Solvable Nonlinear Evolution PDEs in Multidimensional Space

Solvable Nonlinear Evolution PDEs in Multidimensional Space

A class of solvable (systems of) nonlinear evolution PDEs in multidimensional space is discussed. We focus on a rotation-invariant system of PDEs of Schrödinger type and on a relativistically-invariant system of PDEs of Klein–Gordon type. Isochronous variants of these evolution PDEs are also considered.

Diophantine equations of nonlinear physics . Part 1 : Nonlinear evolution PDEs ∗

Image Restoration: Wavelet Frame Shrinkage, Nonlinear Evolution PDEs, and Beyond

In the past few decades, mathematics based approaches have been widely adopted in various image restoration problems, among which the partial differential equation (PDE) based approach (e.g. the total variation model [56] and its generalizations, nonlinear diffusions [15, 52], etc.), and wavelet frame based approach are some of successful examples. These approaches were developed through differ...

Nonlinear Diffusion Pdes

The main theory behind nonlinear diffusion models is to use nonlinear PDEs to create a scale space representation that consists of gradually simplified images where some image features such as edges are maintained or even enhanced. The earliest nonlinear diffusion model proposed in image processing is the so-called anisotropic diffusion1 by Perona and Malik [2]. In their formulation, they repla...

An Eulerian-Lagrangian method for optimization problems governed by multidimensional nonlinear hyperbolic PDEs

We present a numerical method for solving tracking-type optimal control problems subject to scalar nonlinear hyperbolic balance laws in one and two space dimensions. Our approach is based on the formal optimality system and requires numerical solutions of the hyperbolic balance law forward in time and its nonconservative adjoint equation backward in time. To this end, we develop a hybrid method...