7 - Self-similar solutions and travelling waves

Summary

Solutions of travelling-wave type

In various problems in mathematical physics an important role is played by invariant solutions of the travelling-wave type. These are solutions for which the distributions of the properties of the motion at different times can be obtained from one another by a translation rather than by a similarity transformation as in the case of self-similar solutions. In other words, one can always choose a moving Cartesian coordinate system such that the distribution of properties of a motion of travelling-wave type is stationary in that system. One can reduce to a consideration of travelling waves the study of the structure of shock-wave fronts in gas dynamics (see, e.g. Kochin, Kibel' and Roze, 1964; Zeldovich and Raizer, 1966, 1967) and in magneto-hydrodynamics (Kulikovsky and Lyubimov, 1965), the structure of flame fronts (Zeldovich, 1948; Zeldovich, Barenblatt, Librovich and Makhviladze, 1985), the investigation of solitary and periodic waves in a plasma and on the surface of a heavy fluid (Jeffrey and Kakutani, 1972; Whitham, 1974; Karpman, 1975; Lighthill, 1978; Eilenberger, 1981; Drazin and Johnson, 1989; Fordy, 1990), and many other problems. In recent years many processes have been studied involving the effects of the propagation of plasma fronts in electrical, electromagnetic, and light (laser) fields, the so-called waves of discharge propagation. These processes also lead to the consideration of solutions of travelling-wave type (Raizer, 1968, 1977).

In accordance with the definition given above, solutions of travelling-wave type can be expressed in the form

v = V(x − X) + V₀(t). (7.1)

Here v is the property of the phenomenon being considered; x is the spatial Cartesian coordinate, an independent variable of the problem; t is another independent variable, for simplicity identified with time; and X(t) and V₀(t) are time-dependent translations along the x- and v-axes. In particular, if the properties of the process do not depend directly on time, so that the equations governing the process do not contain time explicitly, the travelling-wave propagates uniformly:

v = V(x − λt + c) + μt. (7.2)