# Exact and Asymptotic Methods in Nonlinear Wave Theory

• A. Jeffrey
Chapter
Part of the CISM Courses and Lectures book series (CISM, volume 341)

## Abstract

In Section 1 the concepts of linear dispersive and dissipative wave propagation are reviewed, and then extended to travelling waves characterized by nonlinear evolution equations. The general effect of nonlinearity on the development of a wave is examined and the propagation speed of a discontinuous solution (shock) is derived. Various travelling wave solutions are discussed and soliton solutions of the KdV equation are mentioned. Section 2 reviews different ways of finding travelling wave solutions for the KdVB equation and comments on their equivalence. The ideas of weak and strong dispersion are then defined. The notion of a far field is introduced and hyperbolicity is discussed. Hyperbolic systems and waves form the topic of Section 3, which reviews Riemann invariants and simple waves, and their generalization. Shocks, the Riemann problem and entropy conditions are introduced. Sections 4 and 5 are concerned with the asymptotic derivation of far field equations both for systems and for scalar equations. The reductive perturbation method is described in Section 4 for weakly dispersive systems, while in Section 5 the multiple scale method is introduced and used to derive both the nonlinear Schrödinger and the KdV equation from a model nonlinear dispersive equation. Two physical examples with different evolution equations are given.

## Keywords

Solitary Wave Travel Wave Solution Asymptotic Method Burger Equation Discontinuous Solution
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## References

1. [1]
Zabusky, N.I.: Exact solution for the vibrations of a nonlinear continuous model string. J. Math. Phys., 3 (1962), 1028–1239.
2. [2]
Zabusky, N.I.: A synergetic approach to problems of nonlinear dispersive wave propagation and interaction. Proc. Symp. Nonlinear Partial Differential Equations. W. Ames, ed., Academic Press (1967), 223–258.Google Scholar
3. [3]
Zabusky, N.J. and M.D. Kruskal: Interaction of solitons in a collisionless plasma and the recurrence of initial states. Phys. Rev. Lett., 15 (1965), 240–243.
4. [4]
Jeffrey, A. and T. Kawahara: Asymptotic Methods in Nonlinear Wave Theory. Pitman, 1982.Google Scholar
5. [5]
Lamb, G.L.: Elements of Soliton Theory. Wiley, 1980.Google Scholar
6. [6]
Eckhaus, W. and A. Van Harten: The Inverse Scattering Transformation and the Theory of Solitons. Mathematical Studies 50, North Holland, 1981.Google Scholar
7. [7]
Dodd, R.K., Eilbeck, J.C., Gibbon, J.D. and H.L. Morris: Solitons and Nonlinear Wave Equations. Academic Press, 1982.Google Scholar
8. [8]
Drazin, P.G. and R.S. Johnson: Solitons: an Introduction. Cambridge Texts in Applied Mathematics, Cambridge University Press, 1989.
9. [9]
Ablowitz, M.J. and P.A. Clarkson: Solitons, Nonlinear Evolution Equations and Inverse Scattering. LMS Lecture Note 149, Cambridge University Press, 1991.Google Scholar
10. [10]
Konno, K. and A. Jeffrey: Some remarkable properties of two loop soliton solutions. J. Phys. Soc. Japan, 52 (1983), 1–3.
11. [11]
Konno, K. and A. Jeffrey: The loop soliton. Advances in Nonlinear Waves (Ed. L. Debnath). Pitman Research Note 95 (1984), 162–182.Google Scholar