Abstract
It has already been indicated in Section 2.3 that the nonlinear Schrödinger (NLS) equation arises in a wide variety of physical problems in fluid mechanics, plasma physics, and nonlinear optics. The most common applications of the NLS equation include self-focusing of beams in nonlinear optics, modeling of propagation of electromagnetic pulses in nonlinear optical fibers which act as wave guides, and stability of Stokes waves in water. Some formal derivations of the NLS equation have been obtained by several methods which include the multiple scales expansions, the asymptotic method, Whitham’s (J. Fluid Mech. 22:273–283, 1965a, Proc. R. Soc. Lond. A283:238–261, 1965b) averaged variational equations, and Phillips’ (J. Fluid Mech. 106:215–227, 1981) resonant interaction equations. Zakharov and Shabat (Sov. Phys. JETP 34:62–69, 1972) developed an ingenious inverse scattering method to show that the NLS equation is completely integrable. The NLS equation is of great importance in adding to our fundamental knowledge of the general theory of nonlinear dispersive waves.
…Schrödinger and I both had a very strong appreciation of mathematical beauty, and this appreciation of mathematical beauty dominated all our work. It was a sort of act of faith with us that any equations which describe fundamental laws of Nature must have great mathematical beauty in them. It was like a religion with us. It was a very profitable religion to hold, and can be considered the basis of much of our success.
Paul Dirac
…the progress of physics will to a large extent depend on the progress of nonlinear mathematics, of methods to solve nonlinear equations …and therefore we can learn by comparing different nonlinear problems.
Werner Heisenberg
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Ablowitz, M.J. and Segur, H. (1979). On the evolution of packets of water waves, J. Fluid Mech. 92, 691–715.
Benjamin, T.B. and Feir, J.E. (1967). The disintegration of wavetrains on deep water, Part 1, Theory, J. Fluid Mech. 27, 417–430.
Benney, D.J. and Roskes, G. (1969). Wave instabilities, Stud. Appl. Math. 48, 377–385.
Birkhoff, G.D. (1927). Dynamical Systems, Am. Math. Soc., Providence.
Chu, V.H. and Mei, C.C. (1970). On slowly varying Stokes waves, J. Fluid Mech. 41, 873–887.
Chu, V.H. and Mei, C.C. (1971). The nonlinear evolution of Stokes waves in deep water, J. Fluid Mech. 47, 337–351.
Davey, A. and Stewartson, K. (1974). On three dimensional packets of surface waves, Proc. R. Soc. Lond. A338, 101–110.
Debnath, L. (1994). Nonlinear Water Waves, Academic Press, Boston.
Dutta, M. and Debnath, L. (1965). Elements of the Theory of Elliptic and Associated Functions with Applications, World Press Pub. Ltd., Calcutta.
Fermi, E., Pasta, J., and Ulam, S. (1955). Studies of nonlinear problems I, Los Alamos Report LA 1940, in Lectures in Applied Mathematics (ed. A.C. Newell), Vol. 15, Am. Math. Soc., Providence, 143–156.
Fornberg, B. and Whitham, G.B. (1978). A numerical and theoretical study of certain nonlinear wave phenomena, Philos. Trans. R. Soc. Lond. A289, 373–404.
Gibbons, J., Thornhill, S.G., Wardrop, M.J., and ter Haar, D. (1977). On the theory of Langmuir solitons, J. Plasma Phys. 17, 153–170.
Hasegawa, A. (1990). Optical Solitons in Fibers, 2nd edition, Springer, Berlin.
Hasegawa, A. and Tappert, F. (1973). Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers I: Anomalous dispersion, Appl. Phys. Lett. 23, 142–144.
Hasimoto, H. (1972). A soliton on a vortex filament, J. Fluid Mech. 51, 477–485.
Hasimoto, H. and Ono, H. (1972). Nonlinear modulation of gravity waves, J. Phys. Soc. Jpn. 33, 805–811.
Infeld, E. and Rowlands, G. (1990). Nonlinear Waves, Solitons and Chaos, Cambridge University Press, Cambridge.
Karpman, V.I. (1971). High frequency electromagnetic field in plasma with negative dielectric constant, Plasma Phys. 13, 477–490.
Karpman, V.I. (1975a). Nonlinear Waves in Dispersive Media, Pergamon Press, London.
Karpman, V.I. (1975b). On the dynamics of sonic-Langmuir solitons, Phys. Scr. 11, 263–265.
Konno, K. and Wadati, M. (1975). Simple derivation of the Bäcklund transformation from the Riccati form of the inverse method, Prog. Theor. Phys. 53, 1652–1656.
Lake, B.M., Yuen, H.C., Rundgaldier, H., and Ferguson, W.E. (1977). Nonlinear deep-water waves: Theory and experiment, Part 2, Evolution of a continuous wave train, J. Fluid Mech. 83, 49–74.
Ma, Y.-C. (1979). The perturbed plane-wave solution of the cubic Schrödinger equation, Stud. Appl. Math. 60, 43–58.
Mollenauer, L.F., Gordon, J.P., and Islam, M.N. (1986). Soliton propagation in long fibres with periodically compensated loss, IEEE J. Quantum Electron. 22, 1284–1286.
Oikawa, M. and Yajima, N. (1974a). A class of exactly solvable nonlinear evolution equations, Prog. Theor. Phys. Suppl. 54, 1576–1577.
Oikawa, M. and Yajima, M. (1974b). A perturbation approach to nonlinear systems II. Interaction of nonlinear modulated waves, J. Phys. Soc. Jpn. 37, 486–496.
Peregrine, D.H. (1983). Water waves, nonlinear Schrödinger equations and their solutions, J. Aust. Math. Soc. B25, 16–43.
Phillips, O.M. (1981). Wave interactions—the evolution of an idea, J. Fluid Mech. 106, 215–227.
Rowland, H.L., Lyon, J.G., and Papadopoulus, K. (1981). Strong Langmuir turbulence in one and two dimensions, Phys. Rev. Lett. 46, 346–349.
Sanuki, H., Shimizu, K., and Todoroki, J. (1972). Effects of Landau damping on nonlinear wave modulation in plasma, J. Phys. Soc. Jpn. 33, 198–205.
Schmidt, G. (1975). Stability of envelope solitons, Phys. Rev. Lett. 34, 724–726.
Thyagaraja, A. (1979). Recurrent motions in certain continuum dynamical system, Phys. Fluids 22, 2093–2096.
Thyagaraja, A. (1981). Recurrence, dimensionality, and Lagrange stability of solutions of the nonlinear Schödinger equation, Phys. Fluids 24, 1973–1975.
Thyagaraja, A. (1983). Recurrence phenomena and the number of effective degrees of freedom in nonlinear wave motions, in Nonlinear Waves (ed. L. Debnath), Cambridge University Press, Cambridge, 308–325.
Whitham, G.B. (1965a). A general approach to linear and nonlinear dispersive waves using a Lagrangian, J. Fluid Mech. 22, 273–283.
Whitham, G.B. (1965b). Nonlinear dispersive waves, Proc. R. Soc. Lond. A283, 238–261.
Whitham, G.B. (1974). Linear and Nonlinear Waves, Wiley, New York.
Yajima, N. and Outi, A. (1971). A new example of stable solitary waves, Prog. Theor. Phys. 45, 1997–2005.
Yuen, H.C. and Ferguson, W.E., Jr. (1978a). Relationship between Benjamin–Feir instability and recurrence in the nonlinear Schrödinger equation, Phys. Fluids 21, 1275–1278.
Yuen, H.C. and Ferguson, W.E., Jr. (1978b). Fermi–Pasta–Ulam recurrence in the two space dimensional nonlinear Schrödinger equation, Phys. Fluids 21, 2116–2118.
Yuen, H.C. and Lake, B.M. (1975). Nonlinear deep water waves: Theory and experiment, Phys. Fluids 18, 956–960.
Yuen, H.C. and Lake, B.M. (1980). Instabilities of wave on deep water, Annu. Rev. Fluid Mech. 12, 303–334.
Zabusky, N.J. and Kruskal, M.D. (1965). Interaction of solitons in a collisionless plasma and the recurrence of initial states, Phys. Rev. Lett. 15, 240–243.
Zakharov, V.E. and Shabat, A.B. (1972). Exact theory of two-dimensional self focusing and one-dimensional self modulation of waves in nonlinear 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.
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Debnath, L. (2012). The Nonlinear Schrödinger Equation and Solitary Waves. In: Nonlinear Partial Differential Equations for Scientists and Engineers. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-8265-1_10
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