Fourth-order Derivative Term Research Articles

Deformation of the CP(1) model leading to fixed size solitons in 2+1 dimensions

We discuss static particle-like solitons in the (2+1)-dimensional CP(1) model with a small mass deformation m preserving a U(1) × Z2 symmetry in the Lagrangian. Due to the breaking of scale invariance, the energy function becomes a strictly increasing function of the soliton size ρ, and therefore no classical finite size solution exists in this model. To remedy this we employ a well known technique of introducing a fourth-order derivative term in the Lagrangian to force the soliton action to diverge at small values of ρ. With this additional term the action exhibits a stable minimum at fixed size ρ.

New massive supergravity and auxiliary fields

We construct a supersymmetric formulation of three-dimensional linearized new massive gravity without introducing higher derivatives. Instead, we introduce supersymmetrically a set of bosonic and fermionic auxiliary fields which, upon elimination by their equations of motion, introduce fourth-order derivative terms for the metric and third-order derivative terms for the gravitino. Our construction requires an off-shell formulation of the three-dimensional supersymmetric massive Fierz–Pauli theory. We discuss the nonlinear extension of our results.

閉鎖性水域における汚染源特定のためのリバースシミュレーション

Reverse simulation was proposed to specify the source of pollutant in closed water. One of the major problems of reverse simulation is the numerical instability of computation. This instability is caused by the high-frequency rounding errors. A Gaussian filter was used for the spatial distribution of pollutant or of the flux of pollutant to eliminate the instability. The effects of filtering operation were verified by comparing the results of forward and reverse simulations. Forward simulation was executed assuming that pollutant flows out from the west coast of Lake Biwa as an example of closed water. The final result of forward simulation was used as the initial condition of reverse simulation. The concentration of pollutant was solved inversely using the current velocity obtained in forward simulation. As a result, the instability of reverse simulation was suppressed by applying the Gaussian filter for the spatial distribution of pollutant, while the concentration of pollutant was more diffusive than that of forward simulation. When the Gaussian filter was applied to the special distribution of the flux of pollutant, the accuracy of reverse simulation was improved. This is because the high-frequency rounding errors were mainly suppressed by the fourth-order derivative term. The filter width is a key parameter to determine the accuracy of reverse simulation. It was relatively larger in the present study than that in the existing studies for simpler flows. This may be attributed to the numerical instability caused by complex topography and boundary condition. In addition, the distribution of the concentration of pollutant in reverse simulation was more circular than that in the forward simulation since the filtering operation was more effective where the spatial gradient of pollutant was larger. The filter width should be determined reasonably with examining the sources of the numerical instability as future studies.

The stability and convergence of two linearized finite difference schemes for the nonlinear epitaxial growth model

AbstractThe numerical simulation of the dynamics of the molecular beam epitaxy (MBE) growth is considered in this article. The governing equation is a nonlinear evolutionary equation that is of linear fourth order derivative term and nonlinear second order derivative term in space. The main purpose of this work is to construct and analyze two linearized finite difference schemes for solving the MBE model. The linearized backward Euler difference scheme and the linearized Crank‐Nicolson difference scheme are derived. The unique solvability, unconditional stability and convergence are proved. The linearized Euler scheme is convergent with the convergence order of O(τ + h2) and linearized Crank‐Nicolson scheme is convergent with the convergence order of O(τ2 + h2) in discrete L2‐norm, respectively. Numerical stability with respect to the initial conditions is also obtained for both schemes. Numerical experiments are carried out to demonstrate the theoretical analysis. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011

An Optimal Condition of Compactness for Elasticity Problems Involving one Directional Reinforcement

This paper deals with the homogenization of a homogeneous elastic medium reinforced by very stiff strips in dimension two. We give a general condition linked to the distribution and the stiffness of the strips, under which the nature of the elasticity problem is preserved in the homogenization process. This condition is sharper than the one used in Briane and Camar-Eddine (J. Math. Pures Appl. 88:483–505, 2007) and is shown to be optimal in the case where the strips are periodically arranged. Indeed, a fourth-order derivative term appears in the limit equation as soon as the condition is no more satisfied. In the periodic case the influence of oscillations in the medium surrounding the strips is also considered. The homogenization method is based both on a two-scale convergence for the strips and the use of suitable oscillating test functions. This allows us to obtain a distributional convergence of two of the three entries of the stress tensor contrary to the Γ-convergence approach of Briane and Camar-Eddine (J. Math. Pures Appl. 88:483–505, 2007).

A modified method for a non-standard inverse heat conduction problem

A non-standard inverse heat conduction problem is considered. Data are given along the line x = 1 and the solution at x = 0 is sought. The problem is ill-posed in the sense that the solution (if it exists) does not depend continuously on the data. In order to solve the problem numerically it is necessary to employ some regularization method. In this paper, we study a modification of the equation, where a fourth-order mixed derivative term is added. Error estimates for this equation are given, which show that the solution of the modified equation is an approximation of the heat equation. A numerical implementation is considered and a simple example is given. Some numerical results show the usefulness of the modified method.

Fourth-order modified method for the Cauchy problem for the Laplace equation

We consider the Cauchy problem for the Laplace equation in the half plane x > 0 , y ∈ R where the Cauchy data is given at x = 0 and the solution is sought in the interval 0 < x ⩽ 1 . The problem is ill-posed: the solution (if it exists) does not depend continuously on the data. In order to solve the problem numerically, it is necessary to modify the equation so that a bound on the solution is imposed. We study a modification of the equation, where a fourth-order mixed derivative term is added. Error estimates for this equation are given, which show that the solution of the modified equation is an approximation of the solution of the Cauchy problem for the Laplace equation, and it is shown that when the data error tends to zero, the error in the approximate solution tends to zero logarithmically. Numerical implementation is considered and a simple example is given.

Dirac quantization of the Pais-Uhlenbeck fourth order oscillator

As a model, the Pais-Uhlenbeck fourth order oscillator with equation of motion $d^4q/dt^4+(\omega_1^2+\omega_2^2)d^2q/dt^2 +\omega_1^2\omega_2^2 q=0$ is a quantum-mechanical prototype of a field theory containing both second and fourth order derivative terms. With its dynamical degrees of freedom obeying constraints due to the presence of higher order time derivatives, the model cannot be quantized canonically. We thus quantize it using the method of Dirac constraints to construct the correct quantum-mechanical Hamiltonian for the system, and find that the Hamiltonian diagonalizes in the positive and negative norm states that are characteristic of higher derivative field theories. However, we also find that the oscillator commutation relations become singular in the $\omega_1 \to \omega_2$ limit, a limit which corresponds to a prototype of a pure fourth order theory. Thus the particle content of the $\omega_1 =\omega_2$ theory cannot be inferred from that of the $\omega_1 \neq \omega_2$ theory; and in fact in the $\omega_1 \to \omega_2$ limit we find that all of the $\omega_1 \neq \omega_2$ negative norm states move off shell, with the spectrum of asymptotic in and out states of the equal frequency theory being found to be completely devoid of states with either negative energy or negative norm. As a byproduct of our work we find a Pais-Uhlenbeck analog of the zero energy theorem of Boulware, Horowitz and Strominger, and show how in the equal frequency Pais-Uhlenbeck theory the theorem can be transformed into a positive energy theorem instead.

Structural transition of vortex lattices in d-wave superconductors

We carry out two-dimensional computer simulations of structural phase transition in vortex lattices between a hexagonal symmetry and a square symmetry in d-wave superconductors. Our model is an extended time-dependent Ginzburg–Landau equations including fourth-order derivative terms reflecting the d x 2− y 2 symmetry. The kinetics of the structural transition is studied by analyzing the scattering function of the lattice structures and by extracting the time dependence of the volume fraction of the hexagonal ordered region.

Existence and construction of travelling wavefront solutions of Fisher equations with fourth-order perturbations

We consider a generalized Fisher equation containing a fourth-order spatial derivative term. Of interest is the question of the existence of travelling front solutions of the equation and their qualitative form. When the fourth-order terms has a sufficiently small coefficient existence of such fronts is shown using invariant manifold theory. Asymptotic analysis is employed to construct an expression for the travelling front when its speed is large.

Radiation of Solitons Described by a Higher-Order Nonlinear Schrödinger Equation

The resonant radiation of a soliton described by the nonlinear Schrödinger equation with third and fourth order derivative terms is investigated by different methods. The results are in good agreement between themselves.

Numerical study for vortex lattice transition with extended Ginzburg–Landau model

We study the vortex lattice structure in two dimensional d-wave superconductors in a magnetic field which is perpendicular to the a– b plane. The numerical simulation for the extended Ginzburg–Landau Hamiltonian, which introduces the fourth order derivative terms, is carried out with the Langevin equation. As we increase the effect of these nonlocal terms, the triangular vortex lattice is distorted and vortices form isosceles triangular pattern with no long range order.

Stability criterion for stationary bound states of solitons with radiationless oscillating tails

We investigate stationary bound states of bright solitary waves in an optical fiber with fourth-order dispersion, which is described by a generalized nonlinear Schr\odinger equation with an additional fourth-order derivative term. It is shown that various families of two-soliton (and multisoliton) bound states exist for this equation in the parameter region where single solitons have radiationless oscillatory tails. We analyze and discuss their stability and possible applications. A stability criterion for stationary two-soliton (and multisoliton) states of a conservative Hamiltonian dynamical system is derived.

Energy density formalism and semiclassical nuclear properties

The semiclassical nuclear properties have been studied in the framework of the energy density formalism by choosing two energy densities, one containing derivative terms up to second order and the other also containing fourth-order derivative terms. The surface energy density in each of the two cases can be suitably approximated so that the Euler-Lagrange equations in a semi-infinite nuclear medium (SINM) lead to a pure Fermi distribution (F-1) for the density. The importance of the inclusion of the fourth-order derivative terms in the energy density to reproduce the correct surface energy and the correct behaviour of the density in the outer part of the surface has been shown explicitly. The variations of the central density rho 0 and the surface thickness a with the size of the nucleus have been studied under the assumption that the departures in and lambda of these quantities from their respective SIMN values are small. The restricted simple nature of the surface energy density resulting from the imposition of an F-1 distribution of density in a SINM has the advantage of describing the variations in and lambda in terms of the small ratio alpha between the surface thickness and the half-density radius in a general way, independent of the characteristics of the energy density used. This study shows that the correct behaviour of the variations of in and lambda cannot be reproduced in an expansion in powers of alpha and retaining the first few terms. It is further observed that the lower-order terms in the energy in an expansion in powers of alpha are quite important for a correct description of the departures in and lambda , at least in the region alpha >or=0.1, although these lower-order terms are not so important as far as the total energy is concerned. A comparison of the variations of the surface thickness with the size of the nucleus for the two different energy densities shows that the variation lambda is much smaller for the fourth-order case, in conformity with the fact that the surface thickness is almost independent of the size of the nucleus and thereby favours an energy density containing derivative terms up to fourth order over an energy density having only second-order derivative terms.

The semiclassical description of the nuclear surface properties in the energy density formalism

The semiclassical nuclear surface properties have been studied in the framework of the energy density formalism. The necessity of the inclusion of fourth-order derivative terms in the energy densities has been emphasized to remove the inadequacy of the energy densities containing only second-order derivative terms to reproduce simultaneously the correct surface energy and the correct shape of the density in the tail region. Emphasis has also been given to choose the fourth-order derivative term in a suitable simple form to obtain analytical relations for the various surface characteristics of semi-infinite nuclear matter with the aid of the resulting Euler-Lagrange equations. The justification of using parametrized densities of generalized Fermi shape (F-v) has been shown explicitly by choosing the fourth-order derivative term in the energy density in a convenient form to give exponential behaviour for the density in the extreme interior and exterior parts of the nuclear surface. Simple relations for various surface properties have also been obtained for a number of Skyrme-type interactions for these parametrized densities and these results compare very well with those obtained directly by using the Euler-Lagrange equations.

CP violation in decays from chiral lagrangians with fourth-order derivative terms, including isospin-breaking and rescattering effects

In the framework of recently proposed effective lagrangians for weak nonleptonic meson interactions the amplitudes of the decays K 0 → 3 π have been calculated with inclusion of isospin-breaking and meson rescattering effects. The imaginary part of the penguin diagram contribution, which determines direct CP violation in nonleptonic kaon decays, has been fixed with the help of the measured ratio ϵ′ ϵ of CP -violation parameters. The modification of the Li-Wolfenstein relation for the direct CP -violation parameter in K 0 ( K 0 ) → π + π − π 0 decays is discussed.

Approximations for a Cauchy problem for the heat equation

A Cauchy problem for the heat equation in the quarter plane is considered. Data are given along the line x=1 and the solution at x=0 is sought. The problem is ill-posed: the solution does not depend continuously on the data. In order to solve the problem numerically it is necessary to modify the equation so that a bound on the solution is imposed (explicitly or implicitly). The author studies a modification of the equation, where a fourth-order mixed derivative term is added. Error estimates for this equation are given, which show that the solution of the modified equation is an approximation of the solution of the Cauchy problem for the heat-equation.

Extended skyrmion model from quark flavour dynamics

From the effective chiral meson lagrangian derived recently from Nambu-Jona-Lasinio type of quark flavour dynamics, an extended skyrmion model is extracted. This model contains besides the usual Skyrme term further fourth-order derivative terms of the chiral field and in addition the ω meson field. It is shown that the net effect of all fourth-order derivative terms is to destabilize the soliton and to provide the desired medium-range attraction of the nucleon-nucleon interaction, which is responsible for nuclear binding but missing in the original Skyrme model. The soliton is stabilized by the coupling to the ω meson which produces a short-range repulsion. The corresponding equation of state of skyrmion matter is constructed and found to be in qualitative agreement with the equation of state of nuclear matter obtained from effective density-dependent forces.

Approximate Equations for Long Nonlinear Waves on a Viscous Fluid

A systematic perturbation method is applied to three-dimensional long waves on a viscous liquid film, and the nonlinear evolution equation incorporating the effects of dissipation and dispersion is derived. It is shown that both the fourth-order derivative term as well as the three-dimensionality have stabilizing effects.