%0 Journal Article
%J International Mathematics Research Notices, vol. 22 (2012) , page 5063-5099
%D 2012
%T The KdV hierarchy: universality and a Painleve transcendent
%A Tom Claeys
%A Tamara Grava
%K Small-Dispersion limit
%X We study the Cauchy problem for the Korteweg-de Vries (KdV) hierarchy in the small dispersion limit where $\e\to 0$. For negative analytic initial data with a single negative hump, we prove that for small times, the solution is approximated by the solution to the hyperbolic transport equation which corresponds to $\e=0$. Near the time of gradient catastrophe for the transport equation, we show that the solution to the KdV hierarchy is approximated by a particular Painlev\'e transcendent. This supports Dubrovins universality conjecture concerning the critical behavior of Hamiltonian perturbations of hyperbolic equations. We use the Riemann-Hilbert approach to prove our results.
%B International Mathematics Research Notices, vol. 22 (2012) , page 5063-5099
%I Oxford University Press
%G en
%U http://hdl.handle.net/1963/6921
%1 6902
%2 Mathematics
%4 1
%$ Submitted by Maria Pia Calandra (calapia@sissa.it) on 2013-05-29T08:08:55Z
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%0 Journal Article
%J Comm. Pure Appl. Math. 63 (2010) 203-232
%D 2010
%T PainlevĂ© II asymptotics near the leading edge of the oscillatory zone for the Korteweg-de Vries equation in the small-dispersion limit
%A Tom Claeys
%A Tamara Grava
%X In the small dispersion limit, solutions to the Korteweg-de Vries equation develop an interval of fast oscillations after a certain time. We obtain a universal asymptotic expansion for the Korteweg-de Vries solution near the leading edge of the oscillatory zone up to second order corrections. This expansion involves the Hastings-McLeod solution of the Painlev\\\\\\\'e II equation. We prove our results using the Riemann-Hilbert approach.
%B Comm. Pure Appl. Math. 63 (2010) 203-232
%I Wiley
%G en_US
%U http://hdl.handle.net/1963/3799
%1 527
%2 Mathematics
%3 Mathematical Physics
%$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2009-11-26T16:48:15Z\\nNo. of bitstreams: 1\\n0812.4142v1.pdf: 343118 bytes, checksum: 4bf2fa3751076c18466f29e1163acc09 (MD5)
%R 10.1002/cpa.20277
%0 Journal Article
%J SIAM J. Math. Anal. 42 (2010) 2132-2154
%D 2010
%T Solitonic asymptotics for the Korteweg-de Vries equation in the small dispersion limit
%A Tamara Grava
%A Tom Claeys
%X We study the small dispersion limit for the Korteweg-de Vries (KdV) equation $u_t+6uu_x+\\\\epsilon^{2}u_{xxx}=0$ in a critical scaling regime where $x$ approaches the trailing edge of the region where the KdV solution shows oscillatory behavior. Using the Riemann-Hilbert approach, we obtain an asymptotic expansion for the KdV solution in a double scaling limit, which shows that the oscillations degenerate to sharp pulses near the trailing edge. Locally those pulses resemble soliton solutions of the KdV equation.
%B SIAM J. Math. Anal. 42 (2010) 2132-2154
%G en_US
%U http://hdl.handle.net/1963/3839
%1 488
%2 Mathematics
%3 Mathematical Physics
%$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2010-02-05T10:12:44Z\\nNo. of bitstreams: 1\\n0911.5686v1.pdf: 311708 bytes, checksum: bfe41688febbfb066ebaae202e1e93b6 (MD5)
%R 10.1137/090779103
%0 Journal Article
%J Comm. Math. Phys. 286 (2009) 979-1009
%D 2009
%T Universality of the break-up profile for the KdV equation in the small dispersion limit using the Riemann-Hilbert approach
%A Tamara Grava
%A Tom Claeys
%X We obtain an asymptotic expansion for the solution of the Cauchy problem for the Korteweg-de Vries (KdV) equation in the small dispersion limit near the point of gradient catastrophe (x_c,t_c) for the solution of the dispersionless equation.\\nThe 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 Painleve 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.
%B Comm. Math. Phys. 286 (2009) 979-1009
%G en_US
%U http://hdl.handle.net/1963/2636
%1 1487
%2 Mathematics
%3 Mathematical Physics
%$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2008-04-22T10:17:48Z\\nNo. of bitstreams: 1\\n0801.2326v1.pdf: 375000 bytes, checksum: b00b6e0d823d47a002430b4fdecf8c7c (MD5)
%R 10.1007/s00220-008-0680-5