Nonlinear Scalar Field Research Articles

Global behavior of the ground state energy of the nonlinear scalar field equation

We study the global behavior of the ground state energy for the nonlinear scalar field equation (including the $p$-Laplacian version), and give a complete description of this energy in terms of nonlinearity power. More precisely,let$\mathcal{E}_{\lambda,q}(u)=\frac{1}{p}(\|\nabla~u\|_p^p+\lambda\|u\|_p^p)-\frac{1}{q}\|u\|_q^q$ bethe energy functional in $~W^{1,p}(\mathbb{R}^N)$,where $\lambda>0$, $1<p<N$, $p<q<p^*:=\frac{pN}{N-p}$, and $p^*$ is the critical Sobolev exponent. It is known that $\mathcal{E}_{\lambda,q}(u)$ has a unique radially symmetric mountain-pass critical point $U_{\lambda,q}$, which is called the ground state 解 to the corresponding nonlinear scalar field equation and whose energy $\mathcal{E}_{\lambda,q}(U_{\lambda,q})$ is called the ground state energy $\mathfrak{m}(\lambda,q)$.We show that there is a decreasing function $\lambda_0(q)$ defined in $q\in~[p,p^*]$ with $\lambda_0(p)=1~$ and $\lambda_0(p^*)=0$such that $\mathfrak{m}(\lambda,r)$ is strictly increasing for $r\in(p,~q)$ with $\lambda\in(0,~\lambda_0(q))$ being fixed, $\mathfrak{m}(\lambda,r)$ is strictly decreasing for $r\in(q,~p^*)$ with $\lambda\in(\lambda_0(q),~1)$ being fixed, and $\mathfrak{m}(\lambda,r)$ is strictly decreasing for all $r\in(p,~p^*)$ with $\lambda\in[1,~+\infty)$ being fixed.We also deduce the precise asymptotic behavior of $\lambda_0(q)$ as $q\to~p$ and $q\to~p^*$. This is done by establishing a relation between power-law scalar fieldequations and logarithmic-law scalar field equations,which is of the independent interest.

Non-existence of ground states and gap of variational values for 3D Sobolev critical nonlinear scalar field equations

In this paper, we consider minimization problems related to the combined power-type nonlinear scalar field equations involving the Sobolev critical exponent in three space dimensions. In four and higher space dimensions, it is known that for any frequency and any power of the subcritical nonlinearity, there exists a ground state. In contrast to those cases, when the space dimension is three and the subcritical power is three or less, we can show that there exists a threshold frequency, above which no ground state exists, and below which the ground state exists (see Theorems 1.1 and 1.2). Furthermore, we prove the difference between two typical variational problems used to characterize the ground states (see Theorem 1.3).

An inhomogeneous perturbation for a class of nonlinear scalar field equations

This paper deals with a class of nonlinear scalar field equations with an inhomogeneous perturbation. Two positive solutions were obtained using the variational methods.

Soliton-like solutions of nonlinear scalar and electromagnetic field equations in gravitational theory

In this work, exact analytical static spherical symmetric solutions to the nonlinear interacting electromagnetic and massless scalar field equations have been determined taking into account the proper gravitational field of elementary particles. The obtained results prove that all solutions of the Einstein equation and those of the interacting fields are regular with localized energy density. All non-zero components of the 4-vector potential are solutions to the inverted Painlevé-Gambier XI equation. Moreover, the total energy of interacting fields is limited and the total charge of elementary particles is finite. These solutions obtained are soliton-like and can be used as a model to describe the internal structure of elementary particles.

Small solutions of the Einstein–Boltzmann-scalar field system in a spatially flat FLRW spacetime

The Cauchy problem for the Einstein–Boltzmann-scalar field system is studied. A spatially flat Friedmann-Lemaître-Robertson-Walker spacetime is considered with matter contents described by the relativistic Boltzmann equation for Israel particles and a nonlinear scalar field with an exponential potential. The initial data are assumed to be small in a suitable sense, and we obtain the global existence and asymptotic behavior of small solutions.

Exact Static Plane Symmetric Soliton-Like Solutions to the Nonlinear Interacting Electromagnetic and Scalar Field Equations in General Relativity

This research work is related to soliton solutions considered as models that can describe the complex configuration of elementary particles from the study of the interactions of their fields. It is interested in the interaction of fields between two different elementary particles by expressing their physical properties. For that, we have obtained, exact static plane symmetric soliton-like solutions to the nonlinear equations of interacting electromagnetic and scalar fields taking into account the own gravitational field of elementary particles using the calibrated invariance function P(I). It has been proved that all solutions of the Einstein, nonlinear electromagnetic and scalar field equations are regular with the localized energy density. Moreover, the total charge of particles is finite and the total energy of the interaction fields is bounded. It have been emphasized the importance to the own gravitational field of elementary particles and the role of the nonlinearity of fields in the determination of these solutions. In flat space-time, soliton-like solutions exist but the total energy of the interaction fields is equal to zero. We have also shown that in the linear case, soliton-like solutions are absent.

Thin accretion discs around spherically symmetric configurations with nonlinear scalar fields

We study stable circular orbits (SCO) around static spherically symmetric configuration of general relativity with a nonlinear scalar field (SF). The configurations are described by solutions of the Einstein-SF equations with monomial SF potential $V(\ensuremath{\phi})=|\ensuremath{\phi}{|}^{2n}$, $n>2$, under the conditions of the asymptotic flatness and behavior of SF $\ensuremath{\phi}\ensuremath{\sim}1/r$ at spatial infinity. We proved that under these conditions, the solution exists and is uniquely defined by the configuration mass $M>0$ and scalar ``charge'' $Q$. The solutions and the space-time geodesics have been investigated numerically in the range $n\ensuremath{\le}40$, $|Q|\ensuremath{\le}60$, $M\ensuremath{\le}60$. We focus on how nonlinearity of the field affects properties of SCO distributions (SCOD), which, in turn, affect topological form of the thin accretion disk around the configuration. Maps are presented showing the location of possible SCOD types for different $M$, $Q$, $n$. We found a lot of differences from the Fisher-Janis-Newman-Winicour metric (FJNW) dealing with the linear SF, though basic qualitative properties of the configurations have much in common with the FJNW case. For some values of $n$, a topologically new SCOD type was discovered that is not available for the FJNW metric. Like FJNW, all images of accretion disks have a dark spot in the center (mimicking the ordinary black hole), either because there is no SCO near the center or due to the strong deflection of photons near the singularity, although fine features other than a black hole can appear with a special choice of $M$, $Q$.

Singularities in Static Spherically Symmetric Configurations of General Relativity with Strongly Nonlinear Scalar Fields

There are a number of publications on relativistic objects dealing either with black holes or naked singularities in the center. Here we show that there exist static spherically symmetric solutions of Einstein equations with a strongly nonlinear scalar field, which allow the appearance of singularities of a new type (“spherical singularities”) outside the center of curvature coordinates. As the example, we consider a scalar field potential ∼sinh(ϕ2n),n>2, which grows rapidly for large field values. The space-time is assumed to be asymptotically flat. We fulfill a numerical investigation of solutions with different n for different parameters, which define asymptotic properties at spatial infinity. Depending on the configuration parameters, we show that the distribution of the stable circular orbits of test bodies around the configuration is either similar to that in the case of the Schwarzschild solution (thus mimicking an ordinary black hole), or it contains additional rings of unstable orbits.

Stability analysis of a charged black hole with a nonlinear complex scalar field

It has been shown recently that the charged black hole can be scalarized if Maxwell field minimally couples with a complex scalar which has nonnegative nonlinear potential. We firstly prove that such scalarization cannot be a result of continuous phase transition for general scalar potential. Furthermore, we numerically find that it is possible that the RN black hole will be scalarized by a first order phase transition spontaneously and extreme RN black hole is not stable in micro-canonical ensemble. In addition, considering a massless scalar perturbation, we compute the quasi-normal modes of the scalarized charged black hole and the results imply that the spontaneously scalarized charged black hole is not only favored in thermodynamics but also is kinetically stable against scalar perturbation at linear level. Our numerical results also definitely gives negative answer to Penrose-Gibbons conjecture and two new versions of Penrose inequality in charged case are suggested.

Fifth-force screening around extremely compact sources

Many non-linear scalar field theories possess a screening mechanism that can suppress any associated fifth force in dense environments. As a result, these theories can evade local experimental tests of new forces. Chameleon-like screening, which occurs because of non-linearities in the scalar potential or the coupling to matter, is well understood around extended objects. However, many experimental tests of these theories involve objects with spatial extent much smaller than the scalar field's Compton wavelength, and which could therefore be considered point-like. In this work, we determine how the fifth forces are screened in the limit that the source objects become extremely compact.

Nonlinear scalar field equation with competing nonlocal terms * *d’Avenia and Pomponio are members of GNAMPA of INdAM and are partially supported by PRIN 2017JPCAPN Qualitative and quantitative aspects of nonlinear PDEs. d’Avenia is also supported by FRA2019 of Politecnico di Bari. Mederski was partially supported by the National Science Centre, Poland, Grant No. 2017/26/E/ST1/00817 and by the Deutsche Forschungsgemeinschaft

We find radial and nonradial solutions to the following nonlocal problem under general assumptions, in the spirit of Berestycki and Lions, imposed on f and g, where N ⩾ 3, 0 ⩽ β ⩽ α < N, ω ⩾ 0, are continuous functions with corresponding primitives F, G, and I α , I β are the Riesz potentials. If β > 0, then we deal with two competing nonlocal terms modelling attractive and repulsive interaction potentials.

Normalized solutions for fractional nonlinear scalar field equations via Lagrangian formulation

We study existence of solutions for the fractional problem where N ⩾ 2, s ∈ (0, 1), m > 0, μ is an unknown Lagrange multiplier and satisfies Berestycki–Lions type conditions. Using a Lagrangian formulation of the problem (P m ), we prove the existence of a weak solution with prescribed mass when g has L 2 subcritical growth. The approach relies on the construction of a minimax structure, by means of a Pohozaev’s mountain in a product space and some deformation arguments under a new version of the Palais–Smale condition introduced in Hirata and Tanaka (2019 Adv. Nonlinear Stud. 19 263–90); Ikoma and Tanaka (2019 Adv. Differ. Equ. 24 609–46). A multiplicity result of infinitely many normalized solutions is also obtained if g is odd.

Asymptotics for 1D Klein-Gordon Equations with Variable Coefficient Quadratic Nonlinearities

We initiate the study of the asymptotic behavior of small solutions to one-dimensional Klein-Gordon equations with variable coefficient quadratic nonlinearities. The main discovery in this work is a striking resonant interaction between specific spatial frequencies of the variable coefficient and the temporal oscillations of the solutions. In the resonant case a novel type of modified scattering behavior occurs that exhibits a logarithmic slow-down of the decay rate along certain rays. In the non-resonant case we introduce a new variable coefficient quadratic normal form and establish sharp decay estimates and asymptotics in the presence of a critically dispersing constant coefficient cubic nonlinearity. The Klein-Gordon models considered in this paper are motivated by the study of the asymptotic stability of kink solutions to classical nonlinear scalar field equations on the real line.

Modeling dark matter halos with nonlinear field theories

In the present work, we adopt a nonlinear scalar field theory coupled to the gravity sector to model galactic dark matter. We found analytical solutions for the scalar field coupled to gravity in the Newtonian limit, assuming an isotropic spacetime and a field potential, with a position dependent form of the superpotential, which entails the nonlinear dynamics of the model with self-interactions. The model introduces a position dependent enhancement of the self-interaction of the scalar fields towards the galaxy center, and while going towards the galaxy border the interaction tends to vanish building a non self-interacting DM scenario. The developed approach is able to provide a reasonable analytical description of the rotation curves in both dwarf and low surface brightness late-type galaxies, with parameters associated with the dynamics of the scalar field.

A Sufficient Condition for Asymptotic Stability of Kinks in General (1+1)-Scalar Field Models

We study stability properties of kinks for the (1+1)-dimensional nonlinear scalar field theory models $$\begin{aligned} \partial _t^2\phi -\partial _x^2\phi + W'(\phi ) = 0, \quad (t,x)\in \mathbb {R}\times \mathbb {R}. \end{aligned}$$ The orbital stability of kinks under general assumptions on the potential W is a consequence of energy arguments. Our main result is the derivation of a simple and explicit sufficient condition on the potential W for the asymptotic stability of a given kink. This condition applies to any static or moving kink, in particular no symmetry assumption is required. Last, motivated by the Physics literature, we present applications of the criterion to the $$P(\phi )_2$$ theories and the double sine-Gordon theory.

Hořava-Lifshitz Scalar Field Cosmology: Classical and Quantum Viewpoints

In this paper, we study a projectable Hořava-Lifshitz cosmology without the detailed balance condition minimally coupled to a nonlinear self-coupling scalar field. In the minisuperspace framework, the super-Hamiltonian of the presented model is constructed by means of which some classical solutions for the scale factor and scalar field are obtained. Since these solutions exhibit various types of singularities, we came up with the quantization of the model in the context of the Wheeler-DeWitt approach of quantum cosmology. The resulting quantum wave functions are then used to investigate the possibility of the avoidance of classical singularities due to quantum effects which show themselves important near these singularities.

General class of optimal Sobolev inequalities and nonlinear scalar field equations

We find a class of optimal Sobolev inequalities(∫RN|∇u|2dx)NN−2≥CN,G∫RNG(u)dx,u∈D1,2(RN),N≥3, where the nonlinear function G:R→R of class C1 satisfies general growth assumptions in the spirit of the fundamental works of Berestycki and Lions. We admit, however, a wider class of problems involving zero, positive and infinite mass cases as well as G need not be even. We show that any minimizer is radial up to a translation. Moreover, up to a dilation, it is a least energy solution of the nonlinear scalar field equation−Δu=g(u)in RN,with g=G′. In particular, if G(u)=u2log⁡|u|, then the sharp constant is CN,G:=2⁎(N2)2N−2e2(N−1)N−2(π)NN−2 and uλ(x)=eN−12−λ22|x|2 with λ>0 constitutes the whole family of minimizers up to translations. The optimal inequality provides a new proof of the classical logarithmic Sobolev inequality based on a Pohozaev manifold approach. Moreover, if N≥4, then there is at least one nonradial solution and if, in addition, N≠5, then there are infinitely many nonradial solutions of the nonlinear scalar field equation. The energy functional associated with the problem may be infinite on D1,2(RN) and is not Fréchet differentiable in its domain. We present a variational approach to this problem based on a new variant of Lions' lemma in D1,2(RN).

Topology Density Map for Urban Data Visualization and Analysis

Density map is an effective visualization technique for depicting the scalar field distribution in 2D space. Conventional methods for constructing density maps are mainly based on Euclidean distance, limiting their applicability in urban analysis that shall consider road network and urban traffic. In this work, we propose a new method named Topology Density Map, targeting for accurate and intuitive density maps in the context of urban environment. Based on the various constraints of road connections and traffic conditions, the method first constructs a directed acyclic graph (DAG) that propagates nonlinear scalar fields along 1D road networks. Next, the method extends the scalar fields to a 2D space by identifying key intersecting points in the DAG, dividing the underlying territory into planar regions using a weighted Voronoi diagram, and calculating the scalar fields for every point. Two case studies demonstrate that the Topology Density Map supplies accurate information to users and provides an intuitive visualization for decision making. An interview with domain experts demonstrates the feasibility, usability, and effectiveness of our method.

Nonradial solutions of nonlinear scalar field equations

We prove new results concerning the nonlinear scalar field equation with a nonlinearity g satisfying the general assumptions due to Berestycki and Lions. In particular, we find at least one nonradial solution for any N ⩾ 4 minimizing the energy functional on the Pohozaev constraint in a subspace of consisting of nonradial functions. If in addition N ≠ 5, then there are infinitely many nonradial solutions. These solutions are sign-changing. The results give a positive answer to a question posed by Berestycki and Lions in [, ]. Moreover, we build a critical point theory on a topological manifold, which enables us to solve the above equation as well as to treat new elliptic problems.

A mass supercritical problem revisited

In any dimension $N\geq1$ and for given mass $m>0$, we revisit the nonlinear scalar field equation with an $L^2$ constraint: $$ -\Delta u=f(u)-\mu u, \quad u \in H^1(\mathbb{R}^N) \quad \text{with} \quad \|u\|^2_{L^2(\mathbb{R}^N)}=m. $$ where $\mu\in\mathbb{R}$ will arise as a Lagrange multiplier. Assuming only that the nonlinearity $f$ is continuous and satisfies weak mass supercritical conditions, we show the existence of ground states and reveal the basic behavior of the ground state energy $E_m$ as $m>0$ varies. In particular, to overcome the compactness issue when looking for ground states, we develop robust arguments which we believe will allow treating other $L^2$ constrained problems in general mass supercritical settings. Under the same assumptions, we also obtain infinitely many radial solutions for any $N\geq2$ and establish the existence and multiplicity of nonradial sign-changing solutions when $N\geq4$. Finally we propose two open problems.