Exact and Asymptotic Methods in Nonlinear Wave Theory

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.