The interaction of shocks with dispersive waves I. Weak coupling limit

Studies in Applied Mathematics

Volumn 96, Issue 2, 1996, Pages 201-246

  Axel, Ralph M ^b   Newton, Paul K ^a  

^a UNIVERSITY OF SOUTHERN CALIFORNIA   (United States)

^b NONE

Author keywords

[No Author keywords available]

Indexed keywords

DISPERSIVE WAVES; WEAK-COUPLING LIMITS; LINEAR WAVE EQUATION; NONLINEAR WAVES EQUATION; PHASE GRADIENT; STEADY STATE; TIME-DEPENDENT SOLUTIONS; TURBULENCE THEORY; WAVE ENVELOPE; ZAKHAROV SYSTEM;

DIFFUSION; DIGITAL STORAGE; DISPERSION (WAVES); NONLINEAR EQUATIONS; WAVE EQUATIONS;

EXPANSION;

EID: 0041357239     PISSN: 00222526     EISSN: None     Source Type: Journal    
DOI: 10.1002/sapm1996962201     Document Type: Article

References (23)

1. D. G. CRIGHTON, A. P. DOWLING, J. E. FFOWCS WILLIAMS, M. HECKL and F. G. LEPPINGTON, Modern Methods in Analytical Acoustics, Lecture Notes, Springer-Verlag, 1992.
2. C. S. GARDNER, J. M. GREENE, M. D. KRUSKAL, and R. M. MUIRA, Method for solving the KdV equation, Phys. Rev. Lett. 19: 1095 (1967).
3. M. V. GOLDMAN, Strong turbulence of plasma waves, Rev. Mod. Phys. 56 (4): 709 (1984).
4. J. K. HALE and H. KOÇAK, Dynamics and Bifurcations, Springer-Verlag, 1991.
5. M. HUSSEY, Diagnostic Ultrasound: An Introduction to the Interactions between Ultrasound and Biological Tissues, Wiley, 1975.
6. L. HUTCHINS and S. LEEMAN, Tissue parameter measurement and imaging, in Acoustical Imaging, (J. P. Powers, Ed.), Vol. 11, Plenum Press, 1982.
7. V. I. KARPMAN, Nonlinear Waves in Dispersive Media, Pergamon Press, 1975.
8. J. KEVORKIAN and J. COLE, Perturbation Methods in Applied Mathematics, Applied Math. Sci., Vol. 34, Springer, 1980.
9. P. D. LAX, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, SIAM, 1972.
10. P. D. LAX, Shock waves, increase of entropy and loss of information, in Seminar on Nonlinear PDE (S. S. Chern, Ed.), MSRI Pub. 2, Springer, 1984.
11. P. D. LAX, Integrals of nonlinear equations of evolution and solitary waves, Comm. Pure Appl. Math. 21: 467 (1968).
12. P. D. LAX and C. D. LEVERMORE, The small dispersion limit for the KdV equation I, Comm. Pure Appl. Math. 36: 253 (1983).
13. P. D. LAX and C. D. LEVERMORE, The small dispersion limit for the KdV equation II, Comm. Pure Appl. Math. 36: 571 (1983).
14. P. D. LAX and C. D. LEVERMORE, The small dispersion limit for the KdV equation III, Comm. Pure Appl. Math. 809 (1983).
15. S. LEEMAN, Ultrasound pulse propagation in dispersive media, Phys. Med. Biol. 25:481 (1980).
16. P. K. NEWTON, Wave interactions in the singular Zakharov system, J. Math. Phys. 32 (2): 431 (1991).
17. O. OLEINIK, Discontinuous solutions of nonlinear differential equations, Usp. Mat. Nauk. 12: 3 (1957). [English trans: Amer. Math. Soc. Transl. Ser. 2 26: 95, (1962)].
18. O. OLEINIK, Discontinuous solutions of nonlinear differential equations, Usp. Mat. Nauk. 12: 3 (1957). [English trans: Amer. Math. Soc. Transl. Ser. 2 26: 95, (1962)].
19. S. H. SCHOCHET and M. I. WEINSTEIN, The nonlinear Schrödinger limit of the Zakharov equations governing Langmuir turbulence, Comm. Math. Phys. 106: 569 (1986).
20. M. SLEMROD, Interrelationships among mechanics, numerical analysis, compensated compactness, and oscillation theory, in Oscillation Theory, Computation, and Methods of Compensated Compactness (Dafermos, Eriksen, Kinderlehrer, and Slemrod, Eds.) IMA, Vol. 2, Springer-Verlag, 1986.
21. J. SMOLLER, Shock Waves and Reaction Diffusion Equations, Springer-Verlag, 1983.
22. G. B. WHITHAM, Linear and Nonlinear Waves, Wiley, 1974.
23. V. E. ZAKHAROV, Collapse of Langmuir waves, Sov. Phys. 17: 153 (1977).