The small dispersion limit for a nonlinear semidiscrete system of equations

Studies in Applied Mathematics

Volumn 99, Issue 3, 1997, Pages 205-254

  Turner, Cristina Vilma ^b   Rosales, Rodolfo Rubén ^a  

^a MASSACHUSETTS INSTITUTE OF TECHNOLOGY   (United States)

^b NONE

Author keywords

[No Author keywords available]

Indexed keywords

CLASSICAL SOLUTIONS; DISPERSION LIMIT; GRID SIZE; HOPF EQUATIONS; MODULATION EQUATIONS; NUMERICAL EXPERIMENTS; OSCILLATORY BEHAVIORS; SMALL DISPERSION; SPATIAL PERIODS; SYSTEMS OF EQUATIONS;

NONLINEAR EQUATIONS;

EID: 0039165805     PISSN: 00222526     EISSN: None     Source Type: Journal    
DOI: 10.1111/1467-9590.00060     Document Type: Article

References (36)

1. P. D. LAX and C. D. LEVERMORE, The zero dispersion limit of the Korteweg-de Vries equation, Proc. Natl. Acad. Sci. USA 76(8):3602-3606 (1979); The small dispersion limit of the Korteweg-de Vries equation, I, II, III, Comm. Pure Appl. Math. 36:253-290, 571-593, 809-829 (resp.) (1983).
2. P. D. LAX and C. D. LEVERMORE, The zero dispersion limit of the Korteweg-de Vries equation, Proc. Natl. Acad. Sci. USA 76(8):3602-3606 (1979); The small dispersion limit of the Korteweg-de Vries equation, I, II, III, Comm. Pure Appl. Math. 36:253-290, 571-593, 809-829 (resp.) (1983).
3. S. VENAKIDES, The zero dispersion limit of the Korteweg-de Vries equation with nontrivial reflection coefficient, Comm. Pure Appl. Math. 38:125-155 (1985); The generation of modulated wave trains in the solution of the Korteweg-de Vries equation, Comm. Pure Appl. Math. 38:883-909 (1985); The zero dispersion limit of the Korteweg-de Vries equation with periodic initial data, AMS Trans. 301:189-225 (1987); The continuum limit of theta functions, Comm. Pure Appl. Mam. 42:711-728 (1989); Higher order Lax-Levermore theory, Comm. Pure Appl. Math. 43:335-362 (1990).
4. S. VENAKIDES, The zero dispersion limit of the Korteweg-de Vries equation with nontrivial reflection coefficient, Comm. Pure Appl. Math. 38:125-155 (1985); The generation of modulated wave trains in the solution of the Korteweg-de Vries equation, Comm. Pure Appl. Math. 38:883-909 (1985); The zero dispersion limit of the Korteweg-de Vries equation with periodic initial data, AMS Trans. 301:189-225 (1987); The continuum limit of theta functions, Comm. Pure Appl. Mam. 42:711-728 (1989); Higher order Lax-Levermore theory, Comm. Pure Appl. Math. 43:335-362 (1990).
5. S. VENAKIDES, The zero dispersion limit of the Korteweg-de Vries equation with nontrivial reflection coefficient, Comm. Pure Appl. Math. 38:125-155 (1985); The generation of modulated wave trains in the solution of the Korteweg-de Vries equation, Comm. Pure Appl. Math. 38:883-909 (1985); The zero dispersion limit of the Korteweg-de Vries equation with periodic initial data, AMS Trans. 301:189-225 (1987); The continuum limit of theta functions, Comm. Pure Appl. Mam. 42:711-728 (1989); Higher order Lax-Levermore theory, Comm. Pure Appl. Math. 43:335-362 (1990).
6. S. VENAKIDES, The zero dispersion limit of the Korteweg-de Vries equation with nontrivial reflection coefficient, Comm. Pure Appl. Math. 38:125-155 (1985); The generation of modulated wave trains in the solution of the Korteweg-de Vries equation, Comm. Pure Appl. Math. 38:883-909 (1985); The zero dispersion limit of the Korteweg-de Vries equation with periodic initial data, AMS Trans. 301:189-225 (1987); The continuum limit of theta functions, Comm. Pure Appl. Mam. 42:711-728 (1989); Higher order Lax-Levermore theory, Comm. Pure Appl. Math. 43:335-362 (1990).
7. S. VENAKIDES, The zero dispersion limit of the Korteweg-de Vries equation with nontrivial reflection coefficient, Comm. Pure Appl. Math. 38:125-155 (1985); The generation of modulated wave trains in the solution of the Korteweg-de Vries equation, Comm. Pure Appl. Math. 38:883-909 (1985); The zero dispersion limit of the Korteweg-de Vries equation with periodic initial data, AMS Trans. 301:189-225 (1987); The continuum limit of theta functions, Comm. Pure Appl. Mam. 42:711-728 (1989); Higher order Lax-Levermore theory, Comm. Pure Appl. Math. 43:335-362 (1990).
8. P. D. LAX, C. D. LEVERMORE, and S. VENAKIDES, The generation and propagation of oscillations in dispersive IVPs and their limiting behavior, in Important Developments in Soliton Theory 1980-1990 (T. Fokas and V. E. Zakharov, Eds.), pp. 205-241, Springer Verlag, Berlin, 1992.
9. G. STRANG, Accurate partial differential methods, II, Numer. Math. 6:37-46 (1964).
10. G. B. WHITHAM, Linear and Nonlinear Waves, Wiley, New York, 1974.
11. D. W. MCLAUGHLIN and J. STRAIN, Calculating the weak limit of KdV by quadratic programming, in Singular Limits of Dispersive Waves, (N. Ercolani, I. Gabitov, D. Levermore, and D. Serre, Eds.), NATO ARW Series B320, Plenum, New York, 1994.
12. F. R. TIAN, Oscillations of the zero dispersion limit of the Korteweg-de Vries equation, Comm. Pure Appl. Math. 46:1093-1129 (1993).
13. O. C. WRIGHT, Korteweg-de Vries Zero Dispersion Limit: A Restricted Initial Value Problem, Ph.D. dissertation, Princeton University, 1991.
14. H. FLASCHKA, M. G. FOREST, and D. W. MCLAUGHLIN, Multiple phase averaging and the inverse spectral solution of the Korteweg-de Vries equation, Comm. Pure Appl. Math. 33:739-784 (1980).
15. M. J. ABLOWITZ and H. SEGUR, Solutions and the Inverse Spectral Transform, SIAM, Philadelphia, 1981.
16. A. V. GUREVICH and L. P. PITAEVSKII, Nonstationary structure of a collisionless shock wave, Soviet Phys. JETP 38:291-297 (1974).
17. M. HAYS, C. D. LEVERMORE, and P. D. MILLER, Macroscopic lattice dynamics, Phys. D 79:1-15 (1994).
18. P. ROSENAU and J. M. HYMAN, Compactons: Solitons with finite wavelength, Phys. Rev. Lew. 70:564-567 (1993).
19. H. FLASCHKA, The Toda lattice, I, Phvs. Rev. B 9:1924-1925 (1974); On the Toda lattice, II, Prog. Theoret. Phys. 51:703-716, (1974).
20. H. FLASCHKA, The Toda lattice, I, Phvs. Rev. B 9:1924-1925 (1974); On the Toda lattice, II, Prog. Theoret. Phys. 51:703-716, (1974).
21. M. TODA, Theory of Nonlinear Lattices, Springer Series in Solid-State Sciences, Vol. 20, 2nd enlarged ed., Springer Verlag, New York, 1988.
22. M. KAC and P. VAN MOERBEKE, On an explicitly soluble system of nonlinear differential equations related to certain Toda lattices, Adv. Math. 16:160-169 (1975).
23. J. MOSER, Three integrable Hamiltonian systems connected with isospectral deformations, Adv. Math. 16:197-220 (1975).
24. B. L. HOLIAN, H. FLASCHKA, and D. W. MCLAUGHLIN, Shock waves in the Toda lattice: Analysis, Phys. Rev. A 24:2595-2623 (1981).
25. S. VENAKIDES, P. DEIFT, and R. OBA, The Toda shock problem, Comm. Pure Appl. Math. 44:1171-1242 (1990).
26. J. GOODMAN and P. D. LAX, On dispersive difference schemes, I, Comm. Pure Appl. Math. 41:591-613 (1988).
27. B. T. HAYES, Binary modulated oscillations in a semi-discrete version of Burgers equation, Phys. D, in press (1997).
28. T. Y. Hou and P. D. LAX, Dispersive approximations in fluid dynamics, Comm. Pure Appl. Math. 44:1-40 (1991).
29. C. D. LEVERMORE and J.-G. LIU, Large oscillations arising in a dispersive numerical scheme, Physica D, 99:191-216 (1996). Oscillations arising in numerical experiments, in Singular Limits of Dispersive Wares (N. Ercolani, I. Gabitov, D. Levermore, and D. Serre, Eds.), NATO ARW Series B320, pp. 329-346, Plenum, New York, 1994.
30. C. D. LEVERMORE and J.-G. LIU, Large oscillations arising in a dispersive numerical scheme, Physica D, 99:191-216 (1996). Oscillations arising in numerical experiments, in Singular Limits of Dispersive Wares (N. Ercolani, I. Gabitov, D. Levermore, and D. Serre, Eds.), NATO ARW Series B320, pp. 329-346, Plenum, New York, 1994.
31. J. VON NEUMANN, Proposal and analysis of a new numerical method in the treatment of hydrodynamical shock problems in Collected Works. VI Pergamon Press, New York, 1961.
32. E. FERMI, J. PASTA, and S. ULAM, Studies of Nonlinear Problems, Los Alamos Report LA-1940 (1955), in Collected Papers of Enrico Fermi. II, pp. 978-988 Univ. of Chicago Press, Chicago, 1965.
33. J. C. BRONSKI and D. W. MCLAUGHLIN, Semiclassical behavior in the NLS equation: Optical shocks-focusing instabilities, in Singular Limits of Dispersive Waves (N. Ercolani, I. Gabitov, D. Levermore, and D. Serre, Eds.), NATO ARW Series B320, Plenum, New York, 1994.
34. S. JIN, C. D. LEVERMORE, and D. W. MCLAUGHLIN, The behavior of solutions of the NLS equation in the semiclassical limit, in Singular Limits of Dispersive Waves (N. Ercolani, I. Gabitov, D. Levermore, and D. Serre, Eds.), NATO ARW Series B320, pp. 335-355, Plenum, New York, 1994.
35. P. D. LAX, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, SIAM Regional Conference Series in Applied Mathematics, Vol. 11, 1973.
36. L. F. SHAMPINE and M. K. GORDON, Computer Solution of Ordinary Differential Equations, W. H. Freeman, San Francisco, 1975.