Multi-symplectic methods have recently been considered as a generalization of symplectic ODE methods to the case of Hamiltonian PDEs. The symplectic of Hamiltonian systems is well known, but for Partial Differential Equation (PDEs) this is a global property. In addition, many PDEs can be written as Multisymplectic systems, in which each independent variable has a distinct symplectic structure. ...

Boundary control of nonlinear parabolic PDEs is an open problem with applications that include fluids, thermal, chemically-reacting, and plasma systems. In this paper we present stabilizing control designs for a broad class of nonlinear parabolic PDEs in 1-D. Our approach is a direct infinite dimensional extension of the finite-dimensional feedback linearization/backstepping approaches and empl...

There are two main approaches to the perturbative study of integrable PDEs: 1) perturbations of linear PDEs and 2) perturbations of nonlinear integrable systems of hydrodynamic type. In the talk we will mainly consider the problems and results related to the second approach. After explaining the main geometrical techniques such as the group of generalized Miura transformations, the idea of quas...

We apply a version of the dressing method to a system of four dimensional nonlinear Partial Differential Equations (PDEs), which contains both Pohlmeyer equation (i.e. nonlinear PDE integrable by the Inverse Spectral Transform Method) and nonlinear matrix PDE integrable by the method of characteristics as particular reductions. Some other reductions are suggested.

We consider the nonlinear Neumann problem for fully elliptic PDEs on a quadrant. establish comparison theorem viscosity sub- and supersolutions of problem. The crucial argument in proof is to build $C^{1,1}$ test function which takes care boundary condition. A similar has been treated general $n$-dimensional orthant by Biswas et al. [SIAM J. Control Optim., 55 (2017), pp. 365--396], where funct...

In this paper we present a Krylov acceleration technique for nonlinear PDEs. As a ‘preconditioner’ we use nonlinear multigrid schemes such as the Full Approximation Scheme (FAS) [1]. The benefits of nonlinear multigrid used in combination with the new accelerator are illustrated by difficult nonlinear elliptic scalar problems, such as the Bratu problem, and for systems of nonlinear equations, s...

The roots of the theory of nonlinear dispersive waves date back to hydrodynamics of the 19th century. It was observed, both experimentally and theoretically, that, under certain circumstances, the dissipative effects in nonlinear waves become less important then the dispersive ones. In this way, balance between nonlinearity and dispersion gives rise to formation of stable patterns (solitons, cn...

Title of dissertation: NONLINEAR EVOLUTIONARY PDEs IN IMAGE PROCESSING AND COMPUTER VISION Kexue Liu, Doctor of Philosophy, 2004 Dissertation directed by: Professor Jian-Guo Liu Department of Mathematics Evolutionary PDE-based methods are widely used in image processing and computer vision. For many of these evolutionary PDEs, there is little or no theory on the existence and regularity of solu...