Airy kernel determinant solutions to the KdV equation and integro-differential Painlevé equations

Abstract

We study a family of unbounded solutions to the Korteweg-de Vries equation which can be constructed as log-derivatives of deformed Airy kernel Fredholm determinants, and which are connected to an integro-differential version of the second Painlevé equation. The initial data of the Korteweg-de Vries solutions are well-defined for $x>0$, but not for $x<0$, where the solutions behave like $\frac{x}{2t}$ as $t\to 0$, and hence would be well-defined as solutions of the cylindrical Korteweg-de Vries equation. We provide uniform asymptotics in $x$ as $t\to 0$; for $x>0$ they involve an integro-differential analogue of the Painlevé V equation. A special case of our results yields improved estimates for the {tails} of the narrow wedge solution to the Kardar-Parisi-Zhang equation.

44 pages. V3: Remark 1.2 corrected

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Cited by in corpus (12)

• Uniform tail asymptotics for Airy kernel determinant solutions to KdV and for the narrow wedge solution to KPZ
• Universality for multiplicative statistics of Hermitian random matrices and the integro-differential Painlevé II equation
• Integrability in the weak noise theory
• Generalized Gibbs ensemble of the Ablowitz-Ladik lattice, Circular $\beta$-ensemble and double confluent Heun equation
• The Bessel kernel determinant on large intervals and Birkhoff's ergodic theorem
• Jánossy densities and Darboux transformations for the Stark and cylindrical KdV equations
• Integrable equations associated with the finite-temperature deformation of the discrete Bessel point process
• Bessel kernel determinants and integrable equations
• Large deviations for the $q$-deformed polynuclear growth
• High moments of the SHE in the clustering regimes
• Large time and distance asymptotics of the one-dimensional impenetrable Bose gas and Painlevé IV transition
• The stochastic six-vertex model speed process