Two-timing procedure for higher-order modulations of near-linear dispersive wave trains with an application to plasma waves
Author:
P. Gatignol
Journal:
Quart. Appl. Math. 35 (1977), 357-373
MSC:
Primary 82.65
DOI:
https://doi.org/10.1090/qam/469028
MathSciNet review:
469028
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: An asymptotic procedure is proposed to describe the slow modulations of dispersive wave trains when the dispersive effects and the nonlinear distorsion are of the same order of magnitude. For a model equation, the system of modulation equations is derived up to the second order. At this order of approximation, it is seen that the dispersion relation includes partial derivatives not only of the amplitude but also of the wave vector components. Under some assumptions, a partial differential equation is obtained for the complex amplitude. This equation reveals common features with the modified Korteweg-de Vries equation and with a nonlinear Schrödinger equation. At the first order, it reduces to the cubic Schrödinger equation which had been directly obtained by several authors. Finally, the theory is applied to a plasma wave example.
- G. B. Whitham, Non-linear dispersive waves, Proc. Roy. Soc. London Ser. A 283 (1965), 238–261. MR 176724, DOI https://doi.org/10.1098/rspa.1965.0019
- G. B. Whitham, A general approach to linear and non-linear dispersive waves using a Lagrangian, J. Fluid Mech. 22 (1965), 273–283. MR 182236, DOI https://doi.org/10.1017/S0022112065000745
- David Montgomery and Derek A. Tidman, Secular and nonsecular behavior for the cold plasma equations, Phys. Fluids 7 (1964), 242–249. MR 161690, DOI https://doi.org/10.1063/1.1711139
A. H. Nayfeh, Nonlinear oscillations in a hot electron plasma, Phys. Fluids 8, 1896–1898 (1965)
- P. A. Sturrock, Non-linear effects in electron plasmas, Proc. Roy. Soc. London Ser. A 242 (1957), 277–299. MR 90359, DOI https://doi.org/10.1098/rspa.1957.0176
T. Kawahara, The derivative-expansion method and nonlinear dispersive waves, J. Phys. Soc. Japan 35, 1537–1544 (1973)
- Tosiya Taniuti and Nobuo Yajima, Perturbation method for a nonlinear wave modulation. I, J. Mathematical Phys. 10 (1969), 1369–1372. MR 251353, DOI https://doi.org/10.1063/1.1664975
T. Taniuti, Reductive perturbation method and far fields of wave equations, Prog. Theor. Phys., Supp. 55, 1–35 (1974)
- T. Kakutani and N. Sugimoto, Krylov-Bogoliubov-Mitropolsky method for nonlinear wave modulation, Phys. Fluids 17 (1974), 1617–1625. MR 351257, DOI https://doi.org/10.1063/1.1694942
- J. C. Luke, A perturbation method for nonlinear dispersive wave problems, Proc. Roy. Soc. London Ser. A 292 (1966), 403–412. MR 195289, DOI https://doi.org/10.1098/rspa.1966.0142
- G. B. Whitham, Two-timing, variational principles and waves, J. Fluid Mech. 44 (1970), 373–395. MR 269965, DOI https://doi.org/10.1017/S002211207000188X
- Philippe Gatignol, Sur une méthode asymptotique en théorie des ondes dispersives non linéaires, J. Mécanique 11 (1972), 95–117 (French, with English summary). MR 371244
- Philippe Gatignol, Description asymptotique des solutions d’ondes périodiques lentement variables pour l’équation de Korteweg-de Vries, C. R. Acad. Sci. Paris Sér. A-B 274 (1972), A1861–A1864 (French). MR 298259
W. D. Hayes, Conservation of action and modal wave action, Proc. Roy. Soc. London A 320, 187–208 (1970)
- G. B. Whitham, Linear and nonlinear waves, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. Pure and Applied Mathematics. MR 0483954
- Robert M. Miura, Korteweg-de Vries equation and generalizations. I. A remarkable explicit nonlinear transformation, J. Mathematical Phys. 9 (1968), 1202–1204. MR 252825, DOI https://doi.org/10.1063/1.1664700
- Philippe Gatignol, Sur la forme et la recherche des densités invariantes associées à certaines équations aux dérivées partielles non-linéaires, Bul. Inst. Politehn. Iaşi (N.S.) 14(18) (1968), no. fasc. 1-2, 51–60 (French, with Romanian summary). MR 236501
G. B. Whitham, Nonlinear dispersive waves, Proc. Roy. Soc. London A 283, 238–262 (1965)
G. B. Whitham, A general approach to linear and nonlinear dispersive waves using a Lagrangian, J. Fluid Mech. 22, 273–283 (1965)
D. Montgomery and D. A. Tidman, Secular and non-secular behavior for the cold plasma equations, Phys. Fluids 7, 242–249 (1964)
A. H. Nayfeh, Nonlinear oscillations in a hot electron plasma, Phys. Fluids 8, 1896–1898 (1965)
P. A. Sturrock, Nonlinear effects in electron plasmas, Proc. Roy. Soc. London A 242, 277–299 (1957)
T. Kawahara, The derivative-expansion method and nonlinear dispersive waves, J. Phys. Soc. Japan 35, 1537–1544 (1973)
T. Taniuti and N. Yajima, Perturbation method for a nonlinear wave modulation, J. Math. Phys. 10, 1369–1372 (1969)
T. Taniuti, Reductive perturbation method and far fields of wave equations, Prog. Theor. Phys., Supp. 55, 1–35 (1974)
T. Kakutani and N. Sugimoto, Krylov-Bogoliubov-Mitropolsky method for nonlinear wave modulation, Phys. Fluids 17, 1617–1625 (1974)
J. C. Luke, A perturbation method for nonlinear dispersive wave problems, Proc. Roy. Soc. London A 292, 403–412 (1966)
G. B. Whitham, Two-timing, variational principles and waves, J. Fluid Mech. 44, 373–395 (1970)
P. Gatignol, Sur une méthode asymptotique en théorie des ondes dispersives non-linéaires, J. Mécan. 11, 95–117 (1972)
P. Gatignol, Description asymptotique des solutions d’ondes périodiques lentement variables pour l’ équation de Korteweg-de Vries, C. R. Acad. Sc. Paris A 274, 1861–1864 (1972)
W. D. Hayes, Conservation of action and modal wave action, Proc. Roy. Soc. London A 320, 187–208 (1970)
G. B. Whitham, Linear and nonlinear waves, Part II, J. Wiley & Sons, New York, 1974
R. M. Miura, Korteweg-de Vries equation and generalizations, J. Math. Phys. 9, 1202–1204 (1968)
P. Gatignol, Sur la forme et la recherche des densités invariantes associées à certaines équations aux dérivées partielles non-linéaires, Bul. Inst. Polit. Iasi 14, 51–60 (1968)
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC:
82.65
Retrieve articles in all journals
with MSC:
82.65
Additional Information
Article copyright:
© Copyright 1977
American Mathematical Society