01113nas a2200133   4500008004100000245006400041210005900105260002800164520069000192653002700882100001600909700001800925856003600943       2012                        en  d00aThe KdV hierarchy: universality and a Painleve transcendent0 aKdV hierarchy universality and a Painleve transcendent  bOxford University Press3 aWe 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.10aSmall-Dispersion limit1 aClaeys, Tom1 aGrava, Tamara  uhttp://hdl.handle.net/1963/692100900nas a2200121   4500008004300000245014000043210007000183260001000253520044500263100001600708700001800724856003600742       2010                        en_Ud  00aPainlevé II asymptotics near the leading edge of the oscillatory zone for the Korteweg-de Vries equation in the small-dispersion limit0 aPainlevé II asymptotics near the leading edge of the oscillatory  bWiley3 aIn 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.1 aClaeys, Tom1 aGrava, Tamara  uhttp://hdl.handle.net/1963/379900906nas a2200109   4500008004300000245009100043210006900134520052300203100001800726700001600744856003600760       2010                        en_Ud  00aSolitonic asymptotics for the Korteweg-de Vries equation in the small dispersion limit0 aSolitonic asymptotics for the Kortewegde Vries equation in the s3 aWe 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.1 aGrava, Tamara1 aClaeys, Tom  uhttp://hdl.handle.net/1963/383901175nas a2200109   4500008004300000245012700043210006900170520075600239100001800995700001601013856003601029       2009                        en_Ud  00aUniversality of the break-up profile for the KdV equation in the small dispersion limit using the Riemann-Hilbert approach0 aUniversality of the breakup profile for the KdV equation in the 3 aWe 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.1 aGrava, Tamara1 aClaeys, Tom  uhttp://hdl.handle.net/1963/2636