Related papers: An extensive resonant normal form for an arbitrary…
In this paper we study small amplitude solutions of nonlinear Klein Gordon equations with a potential. Under smoothness and decay assumptions on the potential and a genericity assumption on the nonlinearity, we prove that all small…
The primary objective of this paper is to investigate the well-posedness theories associated with the discrete nonlinear Schr\"odinger equation and Klein-Gordon equation. These theories encompass both local and global well-posedness, as…
We consider the large time behavior of solutions to defocusing nonlinear Schrodinger equation in the presence of a time dependent external potential. The main assumption on the potential is that it grows at most quadratically in space,…
We construct solutions to the eikonal equation for a class of long-range potentials, derive strong radiation condition bounds for the corresponding Schr\"odinger operator and apply them to construct the generalized Fourier transform,…
We numerically solve the Klein-Gordon equation at second order in cosmological perturbation theory in closed form for a single scalar field, describing the method employed in detail. We use the slow-roll version of the second order source…
In this article, we consider nonlinear Schr\"odinger equation with nonlocal nonlinearity which is a generalized model of the Schr\"odinger-Poisson system (Schr\"odinger-Newton equations) in low dimensions. We first prove the global…
We derive a uniform exponential decay of the total energy for the nonlinear Klein-Gordon equation with a damping around spatial infinity in the whole space or in the exterior of a star shaped obstacle.
We discuss normal forms of the completely resonant non-linear Schr\"odinger equation on a torus $\T^n$, with particular applications to quasi periodic solutions.
We consider a parabolic sine-Gordon model with periodic boundary conditions. We prove a fundamental maximum principle which gives a priori uniform control of the solution. In the one-dimensional case we classify all bounded steady states…
We develop a trajectory construction of solutions to the massless wave equation in n+1 dimensions and hence show that the quantum state of a massive relativistic system in 3+1 dimensions may be represented by a stand-alone four-dimensional…
We study an example of exact parametric resonance in a extended system ruled by nonlinear partial differential equations of nonlinear Schr\"odinger type. It is also conjectured how related models not exactly solvable should behave in the…
The covariant Klein-Gordon equation requires twice the boundary conditions of the Schrodinger equation and does not have an accepted single-particle interpretation. Instead of interpreting its solution as a probability wave determined by an…
In the present Chapter, we consider two prototypical Klein-Gordon models: the integrable sine-Gordon equation and the non-integrable $\phi^4$ model. We focus, in particular, on two of their prototypical solutions, namely the kink-like…
The approximate analytic bound state solutions of the Klein-Gordon equation with equal scalar and vector exponential-type potentials including the centrifugal potential term are obtained for any arbitrary orbital angular momentum number l…
In this paper we established the global well-posedness theorem for a special type of wave-Klein-Gordon system that have the strong coupling terms in divergence form on the right hand side of its wave equation. We cope with the problem by…
Formation of the Kondo state in general two-band Anderson model has been investigated within the numerical renormalization group (NRG) calculations. The Abrikosov-Suhl resonance is essentially asymmetric for the model with one electron per…
We study the backstepping stabilization of higher order linear and nonlinear Schr\"odinger equations on a finite interval, where the boundary feedback acts from the left Dirichlet boundary condition. The plant is stabilized with a…
We construct and study relativistic anyons in 1+1 dimensions generalizing well-known models of Dirac fermions. First, a model of free anyons is constructed and then extended in two ways: (i) by adding density-density interactions, as in the…
A resonant chain may be formed in a multi-planetary system when ratios of the orbital periods can be expressed as ratios of small integers $T_1:T_2: \cdots :T_N=k_1: k_2: \cdots: k_N$. We investigate the dynamics and possible formation of…
We propose a Schr\"odinger equation of arbitrary order for modeling charge transport in semiconductors operating in the ballistic regime. This formulation incorporates non-parabolic effects through the Kane dispersion relation, thereby…