Modified radiation laws of a rectangular Kerr nonlinear blackbody

A Kerr nonlinear blackbody (KNB) is a new kind of blackbody in which bare photons with opposite wave vectors and helicities are bound into pairs and unpaired photons are transformed into nonpolaritons. In the present paper, we focus our investigation on the modified radiation laws, such as Planck and Stefan-Boltzmann radiation laws, of a rectangular KNB. Besides, the case of a KNB with no symmetry axes is also discussed. Finally, we make a numerical calculation of modified radiation laws of a cubic KNB under appropriate conditions, and we consider this work may lay the foundation for the experimental verification of the model of a KNB.     

一类高阶非线性波方程的李群分析、最优系统、精确解和守恒律

本文运用李群分析的方法研究了一类高阶非线性波方程,得到了五阶非线性波方程的对称以及方程的最优系统,进而运用幂级数的方法,求得了方程的精确幂级数解.最后,给出了五阶非线性波方程的一些守恒律.     

Lie-symmetry vector fields for linear and nonlinear wave equations

We derive the Lie symmetry vector fields for the linear wave equation u=0 and nonlinear wave equation u=u ³. The conformal vector fields for the underlying metric tensor fieldg are also given. We construct the conservation laws and derive similarity solutions. Furthermore, we perform a Painlevé test for the nonlinear wave equation and discuss whether Lie-Bäcklund vector fields exist.     

Investigation of symmetries of second-order nonlinear susceptibility tensor of GaSe crystals in THz domain

Following our measurements and analysis made on several GaSe crystals, we demonstrated that terahertz (THz) generation from ultrafast laser pulses can be developed into a sensitive technique for investigating symmetries of second-order nonlinear susceptibility tensor of a nonlinear crystal. Indeed, for GaSe crystals, both Kleinman's symmetry condition and spatial symmetry were violated due to the contribution of ionic displacement to nonlinear polarization and deviation of GaSe lattice from hexagonal symmetry. When the pump photon energy was increased from that below the bandgap of GaSe to that above it, the mechanism for the THz generation was switched from optical rectification to photocurrent surge.     

Polarization nonlinear optics of quadratically nonlinear azopolymers

The polarization properties of second harmonic and sum-frequency signals generated by femtosecond laser pulses in films of polymers containing covalent groups of an azobenzothiazole chromophore polarized by an external electric field are investigated. It is shown that the methods of polarization nonlinear optics make it possible to determine the structure of oriented molecular dipoles and reveal important properties of the motion of collectivized π electrons in organic molecules with strong optical nonlinearities. The polarization measurements show that the tensor of quadratic nonlinear optical susceptibility of chromophore fragments oriented by an external field in macromolecules of the noted azopolymers has a degenerate form. This is indicative of a predominantly one-dimensional character of motion of collectivized π electrons along an extended group of atoms in such molecules.     

Infinite sets of conservation laws for linear and nonlinear field equations

The relation between an infinite set of conservation laws of a linear field equation and the enveloping algebra of the space-time symmetry group is established. It is shown that each symmetric element of the enveloping algebra of the space-time symmetry group of a linear field equation generates a one-parameter group of symmetries of the field equation. The cases of the Maxwell and Dirac equations are studied in detail. Then it is shown that (at least in the sense of a power series in the coupling constant) the conservation laws of the linear case can be deformed to conservation laws of a nonlinear field equation which is obtained from the linear one by adding a nonlinear term invariant under the group of space-time symmetries. As an example, our method is applied to the Korteweg-de Vries equation and to the massless Thirring model.     

Reconstruction of the coordinate dependences of quadratic susceptibility tensor \hat \chi ^{(2)} (z, ω1 ± ω2; ω1, ±ω2) components for the one-dimensionally inhomogeneous absorbing medium

A method for the unambiguous reconstruction of the spatial profiles of all components (except for ?? _zzz ) of the quadratic susceptibility complex tensor {ie165-2} (z, ??₁ + ??₂; ??₁, ??₂), which is responsible for the sumfrequency generation in a one-dimensionally inhomogeneous plate is proposed and proven for the first time. Such reconstruction is possible if the symmetry of the medium provides the diagonal character of the linear permittivity tensor {ie165-3} (z, ??). The procedure involves the measurement of the complex amplitude of the new wave with the frequency ??₁ + ??₂ that is reflected from the plate for a certain interval of the angles of incidence of the wave with the frequency ??₂. The reflected wave results from the nonlinear interaction of the wave with frequency ??₂ and the wave with frequency ??₁ that exhibits the normal incidence. A similar approach can be used to determine the profiles of the components of the quadratic susceptibility tensor {ie165-4}(z, ??₁ ? ??₂; ??₁, ? ??₂), which is responsible for the difference-frequency generation.     

Local conserved currents for nonlinear σ-models on symmetric spaces

A general method is presented to construct an infinite series of conserved local charges for a large class of two-dimensional nonlinear σ-models on symmetric spaces. The conservation laws are derived from a couple of first order Ricatti differential equations using the dual symmetry of σ-models on symmetric spaces. The method is exemplified for the case of σ-models on Grassmann manifolds.     

On the non-integrability of the spherically-symmetric nonlinear Schrödinger equation in the Langmuir collapse problem

In the present paper we give some analytic considerations of the spherically-symmetric nonlinear Schrödinger equation arising in the Langmuir collapse problem. We will make a systematic exploration of the various group symmetries of thie equation and show that the latter possesses only a three-parameter symmetry group. We will then give a variational formulation of this equation and use the three-parameter symmetry group to show that the equation in question possesses apparently only two polynomial conservation laws. Finally, we will make a study of the singularity structure of the present equation and show that it does not seem to possess the Painlevé property. The conclusion is that the spherically-symmetric nonlinear Schrödinger equation in question is apparently not integrable.     

Determination of the nonlinear optical susceptibility χ(2) of surface layers by sum and difference frequency generation in reflection and transmission

The theoretical investigation of sum and difference frequency generation in thin surface layers with rotational symmetry leads to formulas which connect the generated light intensities to the surface second order nonlinear susceptibility tensor. A maximum of seven tensor components can be determined in the case of lowest symmetry. Measurements in transmission should be especially useful since they allow easy variation of both polarization and angle of incidence. On the other hand, large signal enhancements are expected for total internal reflection geometries. A consistent set of ⁽²⁾ tensor components for a thin layer of rhodamine-6G adsorbed on fused silica is found based on data from reflection and transmission measurements.     

广义(3+1)维Zakharov-Kuznetsov方程的对称约化、精确解和守恒律

运用李群分析,得到了广义(3+1)维Zakharov-Kuznetsov(ZK)方程的对称及约化方程,结合齐次平衡原理,试探函数法和指数函数法得到了该方程的群不变解和新精确解,包括冲击波解、孤立波解等.进一步给出了广义(3+1)维ZK方程的伴随方程和守恒律.     

On the theory of nonlinear response in a nonideal plasma

A theory of nonlinear response is developed for studying nonlinear phenomena and nonlinear transport processes in nonideal Coulomb systems. Temporal plasma echo and transformation of waves in a nonideal Coulomb system are studied on the basis of the theory of nonlinear response to mechanical perturbations. General constraints imposed on nonlinear response functions are considered, and the model for determining quadratic response functions is formulated. The conditions for the emergence of temporal plasma echo and wave transformation are determined. It is shown that these nonlinear effects in a nonideal plasma can be initiated by ultrashort field pulses. A theory of transport is developed for determining the Burnett transport properties of a nonideal multielement plasma. A procedure is proposed for comparing the phenomenological conservation equations for a charged continuous medium and equations of motion for the operators of corresponding dynamic variables. The Mori algorithm is used for deriving the equations of motion for operators of dynamic variables in the form of generalized Langevin equations. The linearized Burnett approximation is considered in detail. The properties of the matrices of coefficients of higher-order derivatives in the system of conservation equations in the linearized Burnett approximation, which are important for hydrodynamic applications, are discussed. Various versions of the theory of nonlinear response are compared.     

Bäcklund transformations and new solutions of nonlinear wave equations in four-dimensional space-time

Bäcklund transformations for several nonlinear field equations in four-dimensional space-time relating two solutions of the same equation (symmetry), or two different equations (dynamical), are given. These transformations can be used to generate new families of solutions and infinitely many conservation laws for nonlinear equations.Bäcklund transformations and solutions of nonlinear equations have been studied extensively in one-space and one-time dimension. We give here a fairly general method for a class of equations in four-dimensional space-time which paves the way for many further generalizations.Supported in part by the U.S. National Academy of Sciences Foundation Grant No. INT 73-20002 A01 (formerly GF-41958).     

Conservation Laws of a Class of Combined Equations

In this paper, we investigate conservation laws of a class of partial differential equations, which combines the nonlinear telegraph equations and the nonlinear diffusion-convection equations. Moreover, some special conservation laws of the combined equations are obtained by means of symmetry classifications of wave equations uxx = H （x）utt.     

Quantum crossing symmetry as heart of ghost imaging

In this paper it is proved that the keys to understanding the ghost imaging are the crossing symmetric (CS) photon reactions in the nonlinear media. So CS introduced a real optical path between the object and his “ghost” image, making possible to apply the geometric optics for a rigorous proof of the essential laws of the “ghost” imaging phenomena. Hence, the laws of the plane quantum mirror (QM) and that of spherical quantum mirror, observed in the ghost-imaging experiments, are shown that can be obtained as natural consequences of the energy-momentum conservation laws. So, it is proved that the ghost imaging laws depend only on the energy-momentum conservation and not on the photons entanglement. Using DFG-typical features we obtained explicit predictions of the intensities of the idler photons in terms of the intensities of the interacting photon-(p and s)-beams in the nonlinear crystal. Two fundamental experiments for a decisive test of the [SPDC-DFG]-quantum mirrors are suggested.     

Noether-Type Symmetries and Associated Conservation Laws of Some Systems of Nonlinear PDEs

The algorithm for constructing conservation laws of Euler Lagvange type equations via Noether-type symmetry operators associated with partial Lagrangian has been presented. As applications, many new conservation laws of some important systems of nonlinear partial differential equations have been obtained.     

Generation of a Quadphoton via Seventh‐Order Nonlinear Susceptibility

With a wide range of potential applications, the generation of nonclassical multiphoton and number states has attracted renewed interest recently. Here, a correlated quadphoton with seventh‐order nonlinear susceptibility is reported in which all four photons are simultaneously created via an eight‐wave mixing (EWM) process in an atomic medium. The efficiency of the pure EWM process is low, which results from the small order of magnitude of the seventh‐order nonlinear susceptibility (the quadphoton correlation with four periods corresponding to 8 channels and 12 resonance positions). But the EWM process has a strong dressing effect. It not only increases the nonlinear susceptibility, but also produces 15 or 16 resonance positions in the dressing state picture. In addition, with the energy conservation of these positions, 12 coherent channels can be generated in this system. According to the interference between multichannels of the quadphoton, the quadphoton coincidence count appears as a damped Rabi oscillation, which has eight or ten oscillation periods. If only the nonlinear optical response is considered, the coincidence counts of the quadphoton behave as a damped Rabi oscillation with periods that can be controlled by the dressing field. These outcomes may contribute to a new promising method for quantum communication.     

Second-order nonlinear optical susceptibility in chalcone-group crystalline molecular compounds

A study is reported of the second-order nonlinear optical susceptibility in polycrystalline molecular materials of the chalcone group made by second-harmonic generation at a wavelength of 1.06 μm. Some of the compounds studied exhibit a high nonlinear activity and are transparent in the blue region of the spectrum. The measured macroscopic nonlinear polarizability χ ² was compared with the calculated values of β, the second-order molecular polarizability tensor. The high nonlinear optical polarizability observed in some chalcone compounds is shown to be related to crystal symmetry type. Fiz. Tverd. Tela (St. Petersburg) 41, 2084–2087 (November 1999)     

A nonlinear discrete integrable coupling system and its infinite conservation laws

We construct a nonlinear integrable coupling of discrete soliton hierarchy,and establish the infinite conservation laws(CLs) for the nonlinear integrable coupling of the lattice hierarchy.As an explicit application of the method proposed in the paper,the infinite conservation laws of the nonlinear integrable coupling of the Volterra lattice hierarchy are presented.     

The symmetry of the fundamental equations of the dynamics of a Hamiltonian system

This paper constructs covariant defining equations for infinitesimal operators of the Lie symmetry groups of the Hamilton-Jacobi equations, and Hamilton's and Lagrange's systems whose Hamiltonians are a quadratic function of the generalized momenta; a study is made of the relation between the groups, and in particular the relation is considered between the differential laws of conservation and the symmetry of the systems.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 12–16, February, 1977.