NON-NEHARI MANIFOLD METHOD FOR SUPERLINEAR SCHRÖDINGER EQUATION

The Nehari Manifold for Fractional Systems Involving Critical Nonlinearities

We study the combined effect of concave and convex nonlinearities on the number of positive solutions for a fractional system involving critical Sobolev exponents. With the help of the Nehari manifold, we prove that the system admits at least two positive solutions when the pair of parameters (λ, μ) belongs to a suitable subset of R.

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).

The Nehari Manifold for a Class of Indefinite Weight Semilinear Elliptic Equations

An exponential time differencing method for the nonlinear Schrödinger equation

The spectral methods offer very high spatial resolution for a wide range of nonlinear wave equations, so, for the best computational efficiency, it should be desirable to use also high order methods in time but without very strict restrictions on the step size by reason of numerical stability. In this paper we study the exponential time differencing fourth-order Runge-Kutta (ETDRK4) method; thi...

A Spacetime DPG Method for the Schrödinger Equation

A spacetime Discontinuous Petrov Galerkin (DPG) method for the linear timedependent Schrödinger equation is proposed. The spacetime approach is particularly attractive for capturing irregular solutions. Motivated by the fact that some irregular Schrödinger solutions cannot be solutions of certain first order reformulations, the proposed spacetime method uses the second order Schrödinger operato...