The KdV equation: exponential asymptotics, complex singularities and Painlevé II
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Scott W. McCue is Professor of Applied Mathematics at Queensland University of Technology. His research spans interfacial dynamics, water waves, fluid mechanics, mathematical biology, and moving boundary problems. He is widely recognised for his contributions to modelling complex free-boundary phenomena, including thin-film rupture, Hele–Shaw flows, and biological invasion processes.
Abstract
We apply techniques of exponential asymptotics to the KdV equation to derive the small-time behaviour for dispersive waves that propagate in one direction. The results demonstrate how the amplitude, wavelength and speed of these waves depend on the strength and location of complex-plane singularities of the initial condition. Using matched asymptotic expansions, we show how the small-time dynamics of complex singularities of the time-dependent solution are dictated by a Painlevé II problem with decreasing tritronquée solutions. We relate these dynamics to the solution on the real line.
11:15
The KdV equation: exponential asymptotics, complex singularities and Painlevé II
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Scott W. McCue is Professor of Applied Mathematics at Queensland University of Technology. His research spans interfacial dynamics, water waves, fluid mechanics, mathematical biology, and moving boundary problems. He is widely recognised for his contributions to modelling complex free-boundary phenomena, including thin-film rupture, Hele–Shaw flows, and biological invasion processes.
Abstract
We apply techniques of exponential asymptotics to the KdV equation to derive the small-time behaviour for dispersive waves that propagate in one direction. The results demonstrate how the amplitude, wavelength and speed of these waves depend on the strength and location of complex-plane singularities of the initial condition. Using matched asymptotic expansions, we show how the small-time dynamics of complex singularities of the time-dependent solution are dictated by a Painlevé II problem with decreasing tritronquée solutions. We relate these dynamics to the solution on the real line.
Dimension liftings for quantum computation of partial differential equations and related problems
Abstract
Quantum computers are designed based on quantum mechanics principle, they are most suitable to solve the Schrodinger equation, and linear PDEs (and ODEs) evolved by unitary operators. It is important to to explore whether other problems in scientific computing, such as ODEs, PDEs, and linear algebra that arise in both classical and quantum systems which are not unitary evolution, can be handled by quantum computers.
We will present a systematic way to develop quantum simulation algorithms for general differential equations. Our basic framework is dimension lifting, that transfers non-autonomous ODEs/PDEs systems to autonomous ones, nonlinear PDEs to linear ones, and linear ones to Schrodinger type PDEs—coined “Schrodingerization”—with uniform evolutions. Our formulation allows both qubit and qumode (continuous-variable) formulations, and their hybridizations, and provides the foundation for analog quantum computing which are easier to realize in the near term. We will also discuss dimension lifting techniques for quantum simulation of stochastic DEs and PDEs with fractional derivatives.