Nonlinear Scalar Field Research Articles

A scalar field theory of 1+1-dimensional laser wakefield accelerators

Abstract A relativistic non-linear scalar field theory is developed from a 2+2-dimensional decomposition of the cold plasma field equations, and the theory is used to investigate a 1+1-dimensional description of a laser wakefield accelerator. The relationship between the properties of a compact laser pulse and its wake is explored. Non-linear solutions are sought describing a regular (i.e. unbroken) wake driven by a prescribed rectangular circularly-polarised laser pulse. An upper bound on the dimensionless amplitude a 0 of the laser pulse is determined as a function of the phase speed v of the wake. The asymptotic behaviour of the upper bound on a 0 as v → c is shown to agree with well-established, but approximate, results obtained using the conventional encoding of the plasma degrees of freedom. Our approach leads to a closed-form expression for the upper bound on a 0 which is exact for all values of the phase speed of the wake, unlike conventional results that are applicable only when v is sufficiently close to c.

Black bounces in conformal Killing gravity

In this work, we analyse black bounce solutions in the recently proposed “Conformal Killing gravity” (CKG), by coupling the theory to nonlinear electrodynamics (NLED) and scalar fields. The original motivation of the theory was essentially to fulfill specific criteria that are absent in existing gravitational theories, namely, to obtain the cosmological constant as an integration constant, derive the energy–momentum conservation law as a consequence of the gravitational field equations, rather than assuming it, and not necessarily considering conformally flat metrics as vacuum solutions. In this work, we extend the static and spherically symmetric solutions obtained in the literature, and explore the possibility of black bounces in CKG, coupled to NLED and scalar fields. We find novel NLED Lagrangian densities and scalar potentials, and extend the class of black bounce solutions found in the literature. Furthermore, within black bounce geometries, we find generalizations of the Bardeen-type and Simpson–Visser geometries and explore the regularity conditions of the solutions.

Infinitely many free or prescribed mass solutions for fractional Hartree equations and Pohozaev identities

Abstract In this paper we study the following nonlinear fractional Hartree (or Choquard-Pekar) equation ( − Δ ) s u + μ u = ( I α * F ( u ) ) F ′ ( u ) in R N , ${\left(-{\Delta}\right)}^{s}u+\mu u=\left({I}_{\alpha }{\ast}F\left(u\right)\right){F}^{\prime }\left(u\right)\quad \text{in} {\mathbb{R}}^{N},$ (*) where μ > 0, s ∈ (0, 1), N ≥ 2, α ∈ (0, N), I α ∼ 1 | x | N − α ${I}_{\alpha }\sim \frac{1}{\vert x{\vert }^{N-\alpha }}$ is the Riesz potential, and F is a general subcritical nonlinearity. The goal is to prove existence of multiple (radially symmetric) solutions u ∈ H s ( R N ) $u\in {H}^{s}\left({\mathbb{R}}^{N}\right)$ , by assuming F odd or even: we consider both the case μ > 0 fixed and the case ∫ R N u 2 = m > 0 ${\int }_{{\mathbb{R}}^{N}}{u}^{2}=m{ >}0$ prescribed. Here we also simplify some arguments developed for s = 1 (S. Cingolani, M. Gallo, and K. Tanaka, “Multiple solutions for the nonlinear Choquard equation with even or odd nonlinearities,” Calc. Var. Partial Differ. Equ., vol. 61, no. 68, p. 34, 2022). A key point in the proof is given by the research of suitable multidimensional odd paths, which was done in the local case by Berestycki and Lions (H. Berestycki and P.-L. Lions, “Nonlinear scalar field equations II: existence of infinitely many solutions,” Arch. Ration. Mech. Anal., vol. 82, no. 4, pp. 347–375, 1983); for (*) the nonlocalities play indeed a special role. In particular, some properties of these paths are needed in the asymptotic study (as μ varies) of the mountain pass values of the unconstrained problem, then exploited to describe the geometry of the constrained problem and detect infinitely many normalized solutions for any m > 0. The found solutions satisfy in addition a Pohozaev identity: in this paper we further investigate the validity of this identity for solutions of doubly nonlocal equations under a C 1-regularity.

The existence and multiplicity of L 2-normalized solutions to nonlinear Schrödinger equations with variable coefficients

Abstract The existence of L 2–normalized solutions is studied for the equation − Δ u + μ u = f ( x , u ) in R N , ∫ R N u 2 d x = m . $-{\Delta}u+\mu u=f\left(x,u\right)\quad \quad \text{in} {\mathbf{R}}^{N},\quad {\int }_{{\mathbf{R}}^{N}}{u}^{2} \mathrm{d}x=m.$ Here m > 0 and f(x, s) are given, f(x, s) has the L 2-subcritical growth and (μ, u) ∈ R × H 1(R N ) are unknown. In this paper, we employ the argument in Hirata and Tanaka (“Nonlinear scalar field equations with L 2 constraint: mountain pass and symmetric mountain pass approaches,” Adv. Nonlinear Stud., vol. 19, no. 2, pp. 263–290, 2019) and find critical points of the Lagrangian function. To obtain critical points of the Lagrangian function, we use the Palais–Smale–Cerami condition instead of Condition (PSP) in Hirata and Tanaka (“Nonlinear scalar field equations with L 2 constraint: mountain pass and symmetric mountain pass approaches,” Adv. Nonlinear Stud., vol. 19, no. 2, pp. 263–290, 2019). We also prove the multiplicity result under the radial symmetry.

An upper bound for the least energy of a sign-changing solution to a zero mass problem

Abstract We give an upper bound for the least possible energy of a sign-changing solution to the nonlinear scalar field equation − Δ u = f ( u ) , u ∈ D 1,2 ( R N ) , $-{\Delta}u=f\left(u\right), u\in {D}^{1,2}\left({\mathrm{R}}^{N}\right),$ where N ≥ 5 and the nonlinearity f is subcritical at infinity and supercritical near the origin. More precisely, we establish the existence of a nonradial sign-changing solution whose energy is smaller that 12c 0 if N = 5, 6 and smaller than 10c 0 if N ≥ 7, where c 0 is the ground state energy.

(Regular) Black holes in conformal Killing gravity coupled to nonlinear electrodynamics and scalar fields

In this work, we explore new solutions with static and spherical symmetry in 4D for black holes and regular black holes in the recently proposed conformal Killing gravity (CKG). This theory is of third order in the derivatives of the metric tensor and essentially satisfies three theoretical criteria for gravitational theories beyond general relativity (GR). The criteria essentially stipulate the following, that one should: (i) obtain the cosmological constant as an integration constant; (ii) derive the energy conservation law as a consequence of the field equations, rather than assuming it; (iii) and not necessarily consider conformally flat metrics as vacuum solutions. In fact, existing modified theories of gravity, including GR, do not simultaneously fulfil all of these three criteria. Here, we couple CKG to nonlinear electrodynamics (NLED) and scalar fields, and we explore solutions of black holes and regular black holes. More specifically, by solving the field equations of CKG, we find specific forms for the NLED Lagrangian, the scalar field and the field potential, and analyse the regularity of the solutions through the Kretschmann scalar. We find generalizations of the Schwarschild–Reissner-Nordström–AdS solutions, and consequently further extend the class of (regular) black hole solutions found in the literature.

Dynamics of nonlinear scalar field with Robin boundary condition on the Schwarzschild–anti–de Sitter background

Dynamics of nonlinear scalar field with Robin boundary condition on the Schwarzschild–anti–de Sitter background

A critical phenomenon for the least energy solutions to nonlinear scalar field equations

We study the existence of least energy solutions to the nonlinear scalar field equation: (1) − Δ u + λ u + V ( x ) u = Q ( x ) | u | p u , u ∈ H 1 ( R N ) , Where V ( x ) , Q ( x ) ∈ L ∞ ( R N ) are real functions satisfying suitable assumptions. By considering the Nehari type constraint N λ := { u ∈ H 1 ( R N ) ∖ { 0 } : ∫ R N ( | ∇ u | 2 + λ | u | 2 + V ( x ) | u | 2 ) d x = ∫ R N Q ( x ) | u | p + 2 d x } , it is shown that (1) exists at least a nontrivial least energy solutions if \\lambda _{*}:=-\\inf \\sigma (-\\Delta +V) $ ]]> λ > λ ∗ := − inf σ ( − Δ + V ) . In this argument, we point out λ ∗ is a critical value for the existence of least energy solutions restricted to N λ . That is, there exists global least energy solutions in N λ if \\lambda _{*} $ ]]> λ > λ ∗ , but not if λ < λ ∗ . Moreover, the asymptotic behavior of least energy solutions as λ → + ∞ is analyzed.

Quasinormal modes of naked singularities in presence of nonlinear scalar fields

Quasinormal modes of naked singularities in presence of nonlinear scalar fields

Normalized solutions of the Schrödinger equation with potential

AbstractIn this paper, for dimension and prescribed mass , we consider the following nonlinear scalar field equation with constraint: where is a Lagrange multiplier, . In particular, satisfies mass supercritical and Sobolev subcritical growth. We prove the existence results of the normalized solution and infinitely many normalized solutions to the above system under some proper assumptions on the functions by the mountain pass argument.

Spanning the full range of neutron star properties within a microscopic description

The high-density behavior of nuclear matter is analyzed within a relativistic mean-field description with non-linear meson interactions. To assess the model parameters and their output, a Bayesian inference technique is used. The Bayesian setup is limited only by a few nuclear saturation properties, the neutron star maximum mass larger than 2 M$_\odot$, and the low-density pure neutron matter equation of state (EOS) produced by an accurate N$^3$LO calculation in chiral effective field theory. Depending on the strength of the non-linear scalar vector field contribution, we have found three distinct classes of EOSs, each one correlated to different star properties distributions. If the non-linear vector field contribution is absent, the gravitational maximum mass and the sound velocity at high densities are the greatest. However, it also gives the smallest speed of sound at densities below three times saturation density. On the other hand, models with the strongest non-linear vector field contribution predict the largest radii and tidal deformabilities for 1.4 M$_\odot$ stars, together with the smallest mass for the onset of the nucleonic direct Urca processes and the smallest central baryonic densities for the maximum mass configuration. These models have the largest speed of sound below three times saturation density, but the smallest at high densities, in particular, above four times saturation density the speed of sound decreases approaching approximately $\sqrt{0.4}c$ at the center of the maximum mass star. On the contrary, a weak non-linear vector contribution gives a monotonically increasing speed of sound. A 2.75 M$_\odot$ NS maximum mass was obtained in the tail of the posterior with a weak non-linear vector field interaction. We found that pQCD favors models with a large non-linear vector field contribution or hyperons.

Nondegeneracy of ground states for nonlinear scalar field equations involving the Sobolev-critical exponent at high frequencies in three and four dimensions

We consider nonlinear scalar field equations involving the Sobolev-critical exponent at high frequencies ω. Since the limiting profile of the ground states as ω→∞ is the Aubin–Talenti function and degenerate in a certain sense, from the point of view of perturbation methods, the nondegeneracy problem for the ground states at high frequencies is subtle. In addition, since the limiting profile (Aubin–Talenti function) fails to lie in L2(Rd) for d=3,4, the nondegeneracy problem for d=3,4 is more difficult than that for d≥5 and an applicable methodology is not known. In this paper, we solve the nondegeneracy problem for d=3,4 by modifying the arguments in Akahori et al. (2020) and Akahori et al. (2019). We also show that the linearized operator around the ground state has exactly one negative eigenvalue.

Existence of nontrivial solutions for the Klein-Gordon-Maxwell system with Berestycki-Lions conditions

Abstract In this article, we study the following Klein-Gordon-Maxwell system: − Δ u − ( 2 ω + ϕ ) ϕ u = g ( u ) , in R 3 , Δ ϕ = ( ω + ϕ ) u 2 , in R 3 , \left\{\phantom{\rule[-1.25em]{}{0ex}}\begin{array}{l}-\Delta u-\left(2\omega +\phi )\phi u=g\left(u),\hspace{1.0em}{\rm{in}}\hspace{1em}{{\mathbb{R}}}^{3},\hspace{1.0em}\\ \Delta \phi =\left(\omega +\phi ){u}^{2},\hspace{1.0em}{\rm{in}}\hspace{1em}{{\mathbb{R}}}^{3},\hspace{1.0em}\end{array}\right. where ω \omega is a constant that stands for the phase; u u and ϕ \phi are unknowns and g g satisfies the Berestycki-Lions condition [Nonlinear scalar field equations. I. Existence of a ground state, Arch. Rational Mech. Anal. 82 (1983), 313–345; Nonlinear scalar field equations. II. Existence of infinitelymany solutions, Arch. Rational Mech. Anal. 82 (1983), 347–375]. The Klein-Gordon-Maxwell system is a model describing solitary waves for the nonlinear Klein-Gordon equation interacting with an electromagnetic field. By using variational methods and some analysis techniques, the existence of positive solution and multiple solutions can be obtained. Moreover, we study the properties of decay estimates and asymptotic behavior for the positive solution.

Normalized solutions to nonlinear scalar field equations with doubly nonlocal terms and critical exponent

In this paper, we study the existence and regularity of normalized solutions to the following doubly nonlocal equation−Δu=λu+μ1(Iα⁎|u|p)|u|p−2u+μ2(Iβ⁎|u|p)|u|p−2uinR3, having a prescribed mass ∫R3u2=a>0, where λ∈R will arise as a Lagrange multiplier, α,β∈(0,3) with α⩽β, p∈[2⁎β,2α⁎], Iα and Iβ are Riesz potentials, μ1,μ2>0 are constants. In particular, we estimate the energy level ingeniously and consider the existence of normalized solutions to the above equation with Hardy–Littlewood–Sobolev lower critical exponent p=2⁎β or upper critical exponent p=2α⁎. The results are supplements to the works of Cao et al. (2021) [5].

Kirchhoff–Boussinesq-type problems with positive and zero mass

Using variational methods we show the existence of solutions for the following class of elliptic Kirchhoff–Boussinesq-type problems given by Δ 2 u − Δ p u + u = h ( u ) , in R N and Δ 2 u − Δ p u = f ( u ) , in R N , where 2 < p ≤ 2 N N − 2 for N ≥ 3 and 2 ∗ ∗ = ∞ for N = 3, N = 4, 2 ∗ ∗ = 2 N N − 4 for N ≥ 5 and h and f are continuous functions that satisfy hypotheses considered by Berestycki and Lions [Nonlinear scalar field. Arch Rational Mech Anal. 1983;82:313–345]. More precisely, the problem with the nonlinearity h is related to the Positive mass case and the problem with the nonlinearity f is related to the Zero mass case. The main argument is to find a Palais–Smale sequence satisfying a property related to Pohozaev identity, as in Hirata et al. [Nonlinear scalar field equations in RN: mountain pass and symmetric mountain pass approaches. Topol Methods Nonlinear Anal. 2010;35:253–276], which was used for the first time by Jeanjean [On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer- type problem set on R N . Proc R Soc Edinb Sect A. 1999;129:787–809].

Quantum Solitodynamics: Non-linear Wave Mechanics and Pilot-Wave Theory

In 1927 Louis de Broglie proposed an alternative approach to standard quantum mechanics known as the double solution program (DSP) where particles are represented as bunched fields or solitons guided by a base (weaker) wave. DSP evolved as the famous de Broglie-Bohm pilot wave interpretation (PWI) also known as Bohmian mechanics but the general idea to use solitons guided by a base wave to reproduce the dynamics of the PWI was abandoned. Here we propose a nonlinear scalar field theory able to reproduce the PWI for the Schrödinger and Klein–Gordon guiding waves. Our model relies on a relativistic ‘phase harmony’ condition locking the phases of the solitonic particle and the guiding wave. We also discuss an extension of the theory for the N particles cases in presence of entanglement and external (classical) electromagnectic fields.

From vacuum to dark energy. Exact anisotropic cosmological solution of Petrov type D for a nonlinear scalar field

From vacuum to dark energy. Exact anisotropic cosmological solution of Petrov type D for a nonlinear scalar field

Dynamics of strongly interacting kink-antikink pairs for scalar fields on a line

This paper concerns classical nonlinear scalar field models on the real line. If the potential is a symmetric double well, then such a model admits static solutions called kinks and antikinks, which are among the simplest examples of topological solitons. We study pure kink-antikink pairs, which are solutions that converge in one infinite time direction to a superposition of one kink and one antikink, without radiation. Our main result is a complete classification of all kink-antikink pairs in the strongly interacting regime, which means the speeds of the kinks tend asymptotically to zero. We show that up to translation there is exactly one such solution, and we give a precise description of the dynamics of the kink separation.

Biharmonic Nonlinear Scalar Field Equations

Abstract We prove a Brezis–Kato-type regularity result for weak solutions to the biharmonic nonlinear equation $$ \begin{align*} &amp; \Delta^2 u = g(x,u)\qquad\text{in }\mathbb{R}^N\end{align*}$$with a Carathéodory function $g:\mathbb {R}^N\times \mathbb {R}\to \mathbb {R}$, $N\geq 5$. The regularity results give rise to the existence of ground state solutions provided that $g$ has a general subcritical growth at infinity. We also conceive a new biharmonic logarithmic Sobolev inequality $$ \begin{align*} &amp; \int_{\mathbb{R}^N}|u|^2\log |u|\, \text{d}x\leq\frac{N}{8}\log \Big(C\int_{\mathbb{R}^N}|\Delta u|^2\, \text{d}x \Big), \quad\text{for } u \in H^2(\mathbb{R}^N), \; \int_{\mathbb{R}^N}u^2\, \text{d}x = 1, \end{align*}$$for a constant $0&amp;lt;C&amp;lt; \big (\frac {2}{\pi e N}\big )^2$ and we characterize its minimizers.

Uniqueness of ground states for combined power-type nonlinear scalar field equations involving the Sobolev critical exponent at high frequencies in three and four dimensions

Uniqueness of ground states for combined power-type nonlinear scalar field equations involving the Sobolev critical exponent at high frequencies in three and four dimensions