Stabilization and Control for the Biharmonic Schrödinger Equation

Singular Solutions of the Biharmonic Nonlinear Schrödinger Equation

We consider singular solutions of the L 2-critical biharmonic nonlinear Schrödinger equation. We prove that the blowup rate is bounded by a quartic-root, the solution approaches a quasi–self-similar profile, and a finite amount of L 2-norm, which is no less than the critical power, concentrates into the singularity. We also prove the existence of a ground-state solution. We use asymptotic analy...

Enclosure for the Biharmonic Equation

In this paper we give an enclosure for the solution of the biharmonic problem and also for its gradient and Laplacian in the L2-norm, respectively.

Efficient parallel solvers for the biharmonic equation

Fast Multipole Method for the Biharmonic Equation

The evaluation of sums (matrix-vector products) of the solutions of the three-dimensional biharmonic equation can be accelerated using the fast multipole method, while memory requirements can also be significantly reduced. We develop a complete translation theory for these equations. It is shown that translations of elementary solutions of the biharmonic equation can be achieved by considering ...

Schrödinger Equation for Nanoscience

The second order (in time) Schrödinger equation is proposed. The additional term (in comparison to Schrödinger equation) describes the interaction of particles with vacuum filled with virtual particle – antiparticle pairs (zitterbewegung).