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The homotopy continuation method has been widely used to compute multiple solutions of nonlinear differential equations, but the computational cost grows exponentially based on the traditional finite difference and finite element discretizations. In this w ...
Flows of gases and liquids interacting with solid objects are often turbulent within a thin boundary layer. As energy dissipation and momentum transfer are dominated by the boundary layer dynamics, many engineering applications can benefit from an improved ...
We consider the nonlinear Korteweg-de Vries (KdV) equation in a bounded interval equipped with the Dirichlet boundary condition and the Neumann boundary condition on the right. It is known that there is a set of critical lengths for which the solutions of ...
We study the behavior of solutions to the incompressible 2d Euler equations near two canonical shear flows with critical points, the Kolmogorov and Poiseuille flows, with consequences for the associated Navier-Stokes problems. We exhibit a large family of ...
Matrices with hierarchical low-rank structure, including HODLR and HSS matrices, constitute a versatile tool to develop fast algorithms for addressing large-scale problems. While existing software packages for such matrices often focus on linear systems, t ...
Free-electron lasers provide a source of x-ray pulses short enough and intense enough to drive nonlinearities in molecular systems. Impulsive interactions driven by these x-ray pulses provide a way to create and probe valence electron motions with high tem ...
We investigate the stability and stabilization of the cubic focusing Klein-Gordon equation around static solutions on the closed ball in R3. First we show that the system is linearly unstable near the static solution u≡1 for any dissipat ...
While quantum mechanics imposes a fundamental limit on the precision of interferometric measurementsof mechanical motion due to measurement backaction, the nonlinear nature of the coupling also leads toparametric instabilities that place practical limits o ...
Predicting the electrical behavior of the heart, from the cellular scale to the tissue level, relies on the numerical approximation of coupled nonlinear dynamical systems. These systems describe the cardiac action potential, that is the polarization/depola ...
In generalized linear estimation (GLE) problems, we seek to estimate a signal that is observed through a linear transform followed by a component-wise, possibly nonlinear and noisy, channel. In the Bayesian optimal setting, generalized approximate message ...
