Numerical solution of the small dispersion limit of Korteweg-de Vries and whitham equations

Communications on Pure and Applied Mathematics

Volumn 60, Issue 11, 2007, Pages 1623-1664

  Grava, Tamara ^a   Klein, Christian ^b  

^a SISSA   (Italy)

^b MAX PLANCK INSTITUTE FOR MATHEMATICS IN THE SCIENCES   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 34948872169     PISSN: 00103640     EISSN: 00103640     Source Type: Journal    
DOI: 10.1002/cpa.20183     Document Type: Article

References (49)

1. Abramowitz, M.; Stegun, I. A., eds. Handbook of mathematical functions, with formulas, graphs, and mathematical tables. National Bureau of Standards Applied Mathematics Series, 55. Dover, New York, 1965.
2. Acton, F. S. Analysis of straight-line data. Dover, New York, 1966.
3. Avilov, V. V.; Novikov, S. P. Evolution of the Whitham zone in KdV theory. Dokl. Akad. Nauk SSSR 294 (1987), no. 2, 325-329;
4. translation in Soviet Phys. Dokl. 32 (1987), 366-368.
5. Bleher, P.; Its, A. Double scaling limit in the random matrix model: the Riemann-Hilbert approach. Comm. Pure Appl. Math. 56 (2003), no. 4, 433-516.
6. Brézin, E.; Marinari, E.; Parisi, G. A nonperturbative ambiguity free solution of a string model. Phys. Lett. B 242 (1990), no. 1, 35-38.
7. Canuto, C.; Hussaini, M. Y.; Quarteroni, A.; Zang, T. A. Spectral methods in fluid dynamics. Springer Series in Computational Physics. Springer, New York, 1988.
8. Claeys, T.; Kuijlaars, A. B. J.; Vanlessen, M. Multi-critical unitary random matrix ensembles and the general Painleve II equation. arXiv: math-ph/0508062, 2005.
9. Claeys, T.; Vanlessen, M. The existence of a real pole-free solution of the fourth order analogue of the Painleve I equation. arXiv: math-ph/0604046, 2006.
10. Deift, P.; Kriecherbauer, T.; McLaughlin, K. T.-R.; Venakides, S.; Zhou, X. Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory. Comm. Pure Appl. Math. 52 (1999), no. 11, 1335-1425.
11. Deift, P.; Venakides, S.; Zhou, X. New results in small dispersion KdV by an extension of the steepest descent method for Riemann-Hilbert problems. Internat. Math. Res. Notices 1997, no. 6, 286-299.
12. Dubrovin, B. On Hamiltonian perturbations of hyperbolic systems of conservation laws. II. Universality of critical behaviour. Comm. Math. Phys. 267 (2006), no. 1, 117-139. arXiv: math-ph/0510032, 2005.
13. Dubrovin, B. A.; Novikov, S. P. A periodic problem for the Korteweg-de Vries and Sturm-Liouville equations. Their connection with algebraic geometry. Dokl. Akad. Nauk SSSR 219 (1974), 531-534.
14. Dubrovin, B. A.; Novikov, S. P. Hydrodynamics of weakly deformed soliton lattices. Differential geometry and Hamiltonian theory. Uspekhi Mat. Nauk 44 (1989), no. 6 (270), 29-98;
15. translation in Russian Math. Surveys 44 (1989), no. 6, 35-124.
16. El, G. A.; Krylov, A. L.; Venakides, S. Unified approach to KdV modulations. Comm. Pure Appl. Math. 54 (2001), no. 10, 1243-1270.
17. Eynard, B. Universal distribution of random matrix eigenvalues near the 'birth of a cut' transition. J. Stat. Mech. Theory Exp. 2006, no. 7, P07005, 33 pp. (electronic). arXiv: mathph/0605064, 2006.
18. Flaschka, H.; Forest, M. G.; McLaughlin, D. W. Multiphase averaging and the inverse spectral solution of the Korteweg-de Vries equation. Comm. Pure App. Math. 33 (1980), no. 6,739-784.
19. Fornberg, B. A practical guide to pseudospectral methods. Cambridge Monographs on Applied and Computational Mathematics, 1. Cambridge University Press, Cambridge, 1996.
20. Frauendiener, J.; Klein, C. Hyperelliptic theta-functions and spectral methods. J. Comput. Appl. Math. 167 (2004), no. 1, 193-218.
21. Grava, T. From the solution of the Tsarev system to the solution of the Whitham equations. Math. Phys. Anal. Geom. 4 (2001), no. 1, 65-96.
22. Grava, T.; Klein, C. Numerical study of a multiscale expansion of KdV and Camassa-Holm equation. arXiv: math-ph/0702038, 2007.
23. Grava, T.; Tian, F.-R. The generation, propagation, and extinction of multiphases in the KdV zero-dispersion limit. Comm. Pure Appl. Math. 55 (2002), no. 12, 1569-1639.
24. Gurevich, A. V.; Pitaevskii, L. P. Nonstationary structure of a collisionless shock wave. Ž. Èksper. Teoret. Fiz. 65 (1973), no. 8, 590-604;
25. translation in Soviet Physics JETP 38 (1974), no. 2, 291-297.
26. Its, A. R.; Matveev, V. B. Hill operators with a finite number of lacunae. Funkcional. Anal, i Priložen. 9 (1975), no. 1, 69-70.
27. Jin, S.; Levermore, C. D.; McLaughlin, D. W. The semiclassical limit of the defocusing NLS hierarchy. Comm. Pure Appl. Math. 52 (1999), no. 5, 613-654.
28. Jorge, M. C.; Minzoni, A. A.; Smyth, N. F. Modulation solutions for the Benjamin-Ono equation. Phys. D 132 (1999), no. 1-2, 1-18.
29. Kamvissis, S.; McLaughlin, K. D. T.-R.; Miller, P. D. Semiclassical soliton ensembles for the focusing nonlinear Schrödinger equation. Annals of Mathematics Studies, 154. Princeton University Press, Princeton, N.J., 2003.
30. Klein, C. Fourth order time-stepping for low dispersion Korteweg-de Vries and nonlinear Schrödinger equation. Available at: http://www.mis.mpg.de/ preprints/2006/ prepr2006_133.html, 2006.
31. Klein, C.; Richter, O. Ernst equation and Riemann surfaces. Analytical and numerical methods. Lecture Notes in Physics, 685. Springer, Berlin, 2005.
32. Krichever, I. M. The averaging method for two-dimensional "integrable" equations. Funktsional. Anal, i Priložen. 22 (1988), no. 3, 37-52;
33. translation in Funct. Anal. Appl. 22 (1988), no. 3, 200-213.
34. Kudashev, V.; Suleimanov, B. A soft mechanism for the generation of dissipationless shock waves. Phys. Lett. A 111 (1996), 204-208.
35. [31 ] Lagarias, J. C.; Reeds, J. A.; Wright, M. H.; Wright, P. E. Convergence properties of the Nelder-Mead simplex method in low dimensions. SIAM J. Optim. 9 (1999), no. 1, 112-147 (electronic).
36. Lawden, D. F. Elliptic functions and applications. Applied Mathematical Sciences, 80. Springer, New York, 1989.
37. Lax, P. D.; Levermore, C. D. The small dispersion limit of the Korteweg-de Vries equation. I. II. III. Comm. Pure Appl. Math. 36 (1983), no. 3, 253-290;
38. no. 5, 571-593;
39. no. 6, 809-829.
40. Levermore, C. D. The hyperbolic nature of the zero dispersion KdV limit. Comm. Partial Differential Equations 13 (1988), no. 4, 495-514.
41. McLaughlin, D. W.; Strain, J. A. Computing the weak limit of KdV. Comm. Pure Appl. Math. 47 (1994), no. 10, 1319-1364.
42. Tian, F. R. Oscillations of the zero dispersion limit of the Korteweg-de Vries equation. Comm. Pure Appl. Math. 46 (1993), no. 8, 1093-1129.
43. Tian, F. R. The initial value problem for the Whitham averaged system. Comm. Math. Phys. 166 (1994), no. 1, 79-115.
44. Trefethen, L. N. Spectral methods in MATLAB. Software, Environments, and Tools, 10. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 2000. Codes available at: http://web.comlab.ox.ac.uk/oucl/ work/nick.trefethen/spectral.html
45. Tsarëv, S. P. Poisson brackets and one-dimensional Hamiltonian systems of hydrodynamic type. Dokl. Akad. Nauk SSSR 282 ( 1985), no. 3, 534-537;
46. translation in Soviet Math. Dokl. 31 (1985), 488-491.
47. Venakides, S. The zero dispersion limit of the Korteweg-de Vries equation for initial potentials with nontrivial reflection coefficient. Comm. Pure Appl. Math. 38 (1985), no. 2, 125-155.
48. Venakides, S. The Korteweg-de Vries equation with small dispersion: higher order Lax-Lever-more theory. Comm. Pure Appl. Math. 43 (1990), no. 3, 335-361.
49. Whitham, G. B. Linear and nonlinear waves. Pure and Applied Mathematics. Wiley-Interscience, New York-London-Sydney, 1974.