Nonlinear Scalar Field Research Articles

Improved Quark Mass Density-Dependent Model with Non-LinearScalar Interaction

We present an improved quark mass density-dependent model which includesthe quark and non-linear scalar field coupling. The wavefunction ofquark is given. The rms charge radius, the magnetic moment, and theratio between the axial-vector and the vector β-decay couplingconstants of the nucleon are calculated. We find that the results giventhe present model are in agreement with experiments.

The Einstein-Vlasov System with a Scalar Field

We study the Einstein-Vlasov system coupled to a nonlinear scalar field with a nonnegative potential in locally spatially homogeneous space-time, as an expanding cosmological model. It is shown that solutions of this system exist globally in time. When the potential of the scalar field is of an exponential form, the cosmological model corresponds to accelerated expansion. The Einstein-Vlasov system coupled to a nonlinear scalar field whose potential is of an exponential form shows the causal geodesic completeness of the space-time towards the future. The asymptotic behavior of solutions of this system in the future time is analyzed in various aspects, which shows power-law expansion.

Regularization of second-order scalar perturbation produced by a point particle with a nonlinear coupling

Accurate calculation of the motion of a compact object in a background spacetime induced by a supermassive black hole is required for the future detection of such binary systems by the gravitational-wave detector LISA. Reaching the desired accuracy requires calculation of the second-order gravitational perturbations produced by the compact object. At the point-particle limit the second-order gravitational perturbation equations turn out to have highly singular source terms, for which the standard retarded solutions diverge. Here we study a simplified scalar toy-model in which a point particle induces a nonlinear scalar field in a given curved spacetime. The corresponding second-order scalar perturbation equation in this model is found to have a similar singular source term, and therefore its standard retarded solutions diverge. We develop a regularization method for constructing well-defined causal solutions for this equation. Notably these solutions differ from the standard retarded solutions, which are ill defined in this case.

Infinitely many bound states for some nonlinear scalar field equations

In this paper we consider the problem \(-\Delta u + a(x)u = \vert u\vert ^{p-2}u\) in \(\mathbb{R}^N\), where p > 2 and \(p 2. Assuming that the potential a(x) is a regular function such that \(\liminf_{\vert x\vert\rightarrow + \infty} a(x) = a_\infty > 0\) and that verifies suitable decay assumptions, but not requiring any symmetry property on it, we prove that the problem has infinitely many solutions.

Periodic standing-wave approximation: Nonlinear scalar fields, adapted coordinates, and the eigenspectral method

The periodic standing wave (PSW) method for the binary inspiral of black holes and neutron stars computes exact numerical solutions for periodic standing-wave spacetimes and then extracts approximate solutions of the physical problem, with outgoing waves. The method requires solution of a boundary-value problem with a mixed (hyperbolic and elliptic) character. We present here a new numerical method for such problems, based on three innovations: (i) a coordinate system adapted to the geometry of the problem, (ii) an expansion in multipole moments of these coordinates and a filtering out of higher moments, and (iii) the replacement of the continuum multipole moments with their analogs for a discrete grid. We illustrate the efficiency and accuracy of this method with nonlinear scalar model problems. Finally, we take advantage of the ability of this method to handle highly nonlinear models to demonstrate that the outgoing approximations extracted from the standing-wave solutions are highly accurate even in the presence of strong nonlinearities.

Intermediate inflation and the slow-roll approximation

It is shown that spatially homogeneous solutions of the Einstein equations coupled to a nonlinear scalar field and other matter exhibit accelerated expansion at late times for a wide variety of potentials V. These potentials are strictly positive but tend to zero at infinity. They satisfy restrictions on V′/V and V″/V′ related to the slow-roll approximation. These results generalize Wald's theorem for spacetimes with positive cosmological constant to those with accelerated expansion driven by potentials belonging to a large class.

Extended Born-Infeld dynamics and cosmology

We introduce an extension of the Born-Infeld action for a scalar field and show that it can act as unifying dark matter, providing an explanation for both structure formation and the accelerated expansion of the universe. We investigate the cosmological dynamics of this theory in a particular case, referred to as the ``Milne-Born-Infeld'' (MBI) Lagrangian. We show that this model, whose equation of state has effectively a single free parameter, is consistent with recent type Ia supernovae data, providing a fit as good as for the $\ensuremath{\Lambda}\mathrm{C}\mathrm{D}\mathrm{M}$ model with the same number of degrees of freedom. Furthermore, this parameter is tightly constrained by current data, making the model easily testable with other observables. Contrary to previous candidates for unifying dark matter, the sound velocity of the MBI model is vanishing both close to the dark-matter state as well as near the cosmological constant state. This could avoid the problems on the matter power spectrum that were present in previous adiabatic dark-matter/dark-energy unification models. We also present a short discussion on the causal propagation in nonlinear scalar field theories such as the one proposed here.

Multiple Solutions of Nonlinear Scalar Field Equations

We establish existence of multiple solutions of the semilinear elliptic equation where N ≥ 3, 2 < p < 2N/(N − 2), and a(x) and q(x) tend to some positive limits a ∞ and q ∞ respectively as |x| → ∞. Assuming some weak one-sided asymptotic estimates for a(x) and q(x), we prove that (℘) has at least pairs ±u of nontrivial weak solutions.

Generalized Exact Cosmologies with Interacting Yang-Mills and Nonlinear Scalar Fields

It has been shown that the set of the Einstein-Yang-Mills equations and the nonlinear scalar field have generalized solutions with a homogeneous scalar field, which interacts with the YM field in the Friedmann Universe.

Plane-Symmetric Solitons of Spinor and Scalar Fields

We consider a system of nonlinear spinor and scalar fields with minimal coupling in general relativity. The nonlinearity in the spinor field Lagrangian is given by an arbitrary function of the invariants generated from the bilinear spinor forms S=ψψ and P=iψγ5ψ; the scalar Lagrangian is chosen as an arbitrary function of the scalar invariant Ω= ϕ,αϕ,α, that becomes linear at Ω→0. The spinor and the scalar fields in question interact with each other by means of a gravitational field which is given by a plane-symmetric metric. Exact plane-symmetric solutions to the gravitational, spinor and scalar field equations have been obtained. Role of gravitational field in the formation of the field configurations with limited total energy, spin and charge has been investigated. Influence of the change of the sign of energy density of the spinor and scalar fields on the properties of the configurations obtained has been examined. It has been established that under the change of the sign of the scalar field energy density the system in question can be realized physically iff the scalar charge does not exceed some critical value. In case of spinor field no such restriction on its parameter occurs. In general it has been shown that the choice of spinor field nonlinearity can lead to the elimination of scalar field contribution to the metric functions, but leaving its contribution to the total energy unaltered.

Accelerated cosmological expansion due to a scalar field whose potential has a positive lower bound

In many cases a nonlinear scalar field with potential V can lead to accelerated expansion in cosmological models. This paper contains mathematical results on this subject for homogeneous spacetimes. It is shown that, under the assumption that V has a strictly positive minimum, Wald's theorem on spacetimes with positive cosmological constant can be generalized to a wide class of potentials. In some cases, detailed information on late-time asymptotics is obtained. Results on the behaviour in the past time direction are also presented.

A Simple Model of Space-Time Metric Signature Change

The problem of describing the space-time metric signature change is discussed. A simple model with the space-time metric signature determined by a nonlinear scalar field is suggested. It is demonstrated that both classical and quantum descriptions of the metric signature change are possible within the framework of the examined model, and the most characteristic features and differences of the classical and quantum descriptions are briefly discussed.

A Quantum Cosmological Bianchi I Model with Interacting Spinor and Scalar Fields

A Bianchi I model of the Universe filled with interacting nonlinear spinor and scalar fields is studied within quantum geometrodynamics. Three types of interaction are considered: gradient, Yukawa, and axion ones. For massless fermion fields, the variables in the Wheeler – de Witt equation will separate. The solution can be interpreted using a two-component perfect liquid. One component corresponds to a massless scalar field, while the other – to a nonlinear spinor field. The interaction between the spinor and scalar fields can lead to elimination of singularity of the wave function. There is a possibility of existence of a discrete spectrum of the quantum Universe, as well as tunneling from the region with a rigorous equation of state to the region of the de Sitter vacuum.

Nonlinear multidimensional cosmological models with form fields: Stabilization of extra dimensions and the cosmological constant problem

We consider multidimensional gravitational models with a nonlinear scalar curvature term and form fields in the action functional. In our scenario it is assumed that the higher dimensional spacetime undergoes a spontaneous compactification to a warped product manifold. Particular attention is paid to models with quadratic scalar curvature terms and a Freund-Rubin-like ansatz for solitonic form fields. It is shown that for certain parameter ranges the extra dimensions are stabilized. In particular, stabilization is possible for any sign of the internal space curvature, the bulk cosmological constant, and of the effective four-dimensional cosmological constant. Moreover, the effective cosmological constant can satisfy the observable limit on the dark energy density. Finally, we discuss the restrictions on the parameters of the considered nonlinear models and how they follow from the connection between the D-dimensional and the four-dimensional fundamental mass scales.

Classical Euclidean Wormhole Solution and Wave Function for a Nonlinear Scalar Field

In this paper we consider the classical Euclidean wormhole solution of the Born—Infeld scalar field. The corresponding classical Euclidean wormhole solution can be obtained analytically for both very small and large \(\dot \varphi\). At the extreme limit of small \(\dot \varphi\) the wormhole solution has the same format as one obtained by Giddings and Strominger (Nuclear Physics B306, 890, 1988). At the extreme limit of large \(\dot \varphi\) the wormhole solution is a new one. The wormhole wave functions can also be obtained for both very small and large \(\dot \varphi\). These wormhole wave functions are regarded as solutions of quantum-mechanical Wheeler—Dewitt equation with certain boundary conditions.

A remark on least energy solutions in $\mathbf {R}^N$

We study a mountain pass characterization of least energy solutions of the following nonlinear scalar field equation in $\mathbf {R}^N$: \begin{equation*} -\Delta u = g(u), u \in H^1(\mathbf {R}^N), \end{equation*} where $N\geq 2$. Without the assumption of the monotonicity of $t\mapsto \frac {g(t)}{t}$, we show that the mountain pass value gives the least energy level.

Threshold of global existence for 3-D attractive nonlinear Schrödinger equations under confined potentials

Based on a nonlinear critical index, a threshold of global existence on the L 2-norm of initial data for 3-D attractive nonlinear Schrödinger equations under confined potentials is derived. For numerical verification, a newly developed radial basis function method is applied in this paper to solve the related nonlinear scalar field equation in R 3.

No-go theorem forfalse vacuum black holes

We study the possibility of non-singular black holesolutions in the theory of general relativity coupled toa nonlinear scalar field with a positive potentialpossessing two minima: a `false vacuum' with positiveenergy and a `true vacuum' with zero energy. Assumingthat the scalar field starts at the false vacuum at theorigin and comes to the true vacuum at spatial infinity,we prove a no-go theorem by extending a no-hair theoremto the black hole interior: no smooth solutions existwhich interpolate between the local de Sitter solutionnear the origin and the asymptotic Schwarzschild solutionthrough a regular event horizon or several horizons.

The class of exact cosmological models with a scalar-field nonlinearity induced by the Yang-Mills field

A new class of exact solutions of the system of Einstein-Yang-Mills equations and the equations of a nonlinear scalar field interacting with the Yang-Mills SO(3) field is obtained on the basis of the Friedman spatially homogeneous cosmological models.

Approximate particle-like solutions for a nonlinear relativistic scalar complex field model in 3+1 dimensions

A weakly nonlinear Lorentz invariant complex field model in 3+1 dimensions is studied by an asymptotic perturbation method, based on Fourier expansion and spatio-temporal rescaling. It is shown that a nonlinear system of partial differential equations describes oscillation amplitudes of Fourier modes. This system is C-integrable, i.e., can be linearized through a suitable transformation of the dependent and independent variables. We resolve the Cauchy problem and demonstrate that localized nondispersive waves (envelope solitons) with finite energy exist under appropriate initial conditions. These particle-like solutions propagate with the group velocity of their carrier wave. During a collision solitons maintain their shape, because the only change is a phase shift. Energy E and momentum p of solitons are identical to those of a relativistic particle. If the Planck constant is connected to the spatial dimension of the envelope soliton, then we obtain at the lowest order of approximation the quantum relations E=ℏ ω, λ= h/ p, where λ and ω are wavelength and frequency of the carrier wave. This work represents a possible way to achieve the Einstein–de Broglie soliton–particle concept.