Exact sum rules for inhomogeneous strings

We derive explicit expressions for the sum rules of the eigenvalues of inhomogeneous strings with arbitrary density and with different boundary conditions. We show that the sum rule of order N$N$ may be obtained in terms of a diagrammatic expansion, with (N−1)!/2$(N-1)!/2$ independent diagrams. These sum rules are used to derive upper and lower bounds to the energy of the fundamental mode of an inhomogeneous string; we also show that it is possible to improve these approximations taking into account the asymptotic behavior of the spectrum and applying the Shanks transformation to the sequence of approximations obtained to the different orders. We discuss three applications of these results.     

Exact sum rules for inhomogeneous systems containing a zero mode

We show that the formulas for the sum rules for the eigenvalues of inhomogeneous systems that we have obtained in two recent papers are incomplete when the system contains a zero mode. We prove that there are finite contributions of the zero mode to the sum rules and we explicitly calculate the expressions for the sum rules of order one and two. The previous results for systems that do not contain a zero mode are unaffected.     

A new approach of solving Green’s function for wave propagation in an inhomogeneous absorbing medium

A new approach is developed to solve the Green's function that satisfies the Hehmholtz equation with complex refractive index. Especially, the Green's function for the Helmholtz equation can be expressed in terms of a one-dimensional integral, which can convert a Helmholtz equation into a Schr?dinger equation with complex potential. And the Schr?dinger equation can be solved by Feynman path integral. The result is in excellent agreement with the previous work.     

An inhomogeneous theoretical model for analysis of PbSe quantum-dot-doped fiber amplifier

Optical fiber doped with lead salts quantum dots (QDs) of different radii, has previously been proposed as a wide-band amplifier in the optical communication range. The emission and absorption spectra of QDs are size-dependent. Optical fiber amplifier doped with QDs of different sizes is an inhomogeneous gain medium. This inhomogeneity is the principle of operation of wide-band quantum-dot-doped fiber amplifiers (QDFAs). To study the effects of inhomogeneity such as spectral hole burning on the amplifier behavior, a fiber amplifier with an arbitrary size distribution of QDs is taken into consideration. The light propagation equations and rate equations for the inhomogeneous medium are presented. To investigate the gain spectrum, the amplifier bandwidth and the spectral hole burning effect on the amplifier behavior, the system of governing equations for the steady-state regime of operation are solved and numerical results are presented.     

Deterministic inhomogeneous inertia ratchets

We study the deterministic dynamics of a periodically driven particle in the underdamped case in a spatially symmetric periodic potential. The system is subjected to a space-dependent friction coefficient, which is similarly periodic as the potential but with a phase difference. We observe that frictional inhomogeneity in a symmetric periodic potential mimics most of the qualitative features of deterministic dynamics in a homogeneous system with an asymmetric periodic potential. We point out the need of averaging over the initial phase of the external drive at small frictional inhomogeneity parameter values or analogously low potential asymmetry regimes in obtaining ratchet current. We also show that at low amplitudes of the drive, where ratchet current is not possible in the deterministic case, noise plays a significant role in realizing ratchet current.     

New Exact Jacobi Elliptic Function Solutions of Three-Dimensional Nonlinear Helmholtz Equation in a Nonlinear Kerr-Type Medium

In this letter the three-dimensional nonlinear Helmholtz equation is investigated, which describes electro-magnetic wave propagation in a nonlinear Kerr-type medium such that sixteen families of new Jacobi elliptic functionsolutions are obtained, by using our extended Jacobian elliptic function expansion method. When the modulus m → 1or0, the corresponding solitary waves including bright solitons, dark solitons and new line solitons and singly periodicsolutions can be also found.     

Spectral sum rules and search for periodicities in DNA sequences

Periodic patterns play the important regulatory and structural roles in genomic DNA sequences. Commonly, the underlying periodicities should be understood in a broad statistical sense, since the corresponding periodic patterns have been strongly distorted by the random point mutations and insertions/deletions during molecular evolution. The latent periodicities in DNA sequences can be efficiently displayed by Fourier transform. The criteria of significance for observed periodicities are obtained via the comparison versus the counterpart characteristics of the reference random sequences. We show that the restrictions imposed on the significance criteria by the rigorous spectral sum rules can be rationally described with De Finetti distribution. This distribution provides the convenient intermediate asymptotic form between Rayleigh distribution and exact combinatoric theory.     

New Exact Jacobi Elliptic Function Solutions of Three—Dimensional Nonlinear Helmholtz Equation in a Nonlinear Kerr—Type Medium

In this letter the three-dimensional nonlinear Helmholtz equation is investigated.which describes electromagnetic wave propagation in a nonlinear Kerr-type medium such that sixteen families of new Jacobi elliptic function solutions are obtained,by using our extended Jacobian elliptic function expansion method.When the modulus m-→1 or 0,the corresponding solitary waves including bright solitons,dark solitons and new line solitons and singly periodic solutions can be also found.     

The string of variable density: Perturbative and non-perturbative results

We obtain systematic approximations for the modes of vibration of a string of variable density, which is held fixed at its ends. These approximations are obtained iteratively applying three theorems which are proved in the paper and which hold regardless of the inhomogeneity of the string. Working on specific examples we obtain very accurate approximations which are compared both with the results of WKB method and with the numerical results obtained with a collocation approach. Finally, we show that the asymptotic behaviour of the energies of the string obtained with perturbation theory, worked to second order in the inhomogeneities, agrees with that obtained with the WKB method and implies a different functional dependence on the density that in two and higher dimensions.     

The string of variable density: Further results

We analyze the problem of calculating the solutions and the spectrum of a string with arbitrary density and fixed ends. We build a perturbative scheme which uses a basis of WKB-type functions and obtain explicit expressions for the eigenvalues and eigenfunctions of the string. Using this approach we show that it is possible to derive the asymptotic (high energy) behavior of the string, obtaining explicit expressions for the first three coefficients (the first two can also be obtained with the WKB method). Finally, using an iterative approach we also obtain analytical expressions for the low energy behavior of the eigenvalues and eigenfunctions of a string with rapidly oscillating density, recovering (in a simpler way) results in the literature.     

Double-Wronskian solitons and rogue waves for the inhomogeneous nonlinear Schrödinger equation in an inhomogeneous plasma

Plasmas are the main constituent of the Universe and the cause of a vast variety of astrophysical, space and terrestrial phenomena. The inhomogeneous nonlinear Schrödinger equation is hereby investigated, which describes the propagation of an electron plasma wave packet with a large wavelength and small amplitude in a medium with a parabolic density and constant interactional damping. By virtue of the double Wronskian identities, the equation is proved to possess the double-Wronskian soliton solutions. Analytic one- and two-soliton solutions are discussed. Amplitude and velocity of the soliton are related to the damping coefficient. Asymptotic analysis is applied for us to investigate the interaction between the two solitons. Overtaking interaction, head-on interaction and bound state of the two solitons are given. From the non-zero potential Lax pair, the first- and second-order rogue-wave solutions are constructed via a generalized Darboux transformation, and influence of the linear and parabolic density profiles on the background density and amplitude of the rogue wave is discussed.     

The opacity coefficient method—nongray determination of inhomogeneous atmospheric extinction

The computation of radiation transmittance in nongray, inhomogeneous atmospheric models is frequently complicated by complex bands of line spectra which range in value over many orders of magnitudes and depend strongly on either or both of pressure and temperature. We present here a new opacity sampling technique which is shown to determine correctly the wavelength-averaged extinction due to path-dependent realizations of banded line spectra. The technique is easy to implement computationally and is applicable to a wide variety of atmospheric problems in which frequent iteration of the radiative transfer model is required. We consider two such instances: modeling of solar flux attenuation for use in a time-dependent planetary ionosphere model and retrieval from nadir measurements of backscattered solar irradiance. The power of the new method lies in its straightforward analytical treatment of both atmospheric inhomogeneity and spectral complexity. It is thus relevant for both retrieval and radiative transfer modeling purposes.     

New collective modes of interaction nature in inhomogeneous Ising networks

We propose a vertex formulation of the Ising model with inhomogeneous external field on multiconnected networks possessing a superbond structure. The related technique based on gauge degrees of freedom enables us to recognize new collective modes of interaction nature, which provide an exact solution of the inverse profile problem and an explicit form of a local free-energy functional on an extended magnetization-mode space. Application is made to a square strip.     

Momentum sum rules for fragmentation functions

Momentum sum rules for fragmentation functions are considered. In particular, we give a general proof of the Schäfer–Teryaev sum rule for the transverse momentum dependent Collins function. We also argue that corresponding sum rules for related fragmentation functions do not exist. Our model-independent analysis is supplemented by calculations in a simple field-theoretical model.     

Ion acoustic solitary wave in an inhomogeneous plasma with non-uniform temperature

A modifiedK-dV equation, which describes the propagation of an ion acoustic solitary wave in an inhomogeneous plasma with both density and temperature gradients, is derived. It is shown that, the velocity of a solitary wave increases as it propagates towards regions of increasing temperature.     

A new boundary method for electromagnetic scattering from inhomogeneous bodies

A new numerical method for scattering from inhomogeneous bodies is presented. In particular, the 2D case of a TM-polarizated incident wave scattered by an infinite cylinder is considered. The scattered field is sought in two different domains. The first one is a bounded region inside the scattering body with an inhomogeneous permittivity ε(x,y). The second one is an unbounded homogeneous region outside the scatterer. An approximate solution for the scattered field inside the scatterer is sought by applying the QTSM technique. The method of discrete sources is used to approximate the scattered field in the unbounded region outside the scattering body. A comparison of the numerical solution with an analytic solution is performed.     

An efficient, preconditioned, high-order solver for scattering by two-dimensional inhomogeneous media

We consider the problem of evaluating the scattering of TE polarized electromagnetic waves by two-dimensional penetrable inhomogeneities: building upon previous work [IEEE Trans. Antennas Propag. 48 (2000) 1862] we present a practical and general fast integral equation algorithm for this problem. The contributions introduced in this text include: (1) a preconditioner that significantly reduces the number of iterations required by the algorithm in the treatment of electrically large scatterers, (2) a new radial integration scheme based on Chebyshev polynomial approximation, which gives rise to increased accuracy, efficiency and stability, and (3) an efficient and stable method for the evaluation of scaled high-order Bessel functions, which extends the capabilities of the method to arbitrarily high frequencies. These enhancements give rise to an algorithm that is much more accurate and efficient than its previous counterpart, and that allows for treatment of much larger problems than permitted by the previous approach. In one test case, for example, the present algorithm results in far-field errors of 8.9×10⁻¹³ in a 2.12s calculation (on a 1.7 GHz PC) whereas the original algorithm gave rise to far-field errors of 1.1×10⁻⁸ in 88.91s on a 400 MHz PC. In another example, the present algorithm evaluates accurately the scattering by a cylinder of acoustical size κR=256, which is of the order of 20 times larger (400 times larger in square wavelengths) than the largest scatterers that could be treated by the previous approach. Yielding, at worst, third-order far field accuracy (or substantially better, for smooth scatterers) in fast computing times ( operations for an N point mesh) even for discontinuous and complex refractive index distributions (possibly containing severe geometric singularities such as corners and cusps), the proposed approach is the highest-order solver in existence for the problem under consideration.     

Conditional Lie-Bäcklund symmetries and solutions of inhomogeneous nonlinear diffusion equations

The second-order conditional Lie-Bäcklund symmetries of nonlinear diffusion equations with variable coefficients are studied. A number of examples are considered and some exact solutions are constructed via the compatibility of conditional Lie-Bäcklund symmetries and the governing equations. These solutions possess the extended forms of the separation of variables, including the extensions of the instantaneous source solutions of the porous medium equations.     

Nonlocal condensate model for QCD sum rules

We include effects of nonlocal quark condensates into QCD sum rules (QSR) via the Källén–Lehmann representation for a dressed fermion propagator, in which a negative spectral density function manifests their nonperturbative nature. Applying our formalism to the pion form factor as an example, QSR results are in good agreement with data for momentum transfer squared up to Q²≈10 GeV²$Q^2\approx 10{\text{GeV}}^2$. It is observed that the nonlocal quark condensate contribution descends like 1/Q²$1/Q^2$, different from the exponential decrease in Q²$Q^2$ obtained in the literature, and contrary to the linear rise in the local-condensate approximation.