Exact self-consistent plane-symmetric solutions of the spinor-field equation with a nonlinear term that depends on theS 2−P 2 invariant

Exact plane-symmetric solutions of the spinor-field equation with zero mass parameter and nonlinear term that depends arbitrarily on the S²−P² invariant are derived with consideration of an intrinsic gravitational field. The existence of regular solutions with localized energy density among the solutions obtained is investigated. Equations with powerlaw and polynomial nonlinearity types are examined in detail. For the power-law nonlinearity, when the nonlinear term entering into the Lagrangian has the form L_N=γIⁿ, where γ is the nonlinearity parameter and n=const, it is shown that the initial system of Einstein and spinor-field equations has regular solutions with localized energy density only under the conditions λ=−Λ² < 0, n > 1. In this case, the examined field configuration posesses a negative energy. In the case of polynomial nonlinearity, regular solutions with localized energy density T ₀ ⁰ (x), positive energy (upon integration over y and z between finite limits), and an everywhere regular metric that transforms into a two-dimensional space-time metric at spatial infinity are obtained. It is shown that the initial nonlinear spinor-field equations in two-dimensional space-time have no solutions with localized energy density. Thus, it is established that the intrinsic gravitational field plays a regularizing role in the frmation of regular localized solutions to the examined nonlinear spinor-field equations. Russian University of People's Friendship. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 12–19, November, 1999.     

Exact plane-symmetric solutions of the nonlinear equations of a spinor field in the theory of gravitation

We obtain exact plane-symmetric solutions of the spinor field equations with a nonlinear term that is an arbitrary functions of the invariant and with the self-gravitational field taken into account. Conditions are formulated for which the initial system of Einstein's equation and the spinor field equations with a power-law nonlinearity have regular solutions with localized (negative) spinor field energy density: so-called soliton-like solutions. Exact solutions of the spinor field equations are also obtained in flat space—time in this case and it is shown that the initial system of equations does not have soliton-like solutions. Hence the self-gravitational field plays a crucial (regularizing) in soliton-like solutions of the nonlinear spinor field equations.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 63–68, August, 1995.     

Nonlinear spinor fields in Bianchi-I space: Exact self-consistent solutions

Calculations are performed to obtain exact self-consistent solutions of nonlinear spinor-field equations with self-action terms in Bianchi-I space. The latter terms are arbitrary functions of the invariant . A detailed examination is made of equations with exponential nonlinearity, when the nonlinear term in the Lagrangian of the spinor field L_n=sⁿ. Here, is the nonlinearity parameter, n>1. It is shown that these equations have finite solutions and solutions that are singular at the initial moment of time. The singularity is absent in the case of solutions that describe systems of fields for which the energy dominance condition is violated. It is further shown that if the mass parameter m0 in the spinor-field equation, expansion of Bianchi-I space becomes isotropic as t . This does not occur when m=0. Specific examples of solutions of linear and nonlinear spinor-field equations are presented.Russian University of International Fellowship. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 40–45, July, 1994.     

Exact static solutions of nonlinear spinor-field equations in a Hedel universe

Exact static solutions of spinor-field equations with nonlinear terms that are arbitrary functions of the invariant S=ψψ are obtained in the external gravitational field of a Hedel universe. The specific type of nonlinear Lagrangian that produces regular and localized distributions of spinor-field energy density is discussed. Exact solutions of the original equations are also obtained in plane spacetime. Here it is shown that irrespective of the form of the nonlinear Lagrangian, the energy density of the spinor field is constant, i.e., there is no localization. This means that the external gravitational field of a Hedel universe has a definite role in forming soliton-like configurations of the nonlinear spinor field. Russian University of International Amity. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 111–116, July, 1996.     

Static finite energy solutions of a classical field theory with positive mass-square

Static finite energy solutions of the field theory described by are obtained. Some of the interesting features of this model are (1) the mass-square here is positive unlike in the λφ⁴ Higg's model, (2) the potential has three global minimas, (3) the spectrum is bounded from below unlike in the λφ⁴ theory with λ<0, (4) there are two kink and two antikink solutions, (5) unlike sine-Gordon and λφ⁴ models here there are two particles with masses m and 2m. Nontopological finite energy solutions have also been obtained for gφ ⁶ field theory with g < 3λ ² / 16m ².     

Amplitude-squared squeezing in superposed coherent states

We study amplitude-squared squeezing of the Hermitian operator Z_θ=Z₁ cosθ+Z₂ sin θ, in the most general superposition state , of two coherent states and . Here operators Z_1,2 are defined by , a is annihilation operator, θ is angle, and complex numbers C_1,2 , α, β are arbitrary and only restriction on these is the normalization condition of the state . We define the condition for a state to be amplitude-squared squeezed for the operator Z_θ if squeezing parameter , where N=a⁺a and . We find maximum amplitude-squared squeezing of Z_θ in the superposed coherent state with minimum value 0.3268 of the parameter S for an infinite combinations with α- β= 2.16 exp [±i(π/4) + iθ/2], and with arbitrary values of (α+β) and θ. For this minimum value of squeezing parameter S, the expectation value of photon number can vary from the minimum value 1.0481 to infinity. Variations of the parameter S with different variables at maximum amplitude-squared squeezing are also discussed.     

On concentration of positive bound states of nonlinear Schrödinger equations

We study the concentration behavior of positive bound states of the nonlinear Schrödinger equation $$ih\frac{{\partial \psi }}{{\partial t}} = \frac{{ - h^2 }}{{2m}}\Delta \psi + V\left( x \right)\psi - \gamma \left| \psi \right|^{p - 1} \psi .$$ Under certain condition ofV, we show that positive ground state solutions must concentrate at global minimum points ofV ash→0⁺; moreover, a point at which a sequence of positive bound states concentrates must be a critical point ofV. In cases thatV is radial, we prove that the positive radial solutions with least energy among all nontrivial radial solutions must concentrate at the origin ash→0⁺.     

Renormalizable theories from nonrenormalizable interactions

By using Kikkawa’s method the equivalence of the nonrenormalizable pair interaction to a renormalizable theory is proved. Equivalence relationships between a few other nonrenormalizable and renormalizable interactions are also indicated.     

Existence of localized solutions for a classical nonlinear Dirac field

We prove the existence of stationary states for nonlinear Dirac equations of the form: $$i\gamma ^\mu \partial _\mu \psi - m\psi + F(\bar \psi \psi )\psi = 0.$$ We seek solutions which are separable in spherical coordinates and we use a shooting method for solving the associated problem of ordinary differential equations.     

The NLS Equation in Dimension One with Spatially Concentrated Nonlinearities: the Pointlike Limit

In the present paper, we study the following scaled nonlinear Schrödinger equation (NLS) in one space dimension: $$ i\frac{\rm d}{{\rm d}t}\psi^{\varepsilon}(t)=-\Delta\psi^{\varepsilon}(t) +\frac{1}{\varepsilon}V\left(\frac{x}{\varepsilon} \right)|\psi^{\varepsilon}(t)|^{2\mu}\psi^{\varepsilon}(t)\quad \varepsilon > 0\,\quad V\in L^1(\mathbb{R},(1+|x|){\rm d}x) \cap L^\infty(\mathbb{R}).$$ This equation represents a nonlinear Schrödinger equation with a spatially concentrated nonlinearity. We show that in the limit \({\varepsilon\to 0}\) the weak (integral) dynamics converges in \({H^1(\mathbb{R})}\) to the weak dynamics of the NLS with point-concentrated nonlinearity: $$ i\frac{{\rm d}}{{\rm d}t} \psi(t) =H_{\alpha} \psi(t) .$$ where H _α is the Laplacian with the nonlinear boundary condition at the origin \({\psi'(t,0+)-\psi'(t,0-)=\alpha|\psi(t,0)|^{2\mu}\psi(t,0)}\) and \({\alpha=\int_{\mathbb{R}}V{\rm d}x}\) . The convergence occurs for every \({\mu\in \mathbb{R}^+}\) if V ≥  0 and for every  \({\mu\in (0,1)}\) otherwise. The same result holds true for a nonlinearity with an arbitrary number N of concentration points.     

General theory of renormalization of gauge theories in nonlinear gauges

We discuss the general theory of renormalization of unbroken gauge theories in the nonlinear gauges in which the gauge-fixing term is of the form We show that higher loop renormalization modifiesfα [A] to contain ghost terms of the form and show how the corresponding ghost terms are deduced fromfα [A, c, c] uniquely. We show that the theory can be renormalized while preserving a modified form of BRS invariance by multiplicative and independent renormalizations onA, c, g, η, ζ, τ. We briefly discuss the independence of the renormalized S-matrix from η,ζ, τ.     

Nonperturbative Green's function approach to quantum field theory and arbitrariness of solutions

Green's function equations are considered for interacting spinor and (pseudo) scalar fields with interactions . These equations do not determine higher many-point functions if two-point functions are given as “input.” If vertex parts are given as input, two-point functions are determined but higher many-point functions are not determined.     

Asymptotic behavior of solutions to certain nonlinear Schrödinger-Hartree equations

The asymptotic behavior of solutions to the Cauchy problem for the equation $$i\psi _\imath = \frac{1}{2}\Delta \psi - \upsilon (\psi )\psi , \upsilon = r^{ - 1} *|\psi |^2 ,$$ and for systems of similar form, is studied. It is shown that the norms $$\parallel \psi (t)\parallel _{L_2 (|x| \leqq R)}^2 + \parallel \nabla \psi (t)\parallel _{L_2 (|x| \leqq R)}^2 $$ are integrable in time for any fixedR>0, from which it follows that $$\mathop {\lim }\limits_{t \to \infty } \parallel \psi (t)\parallel _{L_2 (|x| \leqq R)} = 0.$$ \] Nevertheless, it is established that anL ₂-scattering theory is impossible.     

Existence of excited states for a nonlinear Dirac field

We prove the existence of infinitely many stationary states for the following nonlinear Dirac equation $$i\gamma ^\mu \partial _\mu \psi - m\psi + (\bar \psi \psi )\psi = 0.$$ Seeking for eigenfunctions splitted in spherical coordinates leads us to analyze a nonautonomous dynamical system inR ². The number of eigenfunctions is given by the number of intersections of the stable manifold of the origin with the curve of admissible datum. This proves the existence of infinitely many stationary states, ordered by the number of nodes of each component.     

Algebraic structure of nonassociative spinor field

An analysis of the algebra of octonions, the algebraic structure of nonassociative spinors, is presented, and a spinor field theory that is completely identical to Dirac theory is constructed in an associative basis. A spinor covariance transformation is introduced, and it is shown that it coincides with the Poincaré group of 4-dimensional space. The field equation is introduced through a spinor invariance transformation. Constraints imposed by the field equation on the eigenvalues of the transformation generators are considered. It is proved that the particles in a system at rest which are nonzero are , the unit; , the energysign of the particle; and s ₆, one of the spin components of the particle. Tbilisi University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 11, pp. 59–65, November, 1998.     

On the Best Constant in the Moser-Onofri-Aubin Inequality

Let S ² be the 2-dimensional unit sphere and let J _α denote the nonlinear functional on the Sobolev space H ¹(S ²) defined by

Stability of chirped bright and dark soliton-like solutions of the cubic complex Ginzburg-Landau equation with variable coefficients

We consider an inhomogeneous optical fiber system described by the generalized cubic complex Ginzburg-Landau (CGL) equation with varying dispersion, nonlinearity, gain (loss), nonlinear gain (absorption) and the effect of spectral limitation. Exact chirped bright and dark soliton-like solutions of the CGL equation were found by using a suitable ansatz. Furthermore, we analyze the features of the solitons and consider the problem of stability of these soliton-like solutions under finite initial perturbations. It is shown by extensive numerical simulations that both bright and dark soliton-like solutions are stable in an inhomogeneous fiber system. Finally, the interaction between two chirped bright and dark soliton-like pulses is investigated numerically.     

A model field theory in which two Nambu-Goldstone symmetries are realised by one massless pseudoscalar boson

We consider a model field theory consisting of two Nambu-Jona-Lasinio spin 1/2 fields interacting via a coupling \(f(\bar \psi _1 \gamma ^\mu \gamma ^5 \psi _1 )(\bar \psi _2 \gamma _\mu \gamma ^5 \psi _2 )\) and which is therefore invariant under the two symmetries \(\psi _1 (x) \to e^{i\alpha _1 } \gamma ^5 \psi _1 (x)\) and \(\psi _2 (x) \to e^{i\alpha _2 } \gamma ^5 \psi _2 (x)\) . We look for solutions in which these symmetries are spontaneously broken by giving the fermions non-zero masses. Each of the two pairs of axial-vector vertex functions in the theory satisfy two coupled integral equations, which are solved in the ‘chain approximation’. We find that all four vertex functions have the same singularity structure, in particular a pole atq ²=0 corresponding to a massless pseudoscalar Nambu-Goldstone boson, and another pole corresponding to an axial-vector boson whose mass is cut-off dependent, but which for a certain range of values off ² is a stable particle. By considering the coupling of the strings of nucleon-antinucleon psuedoscalar ‘bubbles’ which generate the massless Nambu-Goldstone bosons associated with fermions 1 and 2, we show explicitly that there is only one massless Nambu-Goldstone boson in the theory.     

A Centre-Stable Manifold for the Focussing Cubic NLS in $${\mathbb{R}}^{1+3}$$

Consider the focussing cubic nonlinear Schrödinger equation in \({\mathbb{R}}^3\) :

$i\psi_t+\Delta\psi = -|\psi|^2 \psi. \quad (0.1) $

It admits special solutions of the form e ^itα ?, where \(\phi \in {\mathcal{S}}({\mathbb{R}}^3)\) is a positive (? > 0) solution of

$-\Delta \phi + \alpha\phi = \phi^3. \quad (0.2)$

The space of all such solutions, together with those obtained from them by rescaling and applying phase and Galilean coordinate changes, called standing waves, is the 8-dimensional manifold that consists of functions of the form \(e^{i(v \cdot + \Gamma)} \phi(\cdot - y, \alpha)\) . We prove that any solution starting sufficiently close to a standing wave in the \(\Sigma = W^{1, 2}({\mathbb{R}}^3) \cap |x|^{-1}L^2({\mathbb{R}}^3)\) norm and situated on a certain codimension-one local Lipschitz manifold exists globally in time and converges to a point on the manifold of standing waves. Furthermore, we show that \({\mathcal{N}}\) is invariant under the Hamiltonian flow, locally in time, and is a centre-stable manifold in the sense of Bates, Jones [BatJon]. The proof is based on the modulation method introduced by Soffer and Weinstein for the L ²-subcritical case and adapted by Schlag to the L ²-supercritical case. An important part of the proof is the Keel-Tao endpoint Strichartz estimate in \({\mathbb{R}}^3\) for the nonselfadjoint Schrödinger operator obtained by linearizing (0.1) around a standing wave solution. All results in this paper depend on the standard spectral assumption that the Hamiltonian

$\mathcal H = \left ( \begin{array}{cc}\Delta + 2\phi(\cdot, \alpha)^2 - \alpha &;\quad \phi(\cdot, \alpha)^2 \\ -\phi(\cdot, \alpha)^2 &;\quad -\Delta - 2 \phi(\cdot, \alpha)^2 + \alpha \end{array}\right ) \quad (0.3)$

has no embedded eigenvalues in the interior of its essential spectrum \((-\infty, -\alpha) \cup (\alpha, \infty)\) .

     

The Relativistic non-abelian Chern-Simons Equations

We study ther xr system of nonlinear elliptic equations ,a=1,2,...,r,x∈R ², where λ τ 0 is a constant parameter,K = (K_ab) is the Cartan matrix of a semi-simple Lie algebra, and β_p is the Dirac measure concentrated atp R ². This system of equations arises in the relativistic non-Abelian Chern-Simons theory and may be viewed as a nonintegrable deformation of the integrable Toda system. We establish the existence of a class of solutions known as topological multivortices. The crucial step in our method is the use of the decomposition theorem of Cholesky for positive definite matrices so that a variational principle can be formulated. Research supported in part by the National Science Foundation under grant DMS-9596041