This article concerns the existence of multiple non-radial positive solutions L<sup>2</sup>-constrained problem $$\displaylines{-\Delta{u}-Q(\varepsilon x)|u|^{p-2}u=\lambda{u},\quad \text{in }\mathbb{R}^N, \\ \int_{\mathbb{R}^N}|u|^2dx=1,}$$ where \(Q(x)\) is a radially symmetric function, ε>0 small parameter, \(N\geq 2\), and \(p \in (2, 2+4/N)\) assumed to be m...

linear control system, 51-67Cauchy problem, 52-54controllability, 55-62Korteweg-de Vries equation, 65—67transport equation, 62-65adjoint operator, 373a d ^ y (definition), 130admissibility condition, 52, see also regularity propertyapproximate controllability, 55, 57, 61Ascoli's theorem, 152, 170attractor, 280averaging, 332-333 backstepping, 334-337, ...

Free evolution for quantum particle in generic ultrametric space is considered. We prove that if mean zero wave packet is localized in some space domain then its evolution remains localized in the same domain. In the present note we consider ultrametric quantum mechanics with real time and ultrametric space. Quantum mechanics with ultrametric (p–adic and adelic) space was considered in many wor...

The parabolic or forward scattering approximation to the equation describing wave propagation in a random medium leads to a stochastic partial differential equation which has the form of a random SchrOdinger equation. Existence, uniqueness and continuity of solutions to this equation are established. The resulting process is a Markov diffusion process on the unit sphere in complex Hilbert space...

From the discrete nonlinear Schrodinger equation describing transport on a dimer we derive and solve a closed nonlinear equation for the site-occupation probability difference. Our results, which are directly relevant to specific experiments such as neutron scattering in physically realizable dimers, exhibit a transition from "free" to "self-trapped" behavior and illustrate features expected in...

We put a direct new method to construct the rational Jacobi elliptic solutions for nonlinear differential difference equations which may be called the rational Jacobi elliptic functions method. We use the rational Jacobi elliptic function method to construct many new exact solutions for some nonlinear differential difference equations in mathematical physics via the lattice equation and the dis...

We present a family of symplectic splitting methods especially tailored to solve numerically the time-dependent Schrodinger equation. When discretized in time, this equation can be recast in the form of a classical Hamiltonian system with a Hamiltonian function corresponding to a generalized high-dimensional separable harmonic oscillator. The structure of the system allows us to build highly ef...