From the solution of the tsarev system to the solution of the whitham equations

Mathematical Physics, Analysis and Geometry

Volumn 4, Issue 1, 2001, Pages 65-96

  Grava, Tamara ^a,b  

^a University of Maryland   (United States)

^b IMPERIAL COLLEGE LONDON   (United Kingdom)

Author keywords

Hyperelliptic riemann surfaces; Linear overdetermined systems of euler poisson darboux type; Whitham equations

Indexed keywords

EID: 0012307257     PISSN: 13850172     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1011892100149     Document Type: Article

References (20)

1. Whitham, G. B.: Linear and Nonlinear Waves, Wiley, New York, 1974.
2. Flaschka, H., Forest, M. and McLaughlin, D. H.: Multiphase averaging and the inverse spectral solution of the Korteweg-de Vries equations, Comm. Pure Appl. Math. 33 (1980), 739-784.
3. Lax, P. D. and Levermore, C. D.: The small dispersion limit of the Korteweg-de Vries equation, I, II, III, Comm. Pure Appl. Math. 36 (1983), 253-290, 571-593, 809-830.
4. Levermore, C. D.: The hyperbolic nature of the zero dispersion KdV limit, Comm. Partial Differential Equations 13 (1988), 495-514.
5. Fei Ran Tian: Oscillations of the zero dispersion limit of the Korteweg-de Vries equations, Comm. Pure Appl. Math. 46 (1993), 1093-1129.
6. Grava, T.: Existence of a global solution of the Whitham equations, Theoret. Math. Phys. 122(1) (2000), 46-58.
7. Dubrovin, B. and Novikov, S. P.: Hydrodynamic of weakly deformed soliton lattices. Differential geometry and Hamiltonian theory, Russian Math. Surveys 44(6) (1989), 35-124.
8. Tsarev, S. P.: Poisson brackets and one-dimensional Hamiltonian systems of hydrodynamic type, Soviet Math. Dokl. 31 (1985), 488-491.
9. Gurevich, A. G. and Pitaevskii, L. P.: Nonstationary structure of collisionless shock waves, JEPT Lett. 17 (1973), 193-195.
10. Krichever, I. M.: The method of averaging for two dimensional integrable equations, Fund. Anal. Appl. 22 (1988), 200-213.
11. Potemin, G. V.: Algebraic-geometric construction of self-similar solutions of the Whitham equations, Uspekhi Mat. Nauk 43(5) (1988), 211-212.
12. Kudashev, V. R. and Sharapov, S. E.: Inheritance of KdV symmetries under Whitham averaging and hydrodynamic symmetry of the Whitham equations, Theoret. Math. Phys. 87(1) (1991), 40-47.
13. Gurevich, A. V., Krylov, A. L. and El, G. A.: Evolution of a Riemann wave in dispersive hydrodynamics, Soviet Phys. JEPT 74(6) (1992), 957-962.
14. Eisenhart, L. P.: Ann. of Math. 120 (1918), 262.
15. Fei Ran Tian: The Whitham type equations and linear over-determined systems of Euler-Poisson-Darboux type, Duke Math. J. 74 (1994), 203-221.
16. El, G. A.: Generating function of the Whitham-KdV hierarchy and effective solution of the Cauchy problem, Phys. Lett. A 222 (1996), 393-399.
17. Springer, G.: Introduction to Riemann Surfaces, Addison-Wesley, Reading, Mass., 1957.
18. Rodin, Yu. L.: The Riemann Boundary Value Problem on Riemann Surfaces, Math. Appl. Soviet Ser., D. Reidel, Dordrecht, 1987.
19. Dubrovin, B.: Lectures on 2-D Topological Field Theory, Lecture Notes in Math. 1620, Springer-Verlag, Berlin, 1996.
20. Fay, J.: Theta Functions on Riemann Surface, Lecture Notes in Math. 352, Springer-Verlag, Heidelberg, 1973.