Simple Waves and the Formation of Discontinuities

Abstract

For nonlinear systems with lumped parameters the nonlinear oscillator described by the equation ẍ + f(x) = 0 is, as we have seen (Chapter 13), the basic model. The solution of this equation is the basis on which approximate solutions can be constructed to account for disturbances in various factors, such as the external force, positive or negative dissipation (Chapters 15–17), and nonstationary parameters. In the theory of nonlinear waves there are several basic models. Primarily, there is the one-wave approximation model, whose equation is

$$ \partial u/\partial t + V(u)\partial u/\partial x = 0 $$

((18.1))

(18.1)which describes a plane travelling wave in a nonlinear medium without dissipation or dispersion. Then there is Burgers equation for media with damping:

$$ \partial u/\partial t + V(u)\partial u/\partial x - \alpha {\partial^2}u/\partial {x^2} = 0 $$

(18.2)

(18.2) Finally, there is a generalized Korteveg--de Vries equation for a travelling wave in a medium with dispersion in the high-frequency range:

$$ \partial u/\partial t + V(u)\partial u/\partial x + \beta {\partial^3}u/\partial {x^3} = 0 $$

(18.3)

. (18.3)