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This paper is devoted to the study of high-order finite difference scheme for the nonlinear Schrodinger equation with wave operator. The difference scheme is three level and a five point stencil is used for spatial variable. Existence of solutions is shown using a variant of Brouwer fixed point theorem. The unconditional stability as well as uniqueness of the difference scheme are also discussed in detail. The convergence of the difference scheme is proved by utilizing the energy method to be of fourth-order in space and second-order in time in the discrete maximum norm without any restrictions on the mesh sizes. Finally, some numerical experiments and comparisons with other existing methods in the literature are presented. The numerical results show that the high-order difference scheme of this article improves the accuracy of the space and time direction.

In this paper, three energy preserving numerical methods are proposed, including the Crank–Nicolson Galerkin–Legendre spectral (CN–GLS) method, the SAV Galerkin–Legendre spectral (SAV–GLS) method, and the ESAV Galerkin–Legendre spectral (ESAV–GLS) method, for the space fractional nonlinear Schrodinger equation with wave operator. In theoretical analyses, we take the CN–GLS method as an example to analyze the boundness of numerical solution and the unconditional spectral–accuracy convergence in L 2 and L ∞ norms. The effective numerical implementations of the proposed spectral Galerkin methods are discussed in detail. Numerical comparisons are reported to illustrate that the theoretical results are reasonable and the proposed spectral Galerkin methods have high efficiency for energy preservation in long–time computations.

In this work, we consider the numerical computation for the two-dimensional generalized nonlinear Schrodinger equations with wave operator. Based on the scalar auxiliary variable (SAV) approach, the original problem is transformed into an equivalent one, which corresponds to the energy-conservation laws. We present an energy-conserving and linearly implicit three-level scheme for the equivalent system. The energy-conserving property, boundedness of the numerical solution and convergence analysis in the discrete maximum norm are derived, which has not restricted to specific forms of cubic nonlinear term f and not needed the sharply restriction on mesh size. Finally, numerical experiments on several models confirm our theoretical results.

In this paper we introduce a matrix representation of the logarithmic wave operator and we study a second-order semilinear evolution equation governed by this operator. We present a result of local well-posedness for this problem and properties of the logarithmic wave operator in terms of the logarithmic negative Dirichlet Laplacian operator.

We propose a multiscale approach for coupling multi-physics processes across the scales. The physics is based on discrete phenomena, triggered by local thermo-hydro-mechano-chemical (THMC) instabilities, that cause cross-diffusion (quasi-soliton) acceleration waves. These waves nucleate when the overall stress field is incompatible with accelerations from local feedbacks of generalized THMC thermodynamic forces that trigger generalized thermodynamic fluxes of another kind. Cross-diffusion terms in the 4×4 THMC diffusion matrix are shown to lead to multiple diffusional P and S wave equations as coupled THMC solutions. Uncertainties in the location of meso-scale material instabilities are captured by a wave-scale correlation of probability amplitudes. Cross-diffusional waves have unusual dispersion patterns and, although they assume a solitary state, do not behave like solitons but show complex interactions when they collide. Their characteristic wavenumber and constant speed define mesoscopic internal material time–space relations entirely defined by the coefficients of the coupled THMC reaction–cross-diffusion equations. A companion paper proposes an application of the theory to earthquakes showing that excitation waves triggered by local reactions can, through an extreme effect of a cross-diffusional wave operator, lead to an energy cascade connecting large and small scales and cause solid-state turbulence.

In this paper, an efficient semi-implicit difference scheme for solving the fractional nonlinear Schrodinger equation with wave operator are proposed and analyzed. The semi-implicit scheme involves...

Wave equations on flat Minkowski spacetimes are generalized to curved spacetimes. We present two candidates for generalized wave equation, discuss their legitimacy and consider their applications to Klein–Gordon equation.

In this study, we design and analyse an efficient and accurate Fourier pseudo-spectral method for solving the nonlinear Schrodinger equation with wave operator (NLSW). In this method, a modified le...

We consider a boundary-value problem for a semilinear hyperbolic equation with iterated multidimensional wave operator in the principal part. The theorems on existence, uniqueness, and nonexistence of solutions of this problem are established.

In this work, we develop the theory of hyperbolic equations with Bessel operators. We construct and invert hyperbolic potentials generated by multidimensional generalized translation. Chapter 1 contains necessary notation, definitions, auxiliary facts and results. In Chapter 2, we study some generalized weight functions related to a quadratic form. These functions are used below to construct fractional powers of hyperbolic operators and solutions of hyperbolic equations with Bessel operators. Chapter 3 is devoted to hyperbolic potentials generated by multidimensional generalized translation. These potentials express negative real powers of the singular wave operator, i. e. the wave operator where the Bessel operator acts instead of second derivatives. The boundedness of such an operator and its properties are investigated and the inverse operator is constructed. The hyperbolic Riesz B-potential is studied as well in this chapter. In Chapter 4, we consider various methods of solution of the Euler-Poisson-Darboux equation. We obtain solutions of the Cauchy problems for homogeneous and nonhomogeneous equations of this type. In Conclusion, we discuss general methods of solution for problems with arbitrary singular operators.

The topology of closed manifolds forces interacting charges to appear in pairs. We take advantage of this property in the setting of the conformal boundary of $\mathrm{AdS}_5$ spacetime, topologically equivalent to the closed manifold $S^1\times S^3$, by considering the coupling of two massless opposite charges on it. Taking the interaction potential as the analog of Coulomb interaction (derived from a fundamental solution of the $S^3$ Laplace-Beltrami operator), a conformal $S^1\times S^3$ metric deformation is proposed, such that free motion on the deformed metric is equivalent to motion on the round metric in the presence of the interaction potential. We give explicit expressions for the generators of the conformal algebra in the representation induced by the metric deformation. By identifying the charge as the color degree of freedom in QCD, and the two charges system as a quark--anti-quark system, we argue that the associated conformal wave operator equation could provide a realistic quantum mechanical description of the simplest QCD system, the mesons. Finally, we discuss the possibility of employing the compactification radius, $R$, as another scale along $\Lambda_{QCD}$, by means of which, upon reparametrizing $Q^2c^2$ as $\left( Q^2c^2 +\hbar^2 c^2/R^2\right)$, a pertubative treatment of processes in the infrared could be approached., Comment: Minor changes to match the published version

In this paper, we develop two classes of linear high-order conservative numerical schemes for the nonlinear Schrodinger equation with wave operator. Based on the method of order reduction in time and the scalar auxiliary variable technique, we transform the original model into an equivalent system, where the energy is modified as a quadratic form. To construct linear high-order conservative schemes, we first adopt the extrapolation strategy to derive a linearized PDE system, which approximates the transformed model with high precision and inherits the modified energy conservation law. Then we employ the symplectic Runge–Kutta method in time to arrive at a class of linear high-order energy-preserving schemes. This numerical strategy presents a paradigm for developing arbitrarily high-order linear structure-preserving algorithms which could be implemented simply. In order to complement the new linear schemes, the prediction-correction method is presented to obtain another class of energy-preserving algorithms. Furthermore, the trigonometric pseudo-spectral method is applied for the spatial discretization to match the order of accuracy in time. We provide ample numerical results to confirm the convergence, accuracy and conservation property of the proposed schemes.

In this paper, we propose a linearized finite element method (FEM) for solving the cubic nonlinear Schrodinger equation with wave operator. In this method, a modified leap–frog scheme is applied for time discretization and a Galerkin finite element method is applied for spatial discretization. We prove that the proposed method keeps the energy conservation in the given discrete norm. Comparing with non–conservative schemes, our algorithm keeps higher stability. Meanwhile, an optimal error estimate for the proposed scheme is given by an error splitting technique. That is, we split the error into two parts, one from temporal discretization and the other from spatial discretization. First, by introducing a time–discrete system, we prove the uniform boundedness for the solution of this time–discrete system in some strong norms and obtain error estimates in temporal direction. With the help of the preliminary temporal estimates, we then prove the pointwise uniform boundedness of the finite element solution, and obtain the optimal L 2 –norm error estimates in the sense that the time step size is not related to spatial mesh size. Finally, numerical examples are provided to validate the convergence–order, unconditional stability and energy conservation.

In the present paper, the reducibility is derived for the wave equations with finitely smooth and time-quasi-periodic potential subject to periodic boundary conditions. More exactly, the linear wave equation u t t − u x x + M u + e ( V 0 ( ω t ) u x x + V ( ω t , x ) u ) = 0 , x ∈ R / 2 π Z can be reduced to a linear Hamiltonian system with a constant coefficient operator which is of pure imaginary point spectrum set, where V is finitely smooth in ( t , x ) , quasi-periodic in time t with Diophantine frequency ω ∈ R n , and V 0 is finitely smooth and quasi-periodic in time t with Diophantine frequency ω ∈ R n . Moreover, it is proved that the corresponding wave operator possesses the property of pure point spectra and zero Lyapunov exponent.

We prove a global uniqueness theorem of reconstruction of a matrix-potential a ⁢ ( x , t ) {a(x,t)} of one-dimensional wave equation □ ⁢ u + a ⁢ u = 0 {\square u+au=0} , x > 0 , t > 0 {x>0,t>0} , □ = ∂ t 2 - ∂ x 2 {\square=\partial_{t}^{2}-\partial_{x}^{2}} with zero Cauchy data for t = 0 {t=0} and given Cauchy data for x = 0 {x=0} , u ⁢ ( 0 , t ) = 0 {u(0,t)=0} , u x ⁢ ( 0 , t ) = g ⁢ ( t ) {u_{x}(0,t)=g(t)} . Here u , a , f {u,a,f} , and g are n × n {n\times n} smooth real matrices, det ⁡ ( f ⁢ ( 0 ) ) ≠ 0 {\det(f(0))\neq 0} , and the matrix ∂ t ⁡ a {\partial_{t}a} is known.

In this paper, we study a coupled nonlinear Schrödinger system with small initial data in a product space. We establish a modified scattering of the solutions of this system and we construct a modified wave operator. The study of the resonant system, which provides the asymptotic dynamics, allows us to highlight a control of the Sobolev norms and interesting dynamics with the beating effect. The proof uses a recent work of Hani, Pausader, Tzvetkov, and Visciglia for the modified scattering, and a recent work of Grébert, Paturel, and Thomann for the study of the resonant system.

Hertz-Heaviside Field Equations (HHFE) which describe radiation by arbitrary volumetric bodies in arbitrary motion are reviewed. The special case of HHFE for moving electric and magnetic Hertzian dipoles in arbitrary motion are investigated in the sense of Schwartz-Sobolev distributions. In presence of moving non-volumetric radiators HHFE decouple into wave equations expressed by D’Alembert wave operator of stationary media enhanced by convection currents with Diracdelta behavior. The electric and magnetic radiation fields of these dipole sources are discussed and their Doppler analyses are carried out for certain modes of motion.

The paper is devoted to the study of the fractional integral operator which is a negative real power of the singular wave operator generated by Bessel operator, its properties and its inverse using weighted distributions.

We study the problem of wave propagation in the medium whose velocity characteristics change under an external impact. We will study a three-dimensional case. The aim of our study is constructing the asymptotic of the solution for the wave operator with a variable time depended coefficient of the Laplacian at infinity. The case of meromorphic and holomorphic coefficients is considered.

In this paper, we present some new integral and series representations for the eigenfunctions of the multidimensional time‐fractional diffusion‐wave operator with the time‐fractional derivative of order β ∈]1, 2[ defined in the Caputo sense. The integral representations are obtained in form of the inverse Fourier–Bessel transform and as a double contour integrals of the Mellin–Barnes type. Concerning series expansions, the eigenfunctions are expressed as the double generalized hypergeometric series for any β ∈]1, 2[ and as Kampé de Fériet and Lauricella series in two variables for the rational values of β. The limit cases 𝛽=1 (diffusion operator) and 𝛽=2 (wave operator) as well as an intermediate case 𝛽=32 are studied in detail. Finally, we provide several plots of the eigenfunctions to some selected eigenvalues for different particular values of the fractional derivative order β and the spatial dimension n. In this paper, we present some new integral and series representations for the eigenfunctions of the multidimensional time-fractional diffusion-wave operator with the time-fractional derivative of order $\beta \in ]1,2[$ defined in the Caputo sense. The integral representations are obtained in form of the inverse Fourier-Bessel transform and as a double contour integrals of the Mellin-Barnes type. Concerning series expansions, the eigenfunctions are expressed as the double generalized hypergeometric series for any $\beta \in ]1,2[$ and as Kamp\'{e} de F\'{e}riet and Lauricella series in two variables for the rational values of $\beta$. The limit cases $\beta=1$ (diffusion operator) and $\beta=2$ (wave operator) as well as an intermediate case $\beta=\frac{3}{2}$ are studied in detail. Finally, we provide several plots of the eigenfunctions to some selected eigenvalues for different particular values of the fractional derivative order $\beta$ and the spatial dimension $n$. published

We generalize the recent result of Erdogan, Goldberg and Green on the $L^{p}$-boundedness of wave operators for two dimensional Schrodinger operators and prove that they are bounded in $L^{p}(\mathbb{R}^2)$ for all $1 < p < \infty$ if and only if the Schrodinger operator possesses no $p$-wave threshold resonances, viz. Schrodinger equation $(-\Delta + V(x))u(x) = 0$ possesses no solutions which satisfy $u(x) = (a_1 x_1 + a_2 x_2) |x|^{-2} + o(|x|^{-1})$ as $|x| \to \infty$ for an $(a_1, a_2) \in \mathbb{R}^2 \setminus \{(0,0)\}$ and, otherwise, they are bounded in $L^{p}(\mathbb{R}^2)$ for $1 < p \leq 2$ and unbounded for $2 < p < \infty$. We present also a new proof for the known part of the result.

We propose a wave operator method to calculate eigenvalues and eigenvectors of large parameter-dependent matrices, using an adaptative active subspace. We consider a hamiltonian which depends on external adjustable or adiabatic parameters, using adaptative projectors which follow the successive eigenspaces when the adjustable parameters are modified. The method can also handle non-hermitian hamiltonians. An iterative algorithm is derived and tested through comparisons with a standard wave operator algorithm using a fixed active space and with a standard block-Davidson method. The proposed approach is competitive, it converges within a few dozen iterations at constant memory cost. We first illustrate the abilities of the method on a 4-D coupled oscillator model hamiltonian. A more realistic application to molecular photodissociation under intense laser fields with varying intensity or frequency is also presented. Maps of photodissociation resonances of H${}_2^+$ in the vicinity of exceptional points are calculated as an illustrative example., 30 pages, 4 figures

Within the framework of the Duffin-Kemmer-Petiau (DKP) formalism with a deformation, an approach to the construction of the path integral representation in parasuperspace for the Green's function of a spin-1 massive particle in external Maxwell's field is developed. For this purpose a connection between the deformed DKP-algebra and an extended system of the parafermion trilinear commutation relations for the creation and annihilation operators $a^{\pm}_{k}$ and for an additional operator $a_{0}$ obeying para-Fermi statistics of order 2 based on the Lie algebra $\mathfrak{so}(2M+2)$ is established. The representation for the operator $a_{0}$ in terms of generators of the orthogonal group $SO(2M)$ correctly reproducing action of this operator on the state vectors of Fock space is obtained. An appropriate system of the parafermion coherent states as functions of para-Grassmann numbers is introduced. The procedure of the construction of finite-multiplicity approximation for determination of the path integral in the relevant phase space is defined through insertion in the kernel of the evolution operator with respect to para-supertime of resolutions of the identity. In the basis of parafermion coherent states a matrix element of the contribution linear in covariant derivative $\hat{D}_{\mu}$ to the time-dependent Hamilton operator $\hat{\cal H}(\tau)$, is calculated in an explicit form. For this purpose the matrix elements of the operators $a^{\phantom{2}}_0$, $a_{0}^{2}$, the commutators $[\hspace{0.03cm}a^{\phantom{\pm}\!}_{0}, a^{\pm}_{n}\hspace{0.02cm}]$, $[\hspace{0.03cm}a^{2}_{0}, a^{\pm}_{n}\hspace{0.02cm}]$, and the product $\hat{A}\hspace{0.03cm}[\hspace{0.03cm}a^{\phantom{\pm}\!}_{0}, a^{\pm}_{n}\hspace{0.02cm}]$ with $\hat{A} \equiv\exp\hspace{0.02cm}\bigl(-i\frac{2\pi}{3}\,a_{0}\bigr)$, were preliminary defined., Comment: 33 pages, the original text is considerably reduced, version published in JHEP

We consider the class of higher derivative field equations whose wave operator is a square of another self-adjoint operator of lower order. At the free level, the models of this class are shown to admit a two-parameter series of integrals of motion. The series includes the canonical energy. Every conserved quantity is unbounded in this series. The interactions are included into the equations of motion such that a selected representative in conserved quantity series is preserved at the non-linear level. The interactions are not necessarily Lagrangian, but they admit Hamiltonian form of dynamics. The theory is stable if the integral of motion is bounded from below due to the interaction. The motions are finite in the vicinity of the conserved quantity minimum. The equations of motion for fluctuations have the derived form with no resonance. The general constructions are exemplified by the models of the Pais-Uhlenbeck oscillator with multiple frequency and Podolsky electrodynamics. The example is also considered of stable non-abelian Yang-Mills theory with higher derivatives., 21 pages

We extend constructions of classical Clifford analysis to the case of indefinite non-degenerate quadratic forms. We define (p,q)-left- and right-monogenic functions by means of Dirac operators that factor a certain wave operator. We prove two different versions of Cauchy's integral formulas for these functions. The two formulas arise from dealing with singularities in distinct ways, and are inspired by the methods of [L, FL]. These results indicate the merit of these methods for dealing with singularities., Comment: 24 pages, to appear in Complex Analysis and Operator Theory

Combining the compact method with the structure-preserving algorithm, we propose a compact local energy-preserving scheme and a compact local momentum-preserving scheme for the nonlinear Schrödinger equation with wave operator (NSEW). The convergence rates of both schemes are Oh4+τ2. The discrete local conservative properties of the presented schemes are derived theoretically. Numerical experiments are carried out to demonstrate the convergence order and local conservation laws of the developed algorithms.

The paper is devoted to the study of the fractional integral operator which is a negative real power of the singular wave operator generated by Bessel operator and its inverse using weighted generalized functions. Such operators are called hyperbolic B-potentials. Boundedness, Green formula, inversion were proved for hyperbolic B-potentials here.

This paper focuses on the construction and analysis of the energy-conserving numerical schemes for the generalized nonlinear space fractional Schrodinger equations with wave operator. Combining the scalar auxiliary variable (SAV) approach, we present an energy-conserving and linearly implicit scheme, while the previous conservative schemes are generally fully implicit. The energy-conserving property, boundedness and convergence of the numerical solution of the fully discrete scheme are derived for one and multi-dimensional cases. The numerical analysis is also considered. Finally, numerical examples on several fractional models illustrate that the proposed scheme can guarantee conservation of the system energy and confirm our theoretical results.

One of the mathematical problems arising in PhotoAcoustic Tomography (PAT), a novel and promising technology in medical imaging, is to recover the initial function from the solution of the wave equation on a surface, where the detectors of PAT are located. We define the wave operator as the transform assigning to a given function f the solution of the wave equation on the detector surface (where the detectors are located) with the initial function f. Here we study many properties of this wave operator including the inversion formulas and stability estimates assuming that the detector surface is a hyperplane.

In this paper, we study the efficient solution of the nonlinear Schrodinger equation with wave operator, subject to periodic boundary conditions. In such a case, it is known that its solution conserves a related functional. By using a Fourier expansion in space, the problem is at first casted into Hamiltonian form, with the same Hamiltonian functional. A Fourier–Galerkin space semi-discretization then provides a large-size Hamiltonian ODE problem, whose solution in time is carried out by means of energy-conserving methods in the HBVM class (Hamiltonian boundary value methods). The efficient implementation of the methods for the resulting problem is also considered and some numerical examples are reported.

This paper is concerned with the periodic solutions for the one dimensional nonlinear wave equation with either constant or variable coefficients. The constant coefficient model corresponds to the classical wave equation, while the variable coefficient model arises from the forced vibrations of a nonhomogeneous string and the propagation of seismic waves in nonisotropic media. For finding the periodic solutions of variable coefficient wave equation, it is usually required that the coefficient u ( x ) satisfies ess inf η u ( x ) > 0 with η u ( x ) = 1 2 u ″ u − 1 4 ( u ′ u ) 2 , which actually excludes the classical constant coefficient model. For the case η u ( x ) = 0 , it is indicated to remain an open problem by Barbu and Pavel (1997) [6] . In this work, for the periods having the form T = 2 p − 1 q ( p , q are positive integers) and some types of boundary value conditions, we find some fundamental properties for the wave operator with either constant or variable coefficients. Based on these properties, we obtain the existence of periodic solutions when the nonlinearity is monotone and bounded. Such nonlinearity may cross multiple eigenvalues of the corresponding wave operator. In particular, we do not require the condition ess inf η u ( x ) > 0 .

Two-grid algorithms based on two conservative and implicit finite element methods are studied for two-dimensional nonlinear Schrodinger equation with wave operator. The existence and uniqueness of their solutions and the conservative laws are established for both schemes. To linearize the fully discrete nonlinear coupling problem, the original problem is decomposed into two equivalent nonlinear coupled hyperbolic–parabolic equations. Algorithm 1 has three steps based on one Newton iteration on the fine grid and further correction on coarse grid, while Algorithm 2 has three steps based on two Newton iterations on the fine grid. Optimal order L p error estimations of the two-grid algorithms are conducted in detail by optimal order L p error estimates of finite element methods without any time-step size conditions. Both theoretically and numerically are shown that the coarse space can be extremely coarse, with no loss in the order of accuracy, and the two-grid algorithms still achieve the optimal convergence order when the mesh sizes satisfy H = O ( h 1 3 ) for Algorithm 1 and H = O ( h 1 4 ) for Algorithm 2.

We establish the unique solvability and obtain the explicit expression for the classical solution of the mixed problem for multidimensional hyperbolic equations with wave operator.

In this paper, transient surface Green's functions (GFs) excited by a general dynamic pulse buried in a constrained transversely isotropic (TI) half-space are derived in analytical forms. The constrained transverse isotropy is defined by an explicit relation that expresses one of the five elastic constants in terms of the other four independent constants so that no coupled P-SV wave exists. Use is made of the solutions available for general TI materials in Hankel–Laplace transformed domain in order to extract the corresponding response functions for the constrained material. Under this constraint, the coupled fourth-order wave operator, which represents the convolved motion of the quasi-longitudinal and quasi-transverse waves, is decoupled into two separated second-order wave operators describing the motion of the pure dilatational (P) and pure shear (SV) waves. The involved Laplace transform inversion is carried out using the novel Cagniard scheme so that the final displacement GFs are expressed in terms of the elementary line integrals over a finite interval. The GF solution is valid for the entire range of the constrained TI material including material isotropy as its special case, and clearly illustrates the disturbances due to different stress waves including, P, SV, SH, Rayleigh and diffracted SP waves. Moreover, the GFs show that in this constrained TI material, the compressional and horizontally polarized waves, excited by a source located at an arbitrary depth, propagate towards the surface with direction-dependent velocities. The explicit formulas for these velocities of the waves are obtained in terms of the inclination angle of the wave normal with respect to the axis of material symmetry. On the other hand, the vertically polarized wave and the diffracted SP wave propagate with eigen velocity, independent from the direction of the propagation. Two critical epicentral distances, at which mode conversion in ray diagram occurs, are detected for this constrained TI material. One critical distance marks the phenomena of total reflection of SV-waves where the incident SV-wave is totally reflected and the diffracted SP-wave travels along the free surface. The other marks the distance where the SH- and SP-waves become coalescent, and furthermore, before this epicentral distance the SH-wave is faster, and after this distance, the SP-wave is faster. To show the relevance of this constrained TI material in engineering and material modeling applications and its relation to the physics of TI materials, we also demonstrate that, in addition to its mathematical simplification, this reduced anisotropy can well approximate the elastic coefficients of some real TI materials, particularly those encountered in exploration geophysics and seismology.

Based on the invariant energy quadratization approach, we propose a linear implicit and local energy preserving scheme for the nonlinear Schrodinger equation with wave operator, that describes the solitary waves in physics. In order to overcome the difficulty of designing an efficient scheme for the imaginary functions of the nonlinear Schrodinger equation with wave operator, we transform the original problem into its real form. By introducing some auxiliary variables, the real form of nonlinear Schrodinger equation with wave operator is reformulated into an equivalent system, which admits the modified local energy conservation law. Then the equivalent system is discretized by the finite difference method to yield a linear system at each time step, which can be efficiently solved. A numerical analysis of the proposed scheme is conducted to show its uniquely solvability and convergence. Our proposed method is validated by numerical simulations in terms of accuracy, energy conservation law and stability.

As a novel and promising technology in medical imaging, Photoacoustic Tomography (PAT) is based on the generation of acoustic waves inside an object of interest by stimulating electromagnetic waves. This acoustic wave is measured outside the object and converted into a 3-dimensional image. Various shapes of detectors are suggested because of some limitations of the classic detector. Here we study PAT with a linear detector. We define some operator as the transform assigning to a given function f the integral of the solution of the wave equation over the detector line with the initial function f. We provide many properties of this wave operator including the inversion formulas and stability estimates under two different geometric settings.

In this paper, we study the rate of approximation of the combination of some generalized multivariate average on Hardy spaces and obtain its equivalent relation to the K -functionals. The result is an extension of a result in Dai and Ditzian (2004). We also extend and improve Theorem 6.2 in Belinsky et al. (2003).

Within the framework of the Duffin–Kemmer–Petiau (DKP) formalism a consistent approach to derivation of the third-order wave equation is suggested. For this purpose, an additional algebraic object, the so-called q-commutator (q is a primitive cubic root of unity) and a new set of matrices ημ instead of the original matrices βμ of the DKP algebra are introduced. It is shown that in terms of these η-matrices, we have succeeded to reduce the procedure of the construction of cubic root of the third-order wave operator to a few simple algebraic transformations and to a certain operation of passage to the limit z → q, where z is some complex deformation parameter entering into the definition of the ημ-matrices. A corresponding generalization of the result obtained to the case of interaction with an external electromagnetic field introduced through the minimal coupling scheme is performed. The application to the problem of construction within the DKP approach of the path integral representation in parasuperspace for the propagator of a massive vector particle in a background gauge field is discussed.

In this paper, we use the critical point theory for convex, lower semicontinuous perturbations of C 1 {C^{1}} -functionals to obtain the existence of multiple nontrivial solutions for one parameter potential systems involving the operator u ↦ div ⁡ ( ∇ ⁡ u 1 - | ∇ ⁡ u | 2 ) {u\mapsto\operatorname{div}(\frac{\nabla u}{\sqrt{1-|\nabla u|^{2}}})} . The solvability of a general non-potential system is also established.

This paper is a contribution to harmonic analysis of compact solvmanifolds. We consider the four-dimensional oscillator group ${\rm Osc}_1$, which is a semi-direct product of the three-dimensional Heisenberg group and the real line. We classify the lattices of ${\rm Osc}_1$ up to inner automorphisms of ${\rm Osc}_1$. For every lattice $L$ in ${\rm Osc}_1$, we compute the decomposition of the right regular representation of ${\rm Osc}_1$ on $L^2(L\backslash{\rm Osc}_1)$ into irreducible unitary representations. This decomposition allows the explicit computation of the spectrum of the wave operator on the compact locally-symmetric Lorentzian manifold $L\backslash {\rm Osc}_1$.

In container yard, the way of allocating space for outbound containers determines the transport efficiency of containers and the management cost. The existing space allocation methods usually have problems such as low computational efficiency, large number of shifts etc. This paper proposes an outbound containers’ block-location allocation (OCBLA) method to solve this problem, it has two steps, including allocating a block for containers which arrive at the same time and transported by the same vessel and allocating the specific location for each container. The first step aims to balance the load among blocks and reduce the cost of transportation, and last step uses ITO algorithm to reduce the number of shifts. The ITO algorithm regards every optional place as a moving particle and uses drift operator to control particle moving in the direction of the optimal place, wave operator to control particle exploring around. Having a tendency to explore makes the algorithm to find the optimal result in a limited number of tests, thus improving the computational efficiency. Through experiments it can be seen that this method can reduce the number of shifts during the process of container stacking, and get better results while having good real-time performance.

We consider the issue of correspondence between symmetries and conserved quantities in the class of linear relativistic higher-derivative theories of derived type. In this class of models the wave operator is a polynomial in another formally self-adjoint operator, while each isometry of space-time gives rise to the series of symmetries of action functional. If the wave operator is given by n-th-order polynomial then this series includes n independent entries, which can be explicitly constructed. The Noether theorem is then used to construct an n-parameter set of second-rank conserved tensors. The canonical energy-momentum tensor is included in the series, while the other entries define independent integrals of motion. The Lagrange anchor concept is applied to connect the general conserved tensor in the series with the original space-time translation symmetry. This result is interpreted as existence of multiple energy-momentum tensors in the class of derived systems. To study stability we seek for bounded-conserved quantities that are connected with the time translations. We observe that the derived theory is stable if its wave operator is defined by a polynomial with simple and real roots. The general constructions are illustrated by the examples of the Pais&ndash, Uhlenbeck oscillator, higher-derivative scalar field, and extended Chern&ndash, Simons theory.

In this paper we prove global well-posedness and modified scattering for the massive Maxwell–Klein–Gordon equation in the Coulomb gauge on $$\mathbb {R}^{1+d}$$ $$(d \ge 4)$$ for data with small critical Sobolev norm. This extends to the general case $$ m^2 > 0 $$ the results of Krieger–Sterbenz–Tataru ( $$d=4,5 $$ ) and Rodnianski–Tao ( $$ d \ge 6 $$ ), who considered the case $$ m=0$$ . We proceed by generalizing the global parametrix construction for the covariant wave operator and the functional framework from the massless case to the Klein–Gordon setting. The equation exhibits a trilinear cancelation structure identified by Machedon–Sterbenz. To treat it one needs sharp $$L^{2}$$ null form bounds, which we prove by estimating renormalized solutions in null frames spaces similar to the ones considered by Bejenaru–Herr. To overcome logarithmic divergences we rely on an embedding property of $$ \Box ^{-1} $$ in conjunction with endpoint Strichartz estimates in Lorentz spaces.