The Nonlinear Schrödinger Equation and Solitary Waves
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.
KeywordsDispersion Relation Solitary Wave Soliton Solution Solitary Wave Solution Envelope Soliton
- 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. Google Scholar
- Hasegawa, A. (1990). Optical Solitons in Fibers, 2nd edition, Springer, Berlin. Google Scholar
- Karpman, V.I. (1975a). Nonlinear Waves in Dispersive Media, Pergamon Press, London. 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