, N.N., and Ankiewicz, A. 1997. Solitons, Nonlinear Pulses and Beams. London: Chapman & Hall.Google Scholar
Akhmediev, N.N., Soto-Crespo, J.M., and Town, G. 2001. Pulsating solitons, chaotic solitons, period doubling, and pulse coexistence in mode-locked lasers: Complex Ginzburg–Landau equation approach. Phys. Rev. E, 63, 056602.CrossRefGoogle ScholarPubMed
Alfimov, G.L., Kevrekidis, P.G., Konotop, V.V., and Salerno, M. 2002. Wannier functions analysis of the nonlinear Schrödinger equation with a periodic potential. Phys. Rev. E, 66, 046608.CrossRefGoogle ScholarPubMed
Angulo, J., Bona, J.L., Linares, F., and Scialom, F. 2002. Scaling, stability and singularities for nonlinear, dispersive wave equations: the critical case. Nonlinearity, 15, 759–786.CrossRefGoogle Scholar
Batchelor, G.K. 1967. An Introduction to Fluid Dynamics. Cambridge: Cambridge University Press.Google Scholar
Beals, R., and Coifman, R.R. 1984. Scattering and inverse scattering for 1st order systems. Comm. Pure Appl. Math., 37, 39–90.CrossRefGoogle Scholar
Beals, R., and Coifman, R.R. 1985. Inverse scattering and evolution-equations. Comm. Pure Appl. Math., 38, 29–42.CrossRefGoogle Scholar
Beals, R., Deift, P., and Tomei, C. 1988. Direct and Inverse Scattering on the Line. Providence, RI: AMS.CrossRefGoogle Scholar
Bender, C.M., and Orszag, S.A. 1999. Advanced Mathematical Methods for Scientists and Engineers. Berlin: Springer.CrossRefGoogle Scholar
Benjamin, T.B., and Feir, J.F. 1967. The disintegration of wave trains on deep water. Part 1. Theory. J. Fluid Mech., 27, 417–430.CrossRefGoogle Scholar
Benjamin, T.B., Bona, J.L., and Mahony, J.J. 1972. Model equations for long waves in nonlinear dispersive systems. Phil. Trans. Roy. Soc. A, 227, 47–78.CrossRefGoogle Scholar
Benney, D.J. 1966a. Long non-linear waves in fluid flows. J. Math. and Phys., 45, 52–63.CrossRefGoogle Scholar
Benney, D.J. 1966b. Long waves on liquid films. J. Math. and Phys., 45, 150–155.CrossRefGoogle Scholar
Benney, D.J., and Luke, J.C. 1964. Interactions of permanent waves of finite amplitude. J. Math. Phys., 43, 309–313.CrossRefGoogle Scholar
Benney, D.J., and Newell, A.C. 1967. The propagation of nonlinear envelopes. J. Math. and Phys., 46, 133–139.CrossRefGoogle Scholar
Benney, D.J., and Roskes, G.J. 1969. Wave instabilities. Stud. Appl. Math., 48, 377–385.CrossRefGoogle Scholar
Bleistein, N. 1984. Mathematical Methods for Wave Phenomena. New York: Academic Press.Google Scholar
Bleistein, N., and Handelsman, R.A. 1986. Asymptotic Expansions of Integrals. New York: Dover.Google Scholar
Bogoliubov, N.N., and Mitropolsky, Y.A. 1961. Asymptotic Methods in the Theory of Non-linear Oscillations. Second edition. Russian monographs and texts on advanced mathematics and physics, vol. 10. London: Taylor & Francis.Google Scholar
Boussinesq, J.M. 1871. Théorie de l'intumescence appelée onde solitaire ou de translation se propageant dans un canal rectangulaire. Comptes Rendues, Acad. Sci. Paris, 72, 755–759.Google Scholar
Boussinesq, J.M. 1872. Théorie des ondes et des remous qui se propagent le long d'un canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblemant parielles de la surface au fond. J. Math. Pures Appl. Ser. (2), 17, 55–108.Google Scholar
Boussinesq, J.M. 1877. Essai sur la théorie des eaux courantes. Memoires présenté par divers savantes à l'Acad. des Sci. Inst. NAT. France XXIII, 1–680.
Boyd, R.W. 2003. Nonlinear Optics. New York: Academic Press.Google Scholar
Byrd, P.F., and Friedman, M.D. 1971. Handbook of Elliptic Integrals for Engineers and Physicists. Berlin: Springer.CrossRefGoogle Scholar
Calogero, F., and Degasperis, A. 1982. Spectral Transform and Solitons. Amsterdam: Elsevier.Google Scholar
Calogero, F., and Delillo, S. 1989. The Burgers equation on the semi-infinite and finite intervals. Nonlinearity, 2, 37–43.CrossRefGoogle Scholar
Camassa, R., and Holm, D.D. 1993a. An integrable shallow water equation with peaked solitons. Phys. Rev. Lett., 71, 1661–1664.CrossRefGoogle Scholar
Camassa, R., and Holm, D.D. 1993b. An integrable shallow water equation with peaked solitons. Phys. Rev. Lett., 11, 1661–1664.CrossRefGoogle Scholar
Caudrey, P.J. 1982. The inverse problem for a general N × N spectral equation. Physica D, 6, 51–66.CrossRefGoogle Scholar
Chapman, S., and Cowling, T. 1970. Mathematical Theory of Nonuniform Gases. Cambridge: Cambridge University Press.Google Scholar
Chen, M., Tsankov, M.A., Nash, J.M., and Patton, C.E. 1994. Backward volume wave microwave envelope solitons in yttrium iron garnet films. Phys. Rev. B, 49, 12773–12790.CrossRefGoogle ScholarPubMed
Chester, C., Friedman, B., and Ursell, F. 1957. An extension of the method of steepest descents. Proc. Camb. Phil. Soc., 53, 599–611.CrossRefGoogle Scholar
Chong, A., Renninger, W.H., and Wise, F.W. 2008a. Observation of antisymmetric dispersion-managed solitons in a mode-locked laser. Opt. Lett., 33, 1717–1719.CrossRefGoogle Scholar
Chong, A., Renninger, W.H., and Wise, F.W. 2008b. Properties of normal-dispersion femtosecond fiber lasers. J. Opt. Soc. Am. B, 25, 140–148.CrossRefGoogle Scholar
Christodoulides, D.N., and Joseph, R.I. 1998. Discrete self-focusing in nonlinear arrays of coupled waveguides. Opt. Lett., 13(9), 794–796.CrossRefGoogle Scholar
Clarkson, P.A., Fokas, A.S., and Ablowitz, M.J. 1989. Hodograph transformations of linearizable partial differential equations. SIAM J. Appl. Math., 49, 1188–1209.CrossRefGoogle Scholar
Cole, J.D. 1951. On a quasilinear parabolic equation occurring in aerodynamics. Quart. Appl. Math., 9, 225–236.CrossRefGoogle Scholar
Cole, J.D. 1968. Perturbation Methods in Applied Mathematics. London: Ginn and Co.Google Scholar
Copson, E.T. 1965. Asymptotic Expansions. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Courant, R., Friedrichs, K., and Lewy, H. 1928. On the partial differential equations of mathematical physics. Math. Ann., 100, 32–74. English Translation, IBM Journal, 11:215–234, 1967.CrossRefGoogle Scholar
Courant, R., and Hilbert, D. 1989. Methods of Mathematical Physics. New York: John Wiley.Google Scholar
Craig, W., and Groves, M.D. 1994. Hamiltonian long-wave approximations to the water-wave problem. Wave Motion, 19, 367–389.CrossRefGoogle Scholar
Craig, W., and Sulem, C. 1993. Numerical simulation of gravity-waves. J. Comp. Phys., 108, 73–83.CrossRefGoogle Scholar
Crasovan, L.C., Torres, J.P., Mihalache, D., and Torner, L. 2003. Arresting wave collapse by self-rectification. Phys. Rev. Lett., 9, 063904.Google Scholar
Cundiff, S.T. 2005. Soliton dynamics in mode-locked lasers. Lect. Notes Phys., 661, 183–206.CrossRefGoogle Scholar
Cundiff, S.T., Ye, J., and Hall, J. 2008. Rulers of light. Scientific American, April, 74–81.
Davey, A., and Stewartson, K. 1974. On three dimensional packets of surface waves. Proc. Roy. Soc. London A, 338, 101–110.CrossRefGoogle Scholar
Deift, P., and Trubowitz, E. 1979. Inverse scattering on the line. Comm. Pure Appl. Math., 32, 121–251.CrossRefGoogle Scholar
Deift, P., Venakides, S., and Zhou, X. 1994. The collisionless shock region for the long-time behavior of solutions of the KdV equation. Comm. Pure Appl. Math., 47, 199–206.CrossRefGoogle Scholar
Deift, P., and Zhou, X. 1992. A steepest descent method for oscillatory Riemann–Hilbert problems. Bull. Amer. Math. Soc., 26, 119–123.CrossRefGoogle Scholar
Dickey, L.A. 2003. Soliton Equations and Hamiltonian Systems. Singapore: World Scientific.CrossRefGoogle Scholar
Djordjevic, V.D., and Redekopp, L.G. 1977. On two-dimensional packets of capillary–gravity waves. J. Fluid Mech., 79, 703–714.CrossRefGoogle Scholar
Docherty, A. 2003. Collision induced timing shifts in wavelength-division-multiplexed optical fiber communications systems. Ph.D. thesis, University of New South Wales.Google Scholar
Dodd, R.K., Eilbeck, J.C., Gibbon, J.D., and Morris, H.C. 1984. Solitons and Nonlinear Wave Equations. New York: Academic Press.Google Scholar
Douxois, T. 2008. Fermi–Pasta–Ulam and a mysterious lady, Physics Today, Jan. 2008, 55–57.
Dysthe, K.B. 1979. Note on a modification to the nonlinear Schrödinger equation for application to deep water waves. Proc. Roy. Soc. London A, 369, 105–114.CrossRefGoogle Scholar
Elgin, J.N. 1985. Inverse scattering theory with stochastic initial potentials. Phys. Lett. A, 110, 441–443.CrossRefGoogle Scholar
Erdelyi, A. 1956. Asymptotic Expansions. New York: Dover.Google Scholar
Faddeev, L.D. 1963. The inverse problem in the quantum theory of scattering. J. Math. Phys., 4, 72–104.CrossRefGoogle Scholar
Faddeev, L.D., and Takhtajan, L.A. 1987. Hamiltonian Methods in the Theory of Solitons. Berlin: Springer.CrossRefGoogle Scholar
Falcon, E., Laroche, C., and Fauve, S. 2002. Observation of depression solitary surface waves on a thin fluid layer. Phys. Rev. Lett., 89, 204501.CrossRefGoogle ScholarPubMed
Fermi, E., Pasta, S., and Ulam, S. 1955. Studies of nonlinear problems, I. Tech. rept. Los Alamos Report LA1940. [Reproduced in “Nonlinear Wave Motion,” proceedings, Potsdam, New York, 1972, ed. A.C. Newell, Lect. Appl. Math., 15, 143–156, A.M.S., Providence, RI, (1974).].Google Scholar
Fibich, G., and Papanicolaou, G. 1999. Self-focusing in the perturbed and unperturbed nonlinear Schrödinger equation in critical dimension. SIAM J. App. Math., 60, 183–240.CrossRefGoogle Scholar
Fokas, A.S. 2008. A Unified Approach to Boundary Value Problems. Philadelphia: SIAM.CrossRefGoogle Scholar
Fokas, A.S., and Anderson, R.L. 1982. On the use of isospectral eigenvalue problems for obtaining hereditary symmetries for Hamiltonian-systems. J. Math. Phys., 23, 1066–1073.CrossRefGoogle Scholar
Fokas, A.S., and Fuchssteiner, B. 1981. Symplectic structures, their Bäcklund transformations and hereditary symmetries. Physica D, 4, 47–66.Google Scholar
Fokas, A.S., and Santini, P.M. 1989. Coherent structures in multidimensions. Phys. Rev. Lett., 63, 1329–1333.CrossRefGoogle ScholarPubMed
Fokas, A.S., and Santini, P.M. 1990. Dromions and a boundary value problem for the Davey–Stewartson I equation. Physica D, 44, 99–130.CrossRefGoogle Scholar
Forsyth, A.R. 1906. Theory of Differential Equations. Part IV—Partial Differential Equations. Cambridge: Cambridge University Press.Google Scholar
Gabitov, I.R., and Turitsyn, S.K. 1996. Averaged pulse dynamics in a cascaded transmission system with passive dispersion compensation. Opt. Lett., 21, 327–329.CrossRefGoogle Scholar
Garabedian, P. 1984. Partial Differential Equations. New York: Chelsea.Google Scholar
Gardner, C.S., Greene, J., Kruskal, M., and Miura, R.M. 1967. Method for solving the Korteweg–de Vries equation. Phys. Rev. Lett., 19, 1095–1097.CrossRefGoogle Scholar
Gardner, C.S., and Su, C.S. 1969. The Korteweg–de Vries equation and generalizations. III. J. Math. Phys., 10, 536–539.Google Scholar
Gardner, C.S., Greene, J.M., Kruskal, M.D., and Miura, R.M. 1974. Korteweg–de Vries and generalizations. VI. Methods for exact solution. Commun. Pure Appl. Math., 27, 97–133.CrossRefGoogle Scholar
Gel'fand, I.M., and Dickii, L.A. 1977. Resolvants and Hamiltonian systems. Func. Anal. Appl., 11, 93–104.CrossRefGoogle Scholar
Goldstein, H. 1980. Classical Mechanics. Reading, MA: Addison Wesley.Google Scholar
Gordon, J.P., and Haus, H.A. 1986. Random walk of coherently amplified solitons in optical fiber transmission. Opt. Lett., 11, 665–667.CrossRefGoogle ScholarPubMed
Hasegawa, A., and Kodama, Y. 1991a. Guiding-center soliton. Phys. Rev. Lett., 66, 161–164.CrossRefGoogle Scholar
Hasegawa, A., and Kodama, Y. 1991b. Guiding-center soliton in fibers with periodically varying dispersion. Opt. Lett., 16, 1385–1387.CrossRefGoogle Scholar
Hasegawa, A., and Kodama, Y. 1995. Solitons in Optical Communications. Oxford: Oxford University Press.Google Scholar
Hasegawa, A., and Matsumoto, M. 2002. Optical Solitons in Fibers. Berlin: Springer.Google Scholar
Hasegawa, A., and Tappert, F. 1973a. Transmission of stationary nonlinear optical pulses in dispersive dielectric fibres. I. Anomalous dispersion. Appl. Phys. Lett., 23, 142–144.CrossRefGoogle Scholar
Hasegawa, A., and Tappert, F. 1973b. Transmission of stationary nonlinear optical pulses in dispersive dielectric fibres. II. Normal dispersion. Appl. Phys. Lett., 23, 171–172.CrossRefGoogle Scholar
Haus, H.A. 1975. Theory of mode locking with a fast saturable absorber. J. Appl. Phys., 46, 3049–3058.CrossRefGoogle Scholar
Haus, H.A. 2000. Mode-locking of lasers. IEEE J. Sel. Topics Q. Elec., 6, 1173–1185.CrossRefGoogle Scholar
Haus, H.A., Fujimoto, J.G., and Ippen, E.P. 1992. Analytic theory of additive pulse and Kerr lens mode locking. IEEE J. Quant. Elec., 28, 2086–2096.CrossRefGoogle Scholar
Haut, T.S., and Ablowitz, M.J. 2009. A reformulation and applications of interfacial fluids with a free surface. J. Fluid Mech., 631, 375–396.CrossRefGoogle Scholar
Hopf, E. 1950. The partial differential equation ut + uux = μuxx. Comm. Pure Appl. Math., 3, 201–230.CrossRefGoogle Scholar
Ilday, F.Ö., Buckley, J.R., Clark, W.G., and Wise, F.W. 2004b. Self-similar evolution of parabolic pulses in a laser. Phys. Rev. Lett., 92, 213901.CrossRefGoogle Scholar
Ilday, F.Ö., Wise, F.W., and Kaertner, F.X. 2004a. Possibility of self-similar pulse evolution in a Ti:sapphire laser. Opt. Express, 12, 2731–2738.CrossRefGoogle Scholar
Infeld, E., and Rowlands, G. 2000. Nonlinear Waves, Solitons and Chaos. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Ishimori, Y. 1981. On the modified Korteweg–deVries soliton and the loop soliton. J. Phys. Soc. Jpn., 50, 2471–2472.CrossRefGoogle Scholar
Ishimori, Y. 1982. A relationship between the Ablowitz–Kaup–Newell–Segur and Wadati–Konno–Ichikawa schemes of the inverse scattering method. J. Phys. Soc. Jpn., 51, 3036–3041.CrossRefGoogle Scholar
Jackson, J.D. 1998. Classical Electrodynamics. New York: John Wiley.Google Scholar
Jeffreys, H., and Jeffreys, B.S. 1956. Methods of Mathematical Physics. Cambridge: Cambridge University Press.Google Scholar
Kadomtsev, B.B., and Petviashvili, V.I. 1970. On the stability of solitary waves in weakly dispersive media. Sov. Phys. Doklady, 15, 539–541.Google Scholar
Kalinikos, B.A., Kovshikov, N.G., and Patton, C.E. 1997. Decay free microwave envelope soliton pulse trains in yttrium iron garnet thin films. Phys. Rev. Lett., 78, 2827–2830.CrossRefGoogle Scholar
Kalinikos, B.A., Scott, M.M., and Patton, C.E. 2000. Self generation of fundamental dark solitons in magnetic films. Phys. Rev. Lett., 84, 4697–4700.CrossRefGoogle ScholarPubMed
Kapitula, T., Kutz, J.N., and Sandstede, B. 2002. Stability of pulses in the master mode-locking equation. J. Opt. Soc. Am. B, 19, 740–746.CrossRefGoogle Scholar
Karpman, V.I., and Solov'ev, V.V. 1981. A perturbation theory for soliton systems. Physica D, 3, 142–164.CrossRefGoogle Scholar
Kay, I., and Moses, H.E. 1956. Reflectionless transmission through dielectrics and scattering potentials. J. Appl. Phys., 27, 1503–1508.CrossRefGoogle Scholar
Kevorkian, J., and Cole, J.D. 1981. Perturbation Methods in Applied Mathematics. Berlin: Springer.CrossRefGoogle Scholar
Konno, K., and Jeffrey, A. 1983. Some remarkable properties of two loop soliton solutions. J. Phys. Soc. Jpn., 52, 1–3.CrossRefGoogle Scholar
Konopelchenko, B.G. 1993. Solitons in Multidimensions. Singapore: World Scientific.CrossRefGoogle Scholar
Korteweg, D., and de Vries, G. 1895. On the change of a form of long waves advancing in a rectangular canal and a new type of long stationary waves. Phil. Mag., 5th Series, 422–443.
Kruskal, M.D. 1963. Asymptotology. In: Proceedings of Conference on Mathematical Models on Physical Sciences, edited by S., Drobot. Upper Saddle River, NJ: Prentice-Hall.Google Scholar
Kruskal, M.D. 1965. Asymptotology in numerical computation: Progress and plans on the Fermi–Pasta–Ulam problem. In: IBM Scientific Computing Symposium on Large-Scale Problems in Physics, pp. 43–62
Krylov, N.M., and Bogoliubov, N.N. 1949. Introduction to Non-Linear Mechanics. Annals of Mathematics Studies, vol. 11. Princeton: Princeton University Press.Google Scholar
Kutz, J.N. 2006. Mode-locked soliton lasers. SIAM Rev., 48, 629–678.CrossRefGoogle Scholar
Kuzmak, G.E. 1959. Asymptotic solutions of nonlinear second order differential equations with variable coeffcients. PMM, 23(3), 515–526.Google Scholar
Lamb, H. 1945. Hydrodynamics. New York: Dover.Google Scholar
Landau, L.D., and Lifshitz, L.M. 1981. Quantum Mechanics: Non-relativistic Theory. London: Butterworth–Heinemann.Google Scholar
Landau, L.D., Lifshitz, E.M., and Pitaevskii, L.P. 1984. Electrodynamics of Continuous Media. London: Butterworth–Heinemann.Google Scholar
Lax, P.D. 1968. Integrals of nonlinear equations of evolution and solitary waves. Commun. Pure Appl. Math., 21, 467–490.CrossRefGoogle Scholar
Lax, P.D. 1987. Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves. Philadelphia: SIAM.Google Scholar
Lax, P.D., and Levermore, C.D. 1983a. The small dispersion limit of the Korteweg–de Vries equation. III. Commun. Pure Appl. Math., 36, 809–829.CrossRefGoogle Scholar
Lax, P.D., and Levermore, C.D. 1983b. The small dispersion limit of the Korteweg–de Vries equation. I. Commun. Pure Appl. Math., 36, 253–290.CrossRefGoogle Scholar
Lax, P.D., and Levermore, C.D. 1983c. The small dispersion limit of the Korteweg–de Vries equation. II. Commun. Pure Appl. Math., 36, 571–593.CrossRefGoogle Scholar
Leveque, R.J. 2002. Finite Volume Methods for Hyperbolic Problems. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Lighthill, M.J. 1958. Introduction to Fourier Analysis and Generalized Functions. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Lighthill, M.J. 1978. Waves in Fluids. Cambridge: Cambridge University Press.Google Scholar
Lin, C., Kogelnik, H., and Cohen, L.G. 1980. Optical-pulse equalization of low-dispersion transmission in single-mode fibers in the 1.3–1.7μm spectral region. Opt. Lett., 5, 476–478.CrossRefGoogle ScholarPubMed
Luke, J.C. 1966. A perturbation method for nonlinear dispersive wave problems. Proc. R. Soc. Lond. A, 292, 403–412.CrossRefGoogle Scholar
Lushnikov, P.M. 2001. dispersion-managed solitons in the strong dispersion map limit. Opt. Lett., 26, 1535–1537.CrossRefGoogle Scholar
Mamyshev, P.V., and Mollenauer, L.F. 1996. Pseudo-phase-matched four-wave mixing in soliton wavelength-division multiplexing transmission. Opt. Lett., 21, 396–398.CrossRefGoogle ScholarPubMed
Marchenko, V.A. 1986. Sturm–Liouville Operators and Applications. Basel: Birkhauser.CrossRefGoogle Scholar
Maruta, A., Inoue, T., Nonaka, Y., and Yoshika, Y. 2002. Bi-solitons propagating in dispersion-managed transmission systems. IEEE J. Sel. Top. Quant. Electron., 8, 640–650.CrossRefGoogle Scholar
Matveev, V.B., and Salle, M.A. 1991. Darboux Transformations and Solitons. Berlin: Springer.CrossRefGoogle Scholar
Melin, A. 1985. Operator methods for inverse scattering on the real line. Commun. Part. Diff. Eqns., 10, 677–766.CrossRefGoogle Scholar
Merle, F. 2001. Existence of blow-up solutions in the energy space for the critical generalized KdV equation. J. Amer. Math. Soc., 14, 555–578.CrossRefGoogle Scholar
Mikhailov, A.V. 1979. Integrability of a two-dimensional generalization of the Toda chain. Sov. Phys. JETP Lett., 30, 414–418.Google Scholar
Mikhailov, A.V. 1981. The reduction problem and the inverse scattering method. Physica D, 3, 73–117.CrossRefGoogle Scholar
Miles, J.W. 1981. The Korteweg–de Vries equation: An historical essay. J. Fluid Mech., 106 (focus issue), 131–147.CrossRefGoogle Scholar
Miura, R.M. 1968. Korteweg–de Vries equation and generalizations. I. A remarkable explicit nonlinear transformation. J. Math. Phys., 9, 1202–1204.CrossRefGoogle Scholar
Miura, R.M., Gardner, C.S., and Kruskal, M.D. 1968. Korteweg–de Vries equations and generalizations. II. Existence of conservation laws and constants of motion. J. Math. Phys., 9, 1204–1209.CrossRefGoogle Scholar
Mollenauer, L.F., Evangelides, S.G., and Gordon, J.P. 1991. Wavelength division multiplexing with solitons in ultra-long distance transmission using lumped amplifiers. J. Lightwave Technol., 9, 362–367.CrossRefGoogle Scholar
Molleneauer, L.F., and Gordon, J.P. 2006. Solitons in Optical Fibers: Fundamentals and Applications to Telecommunications. New York: Academic Press.Google Scholar
Mollenauer, L.F., Stolen, R.H., and Gordon, J.P. 1980. Experimental observation of picosecond pulse narrowing and solitons in optical fibers. Phys. Rev. Lett., 45, 1095–1098.CrossRefGoogle Scholar
Newell, A.C. 1978. The general structure of integrable evolution equations. Proc. Roy. Soc. Lond. A., 365, 283–311.CrossRefGoogle Scholar
Newell, A.C. 1985. Solitons in Mathematics and Physics. Philadelphia: SIAM.CrossRefGoogle Scholar
Nijhof, J.H.B., Doran, N.J., Forysiak, W., and Knox, F.M. 1997. Stable soliton-like propagation in dispersion-managed systems with net anomalous, zero and normal dispersion. Electron. Lett., 33, 1726–1727.CrossRefGoogle Scholar
Nijhof, J.H.B., Forysiak, W., and Doran, N.J. 2002. The averaging method for finding exactly periodic dispersion-managed solitons. IEEE J. Sel. Topics Q. Elec., 6, 330–336.CrossRefGoogle Scholar
Novikov, S.P., Manakov, S.V., Pitaevskii, L.P., and Zakharov, V.E. 1984. Theory of Solitons: The Inverse Scattering Method. New York: Plenum.Google Scholar
Ostrovsky, L.A., and Potapov, A.S. 1986. Modulated Waves: Theory and Applications. Baltimore: The John Hopkins University Press.Google Scholar
Papanicolaou, G., Sulem, C., Sulem, P.L., and Wang, X.-P. 1994. The focusing singularity of the Davey-Stewartson equations for gravity-capillary surface waves. Physica D, 72, 61–86.CrossRefGoogle Scholar
Patton, C.E., Kabos, P., Xia, H., Kolodin, P.A., Zhang, H.Y., Staudinger, R., Kalinikos, B.A., and Kovshikov, N.G. 1999. Microwave magnetic envelope solitons in thin ferrite films. J. Mag. Soc. Japan, 23, 605–610.CrossRefGoogle Scholar
Pelinovsky, D.E., and Stepanyants, Y.A. 2004. Convergence of Petviashvili's iteration method for numerical approximation of stationary solutions of nonlinear wave equations. SIAM J. Numer. Anal., 42, 1110–1127.CrossRefGoogle Scholar
Pelinovsky, D.E., and Sulem, C. 2000. Spectral decomposition for the Dirac system associated to the DSII equation. Inverse Problems, 16, 59–74.CrossRefGoogle Scholar
Petviashvili, V.I. 1976. Equation of an extraordinary soliton. Sov. J. Plasma Phys., 2, 257–258.Google Scholar
Phillips, O.M. 1977. The Dynamics of the Upper Ocean. Cambridge: Cambridge University Press.Google Scholar
Prinari, B., Ablowitz, M.J., and Biondini, G. 2006. Inverse scattering for the vector nonlinear Schrödinger equation with non-vanishing boundary conditions. J. Math. Phys., 47, 1–33.CrossRefGoogle Scholar
Quraishi, Q., Cundiff, S.T., Ilan, B., and Ablowitz, M.J. 2005. Dynamics of nonlinear and dispersion-managed solitons. Phys.Rev.Lett., 94, 243904.CrossRefGoogle Scholar
Rabinovich, M.I., and Trubetskov, D.I. 1989. Oscillations and Waves in Linear and Nonlinear Systems. Dordsecht: Kluwer Academic.CrossRefGoogle Scholar
Rayleigh, Lord. 1876. On waves. Phil. Mag., 1, 257–279.Google Scholar
Remoissenet, M. 1999. Waves Called Solitons. Berlin: Springer.CrossRefGoogle Scholar
Rogers, C., and Schief, W.K. 2002. Bäcklund and Darboux Transformations. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Russell, J.S. 1844. Report on Waves. In: Report of the 14th meeting of the British Association for the Advancement of Science. London: John Murray, pp. 311–390.Google Scholar
Sanders, J.A., Verhulst, F., and Murdock, J. 2009a. Averaging Methods in Nonlinear Dynamical Systems. Berlin: Springer.Google Scholar
Sanders, M.Y., Birge, J., Benedick, A., Crespo, H.M., and KÄrtner, F.X. 2009b. Dynamics of dispersion-managed octave-spanning titanium: sapphire lasers. J. Opt. Soc. Am. B, 26, 743–749.CrossRefGoogle Scholar
Satsuma, J., and Ablowitz, M.J. 1979. Two-dimensional lumps in nonlinear dispersive systems. J. Math. Phys., 20, 1496–1503.CrossRefGoogle Scholar
Shimizu, T., and Wadati, M. 1980. A new integrable nonlinear evolution equation. Prog. Theor. Phys., 63, 808–820.CrossRefGoogle Scholar
Stokes, G.G. 1847. On the theory of oscillatory waves. Camb. Trans., 8, 441–473.Google Scholar
Sulem, C., and Sulem, P. 1999. The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse. Berlin: Springer.Google Scholar
Tang, D.Y., Man, W.S., Tam, H.Y., and Drummond, P.D. 2001. Observation of bound states of solitons in a passively mode-locked fiber laser. Phys. Rev. A, 64, 033814.CrossRefGoogle Scholar
Taniuti, T., and Wei, C.C. 1968. Reductive perturbation method in nonlinear wave propagation I. J. Phys. Soc. Japan, 24, 941–946.CrossRefGoogle Scholar
Taylor, G.I. 1950. The formation of a blast wave by a very intense explosion. I. Theoretical Discussion. Proc. Roy. Soc. A, 201, 159–174.CrossRefGoogle Scholar
Tsankov, M.A., Chen, M., and Patton, C.E. 1994. Forward volume wave microwave envelope solitons in yttrium iron garnet films-propagation, decay, and collision. J. Appl. Phys., 76, 4274–4289.CrossRefGoogle Scholar
Venakides, S. 1985. The zero dispersion limit of the Korteweg–de Vries equation for initial potentials with non-trivial reflection coeffcient. Commun. Pure Appl. Math., 38, 125–155.CrossRefGoogle Scholar
Villarroel, J., and Ablowitz, M.J. 2002. The Cauchy problem for the Kadomtsev–Petviashili II equation with nondecaying data along a line. Stud. Appl. Math, 109, 151–162.CrossRefGoogle Scholar
Villarroel, J., and Ablowitz, M.J. 2003. On the discrete spectrum of systems in the plane and the Davey-Stewartson II equation. SIAM J. Math. Anal., 34, 1253–1278.CrossRefGoogle Scholar
Vlasov, S., Petrishchev, V., and Talanov, V. 1970. Averaged description of wave beams in linear and nonlinear media. Radiophys. Quantum Electronics, 14, 1062.Google Scholar
Wadati, M. 1974. The modified Korteweg–de Vries equation. J. Phys. Soc. Jpn., 34, 1289–1296.CrossRefGoogle Scholar
Wadati, M., Konno, K., and Ichikawa, Y.-H. 1979a. A generalization of inverse scattering method. J. Phys. Soc. Jpn., 46, 1965–1966.CrossRefGoogle Scholar
Wadati, M., Konno, K., and Ichikawa, Y.H. 1979b. New integrable nonlinear evolution equations. J. Phys. Soc. Jpn., 47, 1698–1700.CrossRefGoogle Scholar
Wadati, M., and Sogo, K. 1983. Gauge transformations in soliton theory. J. Phys. Soc. Jpn., 52, 394–398.CrossRefGoogle Scholar
Weinstein, M. 1983. Nonlinear Schrödinger equations and sharp interpolation estimates. Comm. Math. Phys., 87, 567–576.CrossRefGoogle Scholar
Whitham, G.B. 1965. Non-linear dispersive waves. Proc. R. Soc. Lond. A, 6, 238–261.CrossRefGoogle Scholar
Whitham, G.B. 1974. Linear and Nonlinear Waves. New York: John Wiley.Google Scholar
Wu, M., Kalinikos, B.A., and Patton, C.E. 2004. Generation of dark and bright spin wave envelope soliton trains through self-modulational instability in magnetic films. Phys. Rev. Lett., 93, 157207.CrossRefGoogle Scholar
Yang, T., and Kath, W.L. 1997. Analysis of enhanced-power solitons in dispersion-managed optical fibers. Opt. Lett., 22, 985–987.CrossRefGoogle ScholarPubMed
Zabusky, N.J., and Kruskal, M.D. 1965. Interactions of “solitons” in a collisionless plasma and the recurrence of initial states. Phys.Rev.Lett., 15, 240–243.CrossRefGoogle Scholar
Zakharov, V.E. 1968. Stability of periodic waves of finite amplitude on the surface of a deep fluid. Sov. Phys. J. Appl. Mech. Tech. Phys., 4, 190–194.Google Scholar
Zakharov, V.E., and Rubenchik, A.M. 1974. Instability of waveguides and solitons in nonlinear media. Sov. Phys. JETP, 38, 494–500.Google Scholar
Zakharov, V.E., and Shabat, A. 1972. Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in a non-linear media. Sov. Phys. JETP, 34, 62–69.Google Scholar
Zakharov, V.E., and Shabat, A.B. 1973. Interaction between solitons in a stable medium. Sov. Phys. JETP, 37, 823–828.Google Scholar
Zharnitsky, V., Grenier, E., Turitsyn, S.K., Jones, C., and Hesthaven, J.S. 2000. Ground states of dispersion-managed nonlinear Schrödinger equation. Phys. Rev. E, 62, 7358–7364.CrossRefGoogle ScholarPubMed
Zhou, X. 1989. Direct and inverse scattering transforms with arbitrary spectral singularities. Commun. Pure Appl. Math., 42, 95–938.CrossRefGoogle Scholar
Zvezdin, A.K., and Popkov, A.F. 1983. Contribution to the nonlinear theory of magnetostatic spin waves. Sov. Phys. JETP., 51, 350–354.Google Scholar

Save book to Kindle

To save this book to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

  • References
  • Mark J. Ablowitz, University of Colorado, Boulder
  • Book: Nonlinear Dispersive Waves
  • Online publication: 05 June 2012
  • Chapter DOI: https://doi.org/10.1017/CBO9780511998324.013
Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

  • References
  • Mark J. Ablowitz, University of Colorado, Boulder
  • Book: Nonlinear Dispersive Waves
  • Online publication: 05 June 2012
  • Chapter DOI: https://doi.org/10.1017/CBO9780511998324.013
Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • References
  • Mark J. Ablowitz, University of Colorado, Boulder
  • Book: Nonlinear Dispersive Waves
  • Online publication: 05 June 2012
  • Chapter DOI: https://doi.org/10.1017/CBO9780511998324.013
Available formats
×