Premium
Resonantly Interacting Weakly Nonlinear Hyperbolic Waves in the Presence of Shocks: A Single Space Variable in a Homogeneous, Time Independent Medium
Author(s) -
Cehelsky Priscilla,
Rosales Rodolfo R.
Publication year - 1986
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1002/sapm1986742117
Subject(s) - inviscid flow , mathematical analysis , nonlinear system , mathematics , hyperbolic function , physics , hyperbolic partial differential equation , euler system , classical mechanics , partial differential equation , euler equations , quantum mechanics
We present a systematic asymptotic theory for resonantly interacting weakly nonlinear hyperbolic waves for a single space variable in a homogeneous, time independent medium. This theory extends the results previously presented by A. Majda and R. Rosales, under similar hypotheses, to the case where waves break and shocks form. Similarly the theory of nonresonant interacting waves for general hyperbolic systems developed recently by J. Hunter and J. B. Keller, when specialized to a single space variable, is included as a special case. However, we are also able to treat the general situation when resonances occur in the hyperbolic system. Such resonances are the typical case when the hyperbolic system has at least three equations and when, for example, small‐amplitude periodic initial data are prescribed. In the important physical example of the 3 ✕ 3 system describing compressible fluid flow in a single space variable, the resonant asymptotic theory developed by the authors yields, as limit equations, a pair of inviscid Burgers equations coupled through a linear integral operator with known kernel defined through the initial data for the entropy wave. These are the same equations derived by Majda and Rosales previously. However, the waves are displaced relative to the positions prescribed by them.