Nonlinear Phenomena in Physics and Biology pp 95-123 | Cite as

# The Linearity of Nonlinear Soliton Equations and the Three Wave Resonance Interaction

## Abstract

The purpose of a school such as this is to provide a background of information for those interested in a given particular subject. The basic fundamentals of solitons have already been well expanded on by the previous talks, and in general, the most that I could add without excessive duplication is simply additional references [Kaup, 1977; Kaup and Newell, 1978a, b; Kaup et al, 1979], which describe my viewpoint on these basics. However, there is one basic point that still has not been emphasized here, and which I have always considered to be striking and important. That is the fact that although these systems are indeed nonlinear, their behavior so closely mocks or imitates linear systems, that one is frequently ahead if he simply forgets that it is nonlinear, and looks upon the system as being essentially linear. For example, Professor Newell [Kaup and Newell, 1978b] has demonstrated a striking representation of the general solution for q in terms of the squared eigenstates.

## Keywords

Reflection Coefficient Fundamental Solution Inverse Scattering Scattering Problem Linear Solution## Preview

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## References

- Ablowitz, M.J., Kaup, D.J., Newell, A.C., and Segur, H., 1974, The inverse scattering transform - Fourier analysis for nonlinear problems, Stud. Appl. Math., 53: 249.MathSciNetGoogle Scholar
- Ablowitz, M.J., and Haberman, R., 1975, Resonantly coupled nonlinear evolution equations, J. Math. Phys., 16: 2301.Google Scholar
- Armstrong, J.A., Bloembergen, N., Ducuing, J., and Pershan, P.S., 1962, Interactions between light waves in a nonlinear dielectric, Phys. Rev., 127: 1918.ADSCrossRefGoogle Scholar
- Cornille, H., 1979, Solutions of the nonlinear 3-wave equations in three spatial dimensions, J. Math. Phys., 20: 1653.MathSciNetADSMATHCrossRefGoogle Scholar
- Corones, J., 1976, Solitons and simple pseudopotentials, J. Math. Phys., 17: 756.MathSciNetADSMATHCrossRefGoogle Scholar
- Kaup, D.J., 1977, Coherent pulse propagation: a comparison of the complete solution with the McCall-Hahn theory and others, Phys. Rev., A16: 704.ADSCrossRefGoogle Scholar
- Kaup, D.J., and Newell, A.C., 1978a, Theory of nonlinear oscillating dipolar excitations in one-dimensional condensates, Phys. Rev., B18: 5162.MathSciNetADSCrossRefGoogle Scholar
- Kaup, D.J., and Newell, A.C., 1978b, Solitons as particles, oscillators, and in slowly changing media: a singular perturbation theory, Proc. Roy. Soc. London, A361: 413.ADSCrossRefGoogle Scholar
- Kaup, D.J., 1978, Simple harmonic generation: an exact method of solution, Stud. Appl. Math., 59: 25.MathSciNetGoogle Scholar
- Kaup, B.J., Rieman, A., and Bers, A., 1979, Space-time evolution of nonlinear three-wave interactions. I. Interaction in a homo-geneous medium, Rev. Mod. Phys., 51: 275.ADSCrossRefGoogle Scholar
- Kaup, D.J., 1979, to appear in the Proceedings of the Soviet-American Soliton Symposium, Kiev, USSR, 2-15 September 1979, (Physica D, 1981 ).Google Scholar
- Kaup, D.J., 1980a, A method for solving the separable initial-value problem of the full three-dimensional three-wave interaction, Stud. Appl. Math., 62: 75.MathSciNetADSMATHGoogle Scholar
- Kaup, D.J., 1980b, The inverse scattering solution for the full three-dimensional threeTwave resonant interaction, Physica D, 1: 45.MathSciNetADSMATHCrossRefGoogle Scholar
- Kaup, D.J., 1980c, Determining the final profiles from the initial profiles for the full three-dimensional three-wave resonant interaction, in: “Mathematical Methods and Applications of Scattering Theory,” DeSanto, Saenz and Zachary, ed., Lecture Notes in Physics 130, Springer-Verlag, New York.Google Scholar
- Kaup, D.J., 1981, to appear, The Backlund transformation and N-lump solutions of the three-dimensional three-wave resonant inter-action, J. Math. Phys.Google Scholar
- Zakharov, V.E., and Manakov, S.V., 1973, Resonant interaction of wave packets in nonlinear media, ZhETF Pis. Red., 18: 413. [JETP Letts., 18:243.]Google Scholar
- Zakharov, V.E., and Shabat, A.B., 1974, A scheme for integrating the nonlinear equations of mathematical physics by the method of inverse scattering problem. I., Funkts. Anal. Prilozh., 8: 43.Google Scholar
- Zakharov, V.E., 1976, Exact solutions to the problem of the para-metric interaction of three-dimensional wave packets, Dokl. Akad. Nauk. SSSR, 228: 1314. [Sov. Phys. Dokl., 21:3227]Google Scholar