The derivative nonlinear Schrödinger equation on the half line

We study the initial-boundary value problem for the derivative nonlinear Schrödinger (DNLS) equation. More precisely we study the wellposedness theory and the regularity properties of the DNLS equation on the half line. We prove almost sharp local wellposedness, nonlinear smoothing, and small data global wellposedness in the energy space. One of the obstructions is that the crucial gauge transformation we use replaces the boundary condition with a nonlocal one. We resolve this issue by running an additional fixed point argument. Our method also implies almost sharp local and small energy global wellposedness, and an improved smoothing estimate for the quintic Schrödinger equation on the half line. In the last part of the paper we consider the DNLS equation on $\mathbb{R}$ and prove smoothing estimates by combining the restricted norm method with a normal form transformation.     

Multilinear Eigenfunction Estimates for the Harmonic Oscillator and the Nonlinear Schrödinger Equation with the Harmonic Potential

We consider the Cauchy problem for the cubic nonlinear Schr?dinger equation with the harmonic potential. We prove global well-posedness below the energy class in energy subcritical cases. The main ingredients for the proof are a multilinear eigenfunction estimate for the harmonic oscillator and the I-method. Submitted: January 13, 2008. Accepted: February 11, 2009.     

A special value of the spectral zeta function of the non-commutative harmonic oscillators

The non-commutative harmonic oscillator is a 2×2-system of harmonic oscillators with a non-trivial correlation. We write down explicitly the special value at s=2 of the spectral zeta function of the non-commutative harmonic oscillator in terms of the complete elliptic integral of the first kind, which is a special case of a hypergeometric function. The research of the author is supported in part by a Grant-in-Aid for Scientific Research (B) 15340005 from the Ministry of Education, Culture, Sports, Science and Technology.     

Global wellposedness for a one-dimensional Chern–Simons–Dirac system in Lp

The global wellposedness in L^p(?) for the Chern–Simons–Dirac equation in the 1+1 space and time dimension is discussed. We consider two types of quadratic nonlinearity: the null case and the non-null case. We show the time global wellposedness for the Chern–Simon–Dirac equation in the framework of L^p(?), where 1≤p≤∞ for the null case. For the scaling critical case, p = 1, mass concentration phenomena of the solutions may occur in considering the time global solvability. We invoke the Delgado–Candy estimate which plays a crucial role in preventing concentration phenomena of the global solution. Our method is related to the original work of Candy (2011), who showed the time global wellposedness for the single Dirac equation with cubic nonlinearity in the critical space L²(?).     

Formulation of Quantum Scattering Theory in Terms of Proper Differentials (Stationary Wave Packets)

Constructing the basic operators of scattering theory on and off the mass shell in terms of spatially bounded stationary wave packets or proper differentials is described. For this, we use a technique based on a certain scheme for discretizing the continuum. Finite-dimensional approximations for the Green's functions and T-matrix, which are first found here, are immediately constructed for any energy using a single simple diagonalization of the Hamiltonian matrix in an L ₂-type complete basis. We show that the developed approach leads to a convenient finite-dimensional representation of the scattering operators in the basis of the wave functions of a harmonic oscillator. The method allows an immediate extension to the problem of three and more bodies.     

Can classical equations simulate quantum-mechanical behavior? a molecular dynamics investigation of a diatomic molecule with a morse potential

In a recent paper, we presented a new computational method for molecular dynamics which uses the Backward-Euler scheme to solve the classical Langevin dynamics equations. Parameters for the simulation include a target temperature T, a time step Δt, and a cutoff frequency ω_c. We showed for a harmonic oscillator system that the cutoff frequency can be set as ω_c = kT/h in order to mimic quantum-mechanical behavior. We now continue this investigation for a nonlinear case: a diatomic molecule governed by a Morse bond potential. Since approximate quantum-mechanical energy levels are explicitly known for this model, a comparison of energies can be made with molecular dynamics results. By performing dynamics runs for a wide range of temperatures and calculating mean energies, we find a very good agreement between these energies and quantum mechanical predictions. Vibrational excitation begins at temperatures around 800 K, and for higher temperatures both energy curves (molecular dynamics and quantum mechanics) approach the classical prediction of 7/2kT energy per molecule. Future investigations will focus on more general nonlinear potential functions employed in force fields of nucleic acids and proteins.     

Qualitative analysis of the influence of random perturbations of “white-noise” type applied along the vector of phase velocity on a harmonic oscillator with friction

We consider representations in the phase plane for the harmonic oscillator with friction under random perturbations applied along the vector of phase velocity. We investigate the behavior of the amplitude, phase, and total energy of the damped oscillator.     

Cubic Nonlinear Schrödinger Equation on Three Dimensional Balls with Radial Data

We prove wellposedness of the Cauchy problem for the cubic nonlinear Schrödinger equation with Dirichlet boundary conditions and radial data on 3D balls. The main argument is based on a bilinear eigenfunction estimate and the use of X ^{s, b} spaces. The last part presents a first attempt to study the non radial case. We prove bilinear estimates for the linear Schrödinger flow with particular initial data.     

A note on a discrete model for the harmonic oscillator Schrödinger equation in N-cartesian coordinates

We construct a discrete model for the time-independent harmonic oscillator Schrödinger partial differential equation and demonstrate that it can be separated into N ordinary difference equations for the case of N-cartesian space coordinates.     

The action variable and frequency of a relativistic harmonic oscillator

We present three series representations of the frequency of a relativistic harmonic oscillator. The first two representations use two equivalent forms of the action variable. The third representation involves determining its period by direct integration. The energy dependance of the oscillator frequency is manifestly seen in all three representations. We demonstrate that all three forms yield the same expression for the frequency in the case of the weakly relativistic oscillator and have an identical nonrelativistic limit.     

Endpoint Strichartz estimates and global solutions for the nonlinear Dirac equation

We prove endpoint Strichartz estimates for the Klein-Gordon and wave equations in mixed norms on the polar coordinates in three spatial dimensions. As an application, global wellposedness of the nonlinear Dirac equation is shown for small data in the energy class with some regularity assumption for the angular variable.     

On the classical descriptions of the quantum phenomena in the harmonic oscillator and in a charged particle under the coulomb force

In this work it is shown that the intrinsic phenomenon (the quantization of the energy) that appears in the first and simple systems studied initially by the quantum theory as the harmonic oscillator and the movement of a charged particle under the Coulomb force, can be obtained from the study of dissipative systems. In others words, we show that this phenomenon of the quantization of the energy of a particle which moves as an harmonic oscillator and which loses and wins energy can be obtained via a classical system of equations. The same also applies to the phenomena of the quantization of the energy of a charged particle which moves under the Coulomb force and which loses and wins energy.     

On the classical descriptions of the quantum phenomena in the harmonic oscillator and in a charged particle under the coulomb force

In this work it is shown that the intrinsic phenomenon (the quantization of the energy) that appears in the first and simple systems studied initially by the quantum theory as the harmonic oscillator and the movement of a charged particle under the Coulomb force, can be obtained from the study of dissipative systems. In others words, we show that this phenomenon of the quantization of the energy of a particle which moves as an harmonic oscillator and which loses and wins energy can be obtained via a classical system of equations. The same also applies to the phenomena of the quantization of the energy of a charged particle which moves under the Coulomb force and which loses and wins energy.     

On the wellposedness of the Cauchy problem for weakly hyperbolic equations of higher order

We study the wellposedness in the Gevrey classes G^s and in C^∞ of the Cauchy problem for weakly hyperbolic equations of higher order. In this paper we shall give a new approach to the case that the characteristic roots oscillate rapidly and vanish at an infinite number of points. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Complex Ginzburg-Landau equations in high dimensions and codimension two area minimizing currents

There is an obvious topological obstruction for a finite energy unimodular harmonic extension of a S ¹-valued function defined on the boundary of a bounded regular domain of R ⁿ . When such extensions do not exist, we use the Ginzburg-Landau relaxation procedure. We prove that, up to a subsequence, a sequence of Ginzburg-Landau minimizers, as the coupling parameter tends to infinity, converges to a unimodular harmonic map away from a codimension-2 minimal current minimizing the area within the homology class induced from the S ¹-valued boundary data. The union of this harmonic map and the minimal current is the natural generalization of the harmonic extension. Received December 3, 1998 / final version received May 10, 1999     

The global wellposedness of the 3D heat-conducting viscous incompressible fluids with bounded density

We consider the global wellposedness of the inhomogeneous incompressible heat-conducting viscous fluids in three dimension space. We generalize the result of Fujita & Kato for Navier–Stokes to the heat-conducting inhomogeneous incompressible viscous fluids. The key point is that we get the global wellposedness under the assumption that the initial density has positive lower and upper bound and the initial temperature can be arbitrarily large.     

An energy-based computational method in the analysis of the transmission of energy in a chain of coupled oscillators

In this paper we study the phenomenon of nonlinear supratransmission in a semi-infinite discrete chain of coupled oscillators described by modified sine-Gordon equations with constant external and internal damping, and subject to harmonic external driving at the end. We develop a consistent and conditionally stable finite-difference scheme in order to analyze the effect of damping in the amount of energy injected in the chain of oscillators; numerical bifurcation analyses to determine the dependence of the amplitude at which supratransmission first occurs with respect to the frequency of the driving oscillator are carried out in order to show the consequences of damping on harmonic phonon quenching and the delay of appearance of critical amplitude.     

Wave front set for solutions to Schrödinger equations with long-range perturbed harmonic oscillators

In this paper we consider a Schrödinger operator with variable coefficients and harmonic potential. The perturbation is assumed to be long-range in a sense similar to the work of Nakamura (2009) [13]. We construct a modified propagator, and then by using this propagator and also the propagator of the unperturbed free harmonic oscillator we characterize the propagation of singularities for solutions to the equations.     

Comparing quantum and classical approaches in statistical physics

We discuss hypotheses pertaining to the behavior of the mean energy of an equilibrium system in the quantum and classical theories. We justify the hypotheses for the cases of the harmonic oscillator and the potential well. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 123, No. 3, pp. 516–520, June, 2000.     

Strong solutions for semilinear wave equations with damping and source terms

This article deals with local existence of strong solutions for semilinear wave equations with power-like interior damping and source terms. A long-standing restriction on the range of exponents for the two nonlinearities governs the literature on wellposedness of weak solutions of finite energy. We show that this restriction may be eliminated for the existence of higher regularity solutions by employing natural methods that use the physics of the problem. This approach applies to the Cauchy problem posed on the entire ? ⁿ as well as for initial boundary problems with homogeneous Dirichlet boundary conditions.