Asymptotic solutions of nonlinear wave equations using the methods of averaging and two-timing

Author:
R. W. Lardner

Journal:
Quart. Appl. Math. **35** (1977), 225-238

MSC:
Primary 35B25; Secondary 35L20

DOI:
https://doi.org/10.1090/qam/442443

MathSciNet review:
442443

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The method of averaging and the two-time procedure are outlined for a class of hyperbolic second-order partial differential equations with small nonlinearities. It is shown that they both lead to the same integro-differential equation for the lowest-order approximation to the solution. For the special case of the nonlinear wave equation, this lowest-order solution consists of the superposition of two modulated travelling waves, and separate integro-differential equations are derived for the amplitudes of these two waves. As an example, the wave equation with Van der Pol type of nonlinearity is considered.

**[1]**N. N. Krylov and N. N. Bogoliubov,*Introduction to nonlinear mechanics*, University of Princeton Press, 1947**[2]**N. N. Bogoliubov and Yu. A. Mitropolskii,*Asymptotic methods in the theory of nonlinear oscillations*, Gordon and Breach, 1961**[3]**D. J. Benney,*Nonlinear gravity-wave interactions*, J. Fluid Mech.**14**, 577-584 (1962) MR**0158616****[4]**L. F. McGoldrick,*Resonant interactions among capillary-gravity waves*, J. Fluid Mech.**21**, 305-331 (1965) MR**0174226****[5]**D. Montgomery and D. A. Tidman,*Secular and non-secular behaviour for the cold plasma equations, Phys. Fluids***7**, 242-249 (1964) MR**0161690****[6]**D. Montgomery,*Generalized perturbation expansion for the Klein-Gordon equation with small nonlinearity*, J. Math. and Phys.**5**, 1788-1795 (1964) MR**0170617****[7]**Yu. A. Mitropolskii and B. J. Moseenkov,*Lectures on the application of asymptotic methods to the solution of partial differential equations*, Academy of Sciences of Ukr. SSR, Kiev, 1968 MR**0235259****[8]**G. N. Bojadziev and R. W. Lardner,*Monofrequent oscillations in mechanical systems governed by second-order hyperbolic differential equations with small nonlinearities*, Int. J. Nonlinear Mech.**8**, 289-309 (1973) MR**0334623****[9]**G. N. Bojadziev and R. W. Lardner,*Second-order hyperbolic equations with small nonlinearities in the case of internal resonance*, Int. J. Nonlinear Mech.**9**, 397-407 (1974)**[10]**R. W. Lardner,*The formation of shock waves in KBM solutions of hyperbolic partial differential equations*, J. Sound Vibr.**39**, 489-502 (1975)**[11]**R. W. Lardner,*The development of plane shock-waves in nonlinear viscoelastic media*, Proc. Roy. Soc. A**347**, 329-344 (1976)**[12]**J. D. Cole and J. Kevorkian,*Uniformly valid approximations for certain nonlinear differential equations*, in*Proc. Int. Symp. on Nonlinear Diff. Eqs. and Nonlinear Mechs*., ed. J. P. LaSalle and S. Lefschetz, Academic Press, 1963, pp. 113-120 MR**0147701****[13]**J. A. Morrison,*Comparison of the modified method of averaging and the two variable expansion procedure*, SIAM Review**8**, 66-85 (1966) MR**0196212****[14]**A. H. Nayfeh,*Perturbation methods*, chap. VI, John Wiley, 1973 MR**0404788****[15]**J. B. Keller and S. Kogelman,*Asymptotic solutions of initial value problems for nonlinear partial differential equations*, SIAM J. Appl. Math.**18**, 748-758 (1970) MR**0262653****[16]**D. T. Davy and W. E. Ames,*An asymptotic solution of an initial value problem for a nonlinear viscoelastic rod*, Int. J. Nonlinear Mech.**8**, 59-71 (1973)**[17]**A. H. Nayfeh,*Finite amplitude longitudinal waves in non-uniform bars*, J. Sound Vibr.**42**, 357-361 (1975)**[18]**S. C. Chikwendu and J. Kevorkian,*A perturbation method for hyperbolic equations with small nonlinearities*, SIAM J. Appl. Math.**22**, 235-258 (1972) MR**0374635****[19]**J. P. Fink, W. S. Hall and A. R. Hausrath,*Convergent two-time method for periodic differential equations*, J. Diff. Eqs.**15**, 459-498 (1974) MR**0344620****[20]**W. Eckhaus,*New approach to the asymptotic theory of nonlinear oscillations and wave propagation*, J. Math. Anal. Applic.**49**, 575-611 (1975) MR**0369839****[21]**C. J. Myerscough,*A simple model of the growth of wind-induced oscillations in overhead lines*, J. Sound Vibr.**28**, 699-713 (1973);**39**, 503-517 (1975)**[22]**M. P. Mortell and E. Varley,*Finite amplitude waves in bounded media: nonlinear free vibration of an elastic panel*, Proc. Roy. Soc.*A 318*, 169-196 (1970) MR**0270613****[23]**M. P. Mortell and B. R. Seymour,*Pulse propagation in a nonlinear viscoelastic rod of finite length*, SIAM J. Appl. Math.**22**, 209-223 (1972)**[24]**J. C. Arya and R. W. Lardner,*Plane shock waves in viscoelastic media displaying cubic elasticity*, SFU preprint, 1977 MR**556994**

Retrieve articles in *Quarterly of Applied Mathematics*
with MSC:
35B25,
35L20

Retrieve articles in all journals with MSC: 35B25, 35L20

Additional Information

DOI:
https://doi.org/10.1090/qam/442443

Article copyright:
© Copyright 1977
American Mathematical Society