Universality of the Break-up Profile for the KdV Equation in the Small Dispersion Limit Using the Riemann-Hilbert Approach

Article

Abstract

We obtain an asymptotic expansion for the solution of the Cauchy problem for the Korteweg-de Vries (KdV) equation
$$u_t+6uu_x+\epsilon^{2}u_{xxx}=0,\quad u(x,t=0,\epsilon)=u_0(x),$$
for \({\epsilon}\) small, near the point of gradient catastrophe (xc, tc) for the solution of the dispersionless equation ut + 6uux = 0. The sub-leading term in this expansion is described by the smooth solution of a fourth order ODE, which is a higher order analogue to the Painlevé I equation. This is in accordance with a conjecture of Dubrovin, suggesting that this is a universal phenomenon for any Hamiltonian perturbation of a hyperbolic equation. Using the Deift/Zhou steepest descent method applied on the Riemann-Hilbert problem for the KdV equation, we are able to prove the asymptotic expansion rigorously in a double scaling limit.

References

  1. 1.
    Abramowitz M., Stegun I.A.: Handbook of mathematical functions. Dover Publications, New York (1968)Google Scholar
  2. 2.
    Baik J., Deift P., Johansson K.: On the distribution of the length of the longest increasing subsequence of random permutations. J. Amer. Math. Soc. 12, 1119–1178 (1999)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Beals, R., Deift, P., Tomei, C.: Direct and inverse scattering on the line. In: Mathematical Surveys and Monographs 28, Amer. Math. Soc., Providence, RI: 1988Google Scholar
  4. 4.
    Bressan, A.: One dimensional hyperbolic systems of conservation laws. In: Current developments in mathematics 2002, Somerville, MA: Int. Press, 2003, pp. 1–37Google Scholar
  5. 5.
    Bowick M.J., Brézin E.: Universal scaling of the tail of the density of eigenvalues in random matrix models. Phys. Lett. B 268(1), 21–28 (1991)CrossRefADSMathSciNetGoogle Scholar
  6. 6.
    Brézin E., Marinari E., Parisi G.: A nonperturbative ambiguity free solution of a string model. Phys. Lett. B 242(1), 35–38 (1990)CrossRefADSMathSciNetGoogle Scholar
  7. 7.
    Camassa R., Holm D.D.: An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71, 1661–1664 (1993)MATHCrossRefADSMathSciNetGoogle Scholar
  8. 8.
    Claeys T., Vanlessen M.: The existence of a real pole-free solution of the fourth order analogue of the Painlevé I equation. Nonlinearity 20, 1163–1184 (2007)MATHCrossRefADSMathSciNetGoogle Scholar
  9. 9.
    Claeys T., Vanlessen M.: Universality of a double scaling limit near singular edge points in random matrix models. Commun. Math. Phys. 273, 499–532 (2007)MATHCrossRefADSMathSciNetGoogle Scholar
  10. 10.
    Degiovanni L., Magri F., Sciacca V.: On deformation of Poisson manifolds of hydrodynamic type. Commun. Math. Phys. 253(1), 1–24 (2005)MATHCrossRefADSMathSciNetGoogle Scholar
  11. 11.
    Deift, P.: Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach. Courant Lecture Notes 3, New York: New York University, 1999Google Scholar
  12. 12.
    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, 1335–1425 (1999)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Deift P., Kriecherbauer T., McLaughlin K.T-R, Venakides S., Zhou X.: Strong asymptotics of orthogonal polynomials with respect to exponential weights. Comm. Pure Appl. Math. 52, 1491–1552 (1999)MATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Deift P., Venakides S., Zhou X.: New result in small dispersion KdV by an extension of the steepest descent method for Riemann-Hilbert problems. Internat. Math. Res. Notices 6, 285–299 (1997)Google Scholar
  15. 15.
    Deift P., Venakides S., Zhou X.: An extension of the steepest descent method for Riemann-Hilbert problems: the small dispersion limit of the Korteweg-de Vries. Proc. Natl. Acad. Sc. USA 95(2), 450–454 (1998)MATHCrossRefMathSciNetADSGoogle Scholar
  16. 16.
    Deift P., Zhou X.: A steepest descent method for oscillatory Riemann-Hilbert problems. Asymptotics for the MKdV equation. Ann. Math. 137(2), 295–368 (1993)CrossRefMathSciNetGoogle Scholar
  17. 17.
    Dubrovin B.: On Hamiltonian perturbations of hyperbolic systems of conservation laws, II: universality of critical behaviour. Commun. Math. Phys. 267, 117–139 (2006)MATHCrossRefADSMathSciNetGoogle Scholar
  18. 18.
    Dubrovin, B., Grava, T., Klein, C.: On universality of critical behaviour in the focusing nonlinear Schrödinger equation, elliptic umbilic catastrophe and the tritronquée solution to the Painlevé-I equation. Preprint: http://babbage.sissa.it/abs/0704.0501, 2007
  19. 19.
    Dubrovin, B., Zhang, Y. : Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov-Witten invariants. Preprint:http://xxx.lanl.gov/math.DG/0108160, 2001
  20. 20.
    Fujiié S.: Semiclassical representation of the scattering matrix by a Feynman integral. Commun. Math. Phys. 198(2), 407–425 (1998)MATHCrossRefADSGoogle Scholar
  21. 21.
    Fujiié S., Ramond T.: Matrice de scattering et résonances associées à  une orbite hétérocline (French) [Scattering matrix and resonances associated with a heteroclinic orbit]. Ann. Inst. H. Poincaré Phys. Théor. 69(1), 31–82 (1998)MATHGoogle Scholar
  22. 22.
    Gardner S.C., Greene J.M., Kruskal M.D., Miura R.M.: Korteweg-de Vries equation and generalizations. VI. Methods for exact solution. Comm. Pure Appl. Math. 27, 97–133 (1974)MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Getzler E.: A Darboux theorem for Hamiltonian operators in the formal calculus of variations. Duke Math. J. 111, 535–560 (2002)MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Grava T., Klein C.: Numerical solution of the small dispersion limit of Korteweg de Vries and Whitham equations. Comm. Pure Appl. Math. 60, 1623–1664 (2007)MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Grava, T., Klein, C.: Numerical study of a multiscale expansion of KdV and Camassa-Holm equation. Preprint http://babbage.sissa.it/abs/math-ph/0702038, 2007
  26. 26.
    Gurevich A.G., Pitaevskii L.P.: Non stationary structure of a collisionless shock waves. JEPT Lett. 17, 193–195 (1973)ADSGoogle Scholar
  27. 27.
    Kapaev A.A.: Weakly nonlinear solutions of equation \({P_{I}^{2}}\) . J. Math. Sc. 73(4), 468–481 (1995)CrossRefMathSciNetGoogle Scholar
  28. 28.
    Lax P.D.: Integrals of nonlinear equations of evolution and solitary waves. Comm. Pure Appl. Math. 21, 467–490 (1968)MATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Lax, P.D., Levermore, C.D.: The small dispersion limit of the Korteweg de Vries equation, I,II,III. Comm. Pure Appl. Math. 36, 253–290, 571–593, 809–830 (1983)Google Scholar
  30. 30.
    Lorenzoni P.: Deformations of bi-Hamiltonian structures of hydrodynamic type. J. Geom. Phys. 44(2-3), 331–375 (2002)MATHCrossRefADSMathSciNetGoogle Scholar
  31. 31.
    Agranovich Z.S., Marchenko V.A.: The inverse problem of scattering theory, translated from the Russian by B.D. Seckler. Gordon and Breach Science Publishers, New York-London (1963)MATHGoogle Scholar
  32. 32.
    Menikoff A.: The existence of unbounded solutions of the Korteweg-de Vries equation. Commun. Pure Appl. Math. 25, 407–432 (1972)MATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    Moore G.: Geometry of the string equations. Commun. Math. Phys. 133(2), 261–304 (1990)MATHCrossRefADSGoogle Scholar
  34. 34.
    Ramond T.: Semiclassical study of quantum scattering on the line. Commun. Math. Phys. 177(1), 221–254 (1996)MATHCrossRefADSMathSciNetGoogle Scholar
  35. 35.
    Shabat, A.B.: One dimensional perturbations of a differential operator and the inverse scattering problem. In: Problems in Mechanics and Mathematical Physics, Moscow: Nauka, 1976Google Scholar
  36. 36.
    Venakides S.: The Korteweg de Vries equations with small dispersion: higher order Lax-Levermore theory. Commun. Pure Appl. Math. 43, 335–361 (1990)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Department WiskundeKatholieke Universiteit LeuvenLeuvenBelgium
  2. 2.SISSATriesleItaly

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