Ablowitz, M.J., Bakirtas, I., and Ilan, B. 2005. Wave collapse in nonlocal nonlinear Schrödinger systems. Physica D, 207, 230–253.
Ablowitz, M.J., and Biondini, G. 1998. Multiscale pulse dynamics in communication systems with strong dispersion-management. Opt. Lett., 23, 1668–1670.
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.
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.
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.
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.
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.
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.
Ablowitz, M.J., and Clarkson, P.A. 1991. Solitons, Nonlinear Evolution Equations and Inverse Scattering. Cambridge: Cambridge University Press.
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.
Ablowitz, M.J., and Fokas, A.S. 2003. Complex Variables: Introduction and Applications. Second edition. Cambridge: Cambridge University Press.
Ablowitz, M.J., Fokas, A.S., and Musslimani, Z. 2006. On a new nonlocal formulation of water waves. J. Fluid Mech., 562, 313–344.
Ablowitz, M.J., and Haberman, R. 1975. Resonantly coupled nonlinear evolution equations. Phys. Rev. Lett., 38, 1185–1188.
Ablowitz, M.J., Hammack, J., Henderson, D., and Schober, C.M. 2000a. Modulated periodic waves in deep water. Phys. Rev. Lett., 84, 887–890.
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.
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.
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.
Ablowitz, M.J., and Haut, T.S. 2010. Asymptotic expansions for solitary gravity-capillary waves in two dimensions. J. Phys. Math Theor., 43, 434005.
Ablowitz, M.J., Hirooka, T., and Biondini, G. 2001b. Quasi-linear optical pulses in strongly dispersion-managed transmission systems. Opt. Lett., 26, 459–461.
Ablowitz, M.J., and Hirooka, T. 2002. Managing nonlinearity in strongly dispersion-managed optical pulse transmission. J. Opt. Soc. Am. B, 19, 425–439.
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.
Ablowitz, M.J., and Horikis, T.P. 2008. Pulse dynamics and solitons in mode-locked lasers. Phys. Rev. A, 78, 011802.
Ablowitz, M.J., and Horikis, T.P. 2009a. Solitons and spectral renormalization methods in nonlinear optics. Eur. Phys. J. Special Topics, 173, 147–166.
Ablowitz, M.J., and Horikis, T.P. 2009b. Solitons in normally dispersive mode-locked lasers. Phys. Rev. A, 79, 063845.
Ablowitz, M.J., Horikis, T.P., and Ilan, B. 2008. Solitons in dispersion-managed mode-locked lasers. Phys. Rev. A, 77, 033814.
Ablowitz, M.J., Horikis, T.P., and Nixon, S.D. 2009c. Soliton strings and interactions in mode-locked lasers. Opt. Comm., 282, 4127–4135.
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.
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.
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.
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.
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.
Ablowitz, M.J., Kruskal, M.D., and Segur, H. 1979. A note on Miura's transformation. J. Math. Phys., 20, 999–1003.
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.
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.
Ablowitz, M.J., and Musslimani, Z. 2005. Spectral renormalization method for computing self-localized solutions to nonlinear systems. Opt. Lett., 30, 2140–2142.
Ablowitz, M.J., Nixon, S.D., and Zhu, Y. 2009b. Conical diffraction in honeycomb lattices. Phys. Rev. A, 79, 053830.
Ablowitz, M.J., Prinari, B., and Trubatch, A.D. 2004b. Discrete and Continuous Nonlinear Schrödinger Systems. Cambridge: Cambridge University Press.
Ablowitz, M.J., and Segur, H. 1979. On the evolution of packets of water waves. J. Fluid Mech., 92, 691–715.
Ablowitz, M.J., and Segur, H. 1981a. Solitons and the Inverse Scattering Transform. Philadelphia: SIAM.
Ablowitz, M.J., and Villarroel, J. 1991. On the Kadomtsev–Petviashili equation and associated constraints. Stud. Appl. Math., 85, 195–213.
Ablowitz, M.J., and Wang, X.P. 1997. Initial time layers in Kadomtsev–Petviashili type equations. Stud. Appl. Math., 98, 121–137.
Abramowitz, M., and Stegun, I.R. 1972. Handbook of Mathematical Functions. Tenth edition. New York: Dover.
Agrawal, G.P. 2001. Nonlinear Fiber Optics. New York: Academic Press.
Agrawal, G.P. 2002. Fiber-Optic Communication Systems. New York: Wiley-Interscience.
Ahrens, C. 2006. The asymptotic analysis of communications and wave collapse problems in nonlinear optics. Ph.D. thesis, University of Colorado.
Airy, G.B. 1845. Tides and waves. Encyc. Metrop., 192, 241–396.
Akhmediev, N.N., and Ankiewicz, A. 1997. Solitons, Nonlinear Pulses and Beams. London: Chapman & Hall.
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.
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.
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.
Batchelor, G.K. 1967. An Introduction to Fluid Dynamics. Cambridge: Cambridge University Press.
Beals, R., and Coifman, R.R. 1984. Scattering and inverse scattering for 1st order systems. Comm. Pure Appl. Math., 37, 39–90.
Beals, R., and Coifman, R.R. 1985. Inverse scattering and evolution-equations. Comm. Pure Appl. Math., 38, 29–42.
Beals, R., Deift, P., and Tomei, C. 1988. Direct and Inverse Scattering on the Line. Providence, RI: AMS.
Bender, C.M., and Orszag, S.A. 1999. Advanced Mathematical Methods for Scientists and Engineers. Berlin: Springer.
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.
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.
Benney, D.J. 1966a. Long non-linear waves in fluid flows. J. Math. and Phys., 45, 52–63.
Benney, D.J. 1966b. Long waves on liquid films. J. Math. and Phys., 45, 150–155.
Benney, D.J., and Luke, J.C. 1964. Interactions of permanent waves of finite amplitude. J. Math. Phys., 43, 309–313.
Benney, D.J., and Newell, A.C. 1967. The propagation of nonlinear envelopes. J. Math. and Phys., 46, 133–139.
Benney, D.J., and Roskes, G.J. 1969. Wave instabilities. Stud. Appl. Math., 48, 377–385.
Bleistein, N. 1984. Mathematical Methods for Wave Phenomena. New York: Academic Press.
Bleistein, N., and Handelsman, R.A. 1986. Asymptotic Expansions of Integrals. New York: Dover.
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.
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.
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.
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.
Byrd, P.F., and Friedman, M.D. 1971. Handbook of Elliptic Integrals for Engineers and Physicists. Berlin: Springer.
Calogero, F., and Degasperis, A. 1982. Spectral Transform and Solitons. Amsterdam: Elsevier.
Calogero, F., and Delillo, S. 1989. The Burgers equation on the semi-infinite and finite intervals. Nonlinearity, 2, 37–43.
Camassa, R., and Holm, D.D. 1993a. An integrable shallow water equation with peaked solitons. Phys. Rev. Lett., 71, 1661–1664.
Camassa, R., and Holm, D.D. 1993b. An integrable shallow water equation with peaked solitons. Phys. Rev. Lett., 11, 1661–1664.
Caudrey, P.J. 1982. The inverse problem for a general N × N spectral equation. Physica D, 6, 51–66.
Chapman, S., and Cowling, T. 1970. Mathematical Theory of Nonuniform Gases. Cambridge: Cambridge University Press.
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.
Chester, C., Friedman, B., and Ursell, F. 1957. An extension of the method of steepest descents. Proc. Camb. Phil. Soc., 53, 599–611.
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.
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.
Christodoulides, D.N., and Joseph, R.I. 1998. Discrete self-focusing in nonlinear arrays of coupled waveguides. Opt. Lett., 13(9), 794–796.
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.
Cole, J.D. 1951. On a quasilinear parabolic equation occurring in aerodynamics. Quart. Appl. Math., 9, 225–236.
Cole, J.D. 1968. Perturbation Methods in Applied Mathematics. London: Ginn and Co.
Copson, E.T. 1965. Asymptotic Expansions. Cambridge: Cambridge University Press.
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.
Courant, R., and Hilbert, D. 1989. Methods of Mathematical Physics. New York: John Wiley.
Craig, W., and Groves, M.D. 1994. Hamiltonian long-wave approximations to the water-wave problem. Wave Motion, 19, 367–389.
Craig, W., and Sulem, C. 1993. Numerical simulation of gravity-waves. J. Comp. Phys., 108, 73–83.
Crasovan, L.C., Torres, J.P., Mihalache, D., and Torner, L. 2003. Arresting wave collapse by self-rectification. Phys. Rev. Lett., 9, 063904.
Cundiff, S.T. 2005. Soliton dynamics in mode-locked lasers. Lect. Notes Phys., 661, 183–206.
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.
Deift, P., and Trubowitz, E. 1979. Inverse scattering on the line. Comm. Pure Appl. Math., 32, 121–251.
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.
Deift, P., and Zhou, X. 1992. A steepest descent method for oscillatory Riemann–Hilbert problems. Bull. Amer. Math. Soc., 26, 119–123.
Dickey, L.A. 2003. Soliton Equations and Hamiltonian Systems. Singapore: World Scientific.
Djordjevic, V.D., and Redekopp, L.G. 1977. On two-dimensional packets of capillary–gravity waves. J. Fluid Mech., 79, 703–714.
Docherty, A. 2003. Collision induced timing shifts in wavelength-division-multiplexed optical fiber communications systems. Ph.D. thesis, University of New South Wales.
Dodd, R.K., Eilbeck, J.C., Gibbon, J.D., and Morris, H.C. 1984. Solitons and Nonlinear Wave Equations. New York: Academic Press.
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.
Elgin, J.N. 1985. Inverse scattering theory with stochastic initial potentials. Phys. Lett. A, 110, 441–443.
Erdelyi, A. 1956. Asymptotic Expansions. New York: Dover.
Faddeev, L.D. 1963. The inverse problem in the quantum theory of scattering. J. Math. Phys., 4, 72–104.
Faddeev, L.D., and Takhtajan, L.A. 1987. Hamiltonian Methods in the Theory of Solitons. Berlin: Springer.
Falcon, E., Laroche, C., and Fauve, S. 2002. Observation of depression solitary surface waves on a thin fluid layer. Phys. Rev. Lett., 89, 204501.
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).].
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.
Fokas, A.S. 2008. A Unified Approach to Boundary Value Problems. Philadelphia: SIAM.
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.
Fokas, A.S., and Fuchssteiner, B. 1981. Symplectic structures, their Bäcklund transformations and hereditary symmetries. Physica D, 4, 47–66.
Fokas, A.S., and Santini, P.M. 1989. Coherent structures in multidimensions. Phys. Rev. Lett., 63, 1329–1333.
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.
Forsyth, A.R. 1906. Theory of Differential Equations. Part IV—Partial Differential Equations. Cambridge: Cambridge University Press.
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.
Garabedian, P. 1984. Partial Differential Equations. New York: Chelsea.
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.
Gardner, C.S., and Su, C.S. 1969. The Korteweg–de Vries equation and generalizations. III. J. Math. Phys., 10, 536–539.
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.
Gel'fand, I.M., and Dickii, L.A. 1977. Resolvants and Hamiltonian systems. Func. Anal. Appl., 11, 93–104.
Goldstein, H. 1980. Classical Mechanics. Reading, MA: Addison Wesley.
Gordon, J.P., and Haus, H.A. 1986. Random walk of coherently amplified solitons in optical fiber transmission. Opt. Lett., 11, 665–667.
Hasegawa, A., and Kodama, Y. 1991a. Guiding-center soliton. Phys. Rev. Lett., 66, 161–164.
Hasegawa, A., and Kodama, Y. 1991b. Guiding-center soliton in fibers with periodically varying dispersion. Opt. Lett., 16, 1385–1387.
Hasegawa, A., and Kodama, Y. 1995. Solitons in Optical Communications. Oxford: Oxford University Press.
Hasegawa, A., and Matsumoto, M. 2002. Optical Solitons in Fibers. Berlin: Springer.
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.
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.
Haus, H.A. 1975. Theory of mode locking with a fast saturable absorber. J. Appl. Phys., 46, 3049–3058.
Haus, H.A. 2000. Mode-locking of lasers. IEEE J. Sel. Topics Q. Elec., 6, 1173–1185.
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.
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.
Hopf, E. 1950. The partial differential equation ut + uux = μuxx. Comm. Pure Appl. Math., 3, 201–230.
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.
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.
Infeld, E., and Rowlands, G. 2000. Nonlinear Waves, Solitons and Chaos. Cambridge: Cambridge University Press.
Ishimori, Y. 1981. On the modified Korteweg–deVries soliton and the loop soliton. J. Phys. Soc. Jpn., 50, 2471–2472.
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.
Jackson, J.D. 1998. Classical Electrodynamics. New York: John Wiley.
Jeffreys, H., and Jeffreys, B.S. 1956. Methods of Mathematical Physics. Cambridge: Cambridge University Press.
Kadomtsev, B.B., and Petviashvili, V.I. 1970. On the stability of solitary waves in weakly dispersive media. Sov. Phys. Doklady, 15, 539–541.
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.
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.
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.
Karpman, V.I., and Solov'ev, V.V. 1981. A perturbation theory for soliton systems. Physica D, 3, 142–164.
Kay, I., and Moses, H.E. 1956. Reflectionless transmission through dielectrics and scattering potentials. J. Appl. Phys., 27, 1503–1508.
Kevorkian, J., and Cole, J.D. 1981. Perturbation Methods in Applied Mathematics. Berlin: Springer.
Konno, K., and Jeffrey, A. 1983. Some remarkable properties of two loop soliton solutions. J. Phys. Soc. Jpn., 52, 1–3.
Konopelchenko, B.G. 1993. Solitons in Multidimensions. Singapore: World Scientific.
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.
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.
Kutz, J.N. 2006. Mode-locked soliton lasers. SIAM Rev., 48, 629–678.
Kuzmak, G.E. 1959. Asymptotic solutions of nonlinear second order differential equations with variable coeffcients. PMM, 23(3), 515–526.
Lamb, H. 1945. Hydrodynamics. New York: Dover.
Landau, L.D., and Lifshitz, L.M. 1981. Quantum Mechanics: Non-relativistic Theory. London: Butterworth–Heinemann.
Landau, L.D., Lifshitz, E.M., and Pitaevskii, L.P. 1984. Electrodynamics of Continuous Media. London: Butterworth–Heinemann.
Lax, P.D. 1968. Integrals of nonlinear equations of evolution and solitary waves. Commun. Pure Appl. Math., 21, 467–490.
Lax, P.D. 1987. Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves. Philadelphia: SIAM.
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.
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.
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.
Leveque, R.J. 2002. Finite Volume Methods for Hyperbolic Problems. Cambridge: Cambridge University Press.
Lighthill, M.J. 1958. Introduction to Fourier Analysis and Generalized Functions. Cambridge: Cambridge University Press.
Lighthill, M.J. 1978. Waves in Fluids. Cambridge: Cambridge University Press.
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.
Luke, J.C. 1966. A perturbation method for nonlinear dispersive wave problems. Proc. R. Soc. Lond. A, 292, 403–412.
Lushnikov, P.M. 2001. dispersion-managed solitons in the strong dispersion map limit. Opt. Lett., 26, 1535–1537.
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.
Marchenko, V.A. 1986. Sturm–Liouville Operators and Applications. Basel: Birkhauser.
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.
Matveev, V.B., and Salle, M.A. 1991. Darboux Transformations and Solitons. Berlin: Springer.
Melin, A. 1985. Operator methods for inverse scattering on the real line. Commun. Part. Diff. Eqns., 10, 677–766.
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.
Mikhailov, A.V. 1979. Integrability of a two-dimensional generalization of the Toda chain. Sov. Phys. JETP Lett., 30, 414–418.
Mikhailov, A.V. 1981. The reduction problem and the inverse scattering method. Physica D, 3, 73–117.
Miles, J.W. 1981. The Korteweg–de Vries equation: An historical essay. J. Fluid Mech., 106 (focus issue), 131–147.
Miura, R.M. 1968. Korteweg–de Vries equation and generalizations. I. A remarkable explicit nonlinear transformation. J. Math. Phys., 9, 1202–1204.
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.
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.
Molleneauer, L.F., and Gordon, J.P. 2006. Solitons in Optical Fibers: Fundamentals and Applications to Telecommunications. New York: Academic Press.
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.
Newell, A.C. 1978. The general structure of integrable evolution equations. Proc. Roy. Soc. Lond. A., 365, 283–311.
Newell, A.C. 1985. Solitons in Mathematics and Physics. Philadelphia: SIAM.
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.
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.
Novikov, S.P., Manakov, S.V., Pitaevskii, L.P., and Zakharov, V.E. 1984. Theory of Solitons: The Inverse Scattering Method. New York: Plenum.
Ostrovsky, L.A., and Potapov, A.S. 1986. Modulated Waves: Theory and Applications. Baltimore: The John Hopkins University Press.
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.
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.
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.
Pelinovsky, D.E., and Sulem, C. 2000. Spectral decomposition for the Dirac system associated to the DSII equation. Inverse Problems, 16, 59–74.
Petviashvili, V.I. 1976. Equation of an extraordinary soliton. Sov. J. Plasma Phys., 2, 257–258.
Phillips, O.M. 1977. The Dynamics of the Upper Ocean. Cambridge: Cambridge University Press.
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.
Quraishi, Q., Cundiff, S.T., Ilan, B., and Ablowitz, M.J. 2005. Dynamics of nonlinear and dispersion-managed solitons. Phys.Rev.Lett., 94, 243904.
Rabinovich, M.I., and Trubetskov, D.I. 1989. Oscillations and Waves in Linear and Nonlinear Systems. Dordsecht: Kluwer Academic.
Rayleigh, Lord. 1876. On waves. Phil. Mag., 1, 257–279.
Remoissenet, M. 1999. Waves Called Solitons. Berlin: Springer.
Rogers, C., and Schief, W.K. 2002. Bäcklund and Darboux Transformations. Cambridge: Cambridge University Press.
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.
Sanders, J.A., Verhulst, F., and Murdock, J. 2009a. Averaging Methods in Nonlinear Dynamical Systems. Berlin: Springer.
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.
Satsuma, J., and Ablowitz, M.J. 1979. Two-dimensional lumps in nonlinear dispersive systems. J. Math. Phys., 20, 1496–1503.
Shimizu, T., and Wadati, M. 1980. A new integrable nonlinear evolution equation. Prog. Theor. Phys., 63, 808–820.
Stokes, G.G. 1847. On the theory of oscillatory waves. Camb. Trans., 8, 441–473.
Sulem, C., and Sulem, P. 1999. The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse. Berlin: Springer.
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.
Taniuti, T., and Wei, C.C. 1968. Reductive perturbation method in nonlinear wave propagation I. J. Phys. Soc. Japan, 24, 941–946.
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.
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.
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.
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.
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.
Vlasov, S., Petrishchev, V., and Talanov, V. 1970. Averaged description of wave beams in linear and nonlinear media. Radiophys. Quantum Electronics, 14, 1062.
Wadati, M. 1974. The modified Korteweg–de Vries equation. J. Phys. Soc. Jpn., 34, 1289–1296.
Wadati, M., Konno, K., and Ichikawa, Y.-H. 1979a. A generalization of inverse scattering method. J. Phys. Soc. Jpn., 46, 1965–1966.
Wadati, M., Konno, K., and Ichikawa, Y.H. 1979b. New integrable nonlinear evolution equations. J. Phys. Soc. Jpn., 47, 1698–1700.
Wadati, M., and Sogo, K. 1983. Gauge transformations in soliton theory. J. Phys. Soc. Jpn., 52, 394–398.
Weinstein, M. 1983. Nonlinear Schrödinger equations and sharp interpolation estimates. Comm. Math. Phys., 87, 567–576.
Whitham, G.B. 1965. Non-linear dispersive waves. Proc. R. Soc. Lond. A, 6, 238–261.
Whitham, G.B. 1974. Linear and Nonlinear Waves. New York: John Wiley.
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.
Yang, T., and Kath, W.L. 1997. Analysis of enhanced-power solitons in dispersion-managed optical fibers. Opt. Lett., 22, 985–987.
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.
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.
Zakharov, V.E., and Rubenchik, A.M. 1974. Instability of waveguides and solitons in nonlinear media. Sov. Phys. JETP, 38, 494–500.
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.
Zakharov, V.E., and Shabat, A.B. 1973. Interaction between solitons in a stable medium. Sov. Phys. JETP, 37, 823–828.
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.
Zhou, X. 1989. Direct and inverse scattering transforms with arbitrary spectral singularities. Commun. Pure Appl. Math., 42, 95–938.
Zvezdin, A.K., and Popkov, A.F. 1983. Contribution to the nonlinear theory of magnetostatic spin waves. Sov. Phys. JETP., 51, 350–354.
