Galerkin-Bernstein Approximations for the System of Third-Order Nonlinear Boundary Value Problems

This paper is devoted to find the numerical solutions of one dimensional general nonlinear system of third-order boundary value problems (BVPs) for the pair of functions using Galerkin weighted residual method. We derive mathematical formulations in matrix form, in detail, by exploiting Bernstein polynomials as basis functions. A reasonable accuracy is found when the proposed method is used on few examples. At the end of the study, a comparison is made between the approximate and exact solutions, and also with the solutions of the existing methods. Our results converge monotonically to the exact solutions. In addition, we show that the derived formulations may be applicable by reducing higher order complicated BVP into a lower order system of BVPs, and the performance of the numerical solutions is satisfactory. .

THE ASYMPTOTIC BEHAVIOR AND OSCILLATION FOR A CLASS OF THIRD-ORDER NONLINEAR DELAY DYNAMIC EQUATIONS

In this paper,we consider a class of third-order nonlinear delay dynamic equations.First,we establish a Kiguradze-type lemma and some useful estimates.Second,we give a sufficient and necessary condition for the existence of eventually positive solutions having upper bounds and tending to zero.Third,we obtain new oscillation criteria by employing the Potzsche chain rule.Then,using the generalized Riccati transformation technique and averaging method,we establish the Philos-type oscillation criteria.Surprisingly,the integral value of the Philos-type oscillation criteria,which guarantees that all unbounded solutions oscillate,is greater than θ_(4)(t_(1),T).The results of Theorem 3.5 and Remark 3.6 are novel.Finally,we offer four examples to illustrate our results.

Synthesis of Poly (3-acetylpyrrolyl methine) with Azobenzene Side Groups and Study on Its Third-order Nonlinear Optical Property

A novel soluble π-conjugated polymer, poly [（3-acetylpyrrole-2, 5-diyl） p-（N, N-dimethylamino） azobenzylidene] （PAPDMAABE）, was synthesized by condensation of 3-acetylpyrrole with 4-aldehyde-4＇-dimethylaminoazobenzene （ADMAA）. The chemical structure of PAPDMAABE was characterized by Fourier transform infrared spectroscopy （FTIR）, ^1H-NMR, and UV-Vis-NIR spectra. Transmission electron microscope （TEM） analysis for PAPDMAABE indicates that part of PAPDMAABE is in crystal state, due to the short-range order of the polymer. Thermogravimetric analysis （TGA） curve shows that the polymer has good thermal stability and its decomposition temperature is 248℃. The optical band gap of PAPDMAABE obtained from the optical absorption spectrum is about 1.73 eV. The resonant third-order nonlinear optical property of PAPDMAABE at 532 nm was studied using degenerate four-wave mixing （DFWM） technique. The resonant third-order nonlinear optical susceptibility of the polymer is about 7.48×10^-8 esu.

UV-Vis Spectrum and the Third-order Nonlinear Optical Properties of the Chiral Camphorderived β-diketonate Platinum Complexes

UV-Vis spectrum and the third-order nonlinear optical properties of the chiral camphor-derived β-diketonate have been studied at the B3LYP/6-31G＊ level. The results showed that the introduction of electron-drawing group -CF3 and -C3F7 on β-diketonate made the strongest absorption peak red-shift and the lowest energy absorption blue-shied. Introduction of -OC2H5 on the benzene or pyridine ring made the lowest energy absorption blue-shift. When the -C2H3 was introduced on the benzene or pyridine ring, the lowest energy absorption was red-shifted. Introduction of electron-donating group on β-diketonate can enlarge their nonlinear optical properties. On the contrary, the introduction of electron-drawing group dropped it down.

The Third-Order Nonlinear Optical Properties in Cobalt-Doped ZnO Films

We quantitatively investigate the third-order optical nonlinear response of Co-doped ZnO thin films prepared by magnetron sputtering using the Z-scan method. The two-photon absorption and optical Kerr effect are revealed to contribute to the third-order nonlinear response of the Co-doped ZnO films. The nonlinear absorption coefficient β is determined to be approximately 8.8 × 10-5 cm/W and the third-order nonlinear susceptibility X（3） is 2.93 × 10-6 esu. The defect-associated energy levels within the band gap are suggested to be responsible for the enhanced nonlinear response observed in Co-doped ZnO films.

Electronic Absorption Spectra and Third-Order Nonlinear Optical Property of Dinaphtho[2,3-b:2’,3’-d]Thiophene-5,7,12,13- Tetraone (DNTTRA) and Its Phenyldiazenyl Derivatives: DFT Calculations

Third-order nonlinear optical (NLO) materials have broad application prospects in high-density data storage, optical computer, modern laser technology, and other high-tech industries. The structures and frequencies of Dinaphtho[2,3-b:2’,3’-d]thiophene-5,7,12,13-tetraone (DNTTRA) and its 36 derivatives containing azobenzene were calculated by using density functional theory B3LYP and M06-2X methods at 6-311++g(d, p) level, respectively. Besides, the atomic charges of natural bond orbitals (NBO) were analyzed. The frontier orbitals and electron absorption spectra of A-G5 molecule were calculated by TD-DFT (TD-B3LYP/6-311++g(d, p) and TD-M06-2X/6-311++g(d, p)). The NLO properties were calculated by effective finite field FF method and self-compiled program. The results show that 36 molecules of these six series are D-π-A-π-D structures. The third-order NLO coefficients γ (second-order hyperpolarizability) of the D series molecules are the largest among the six series, reaching 107 atomic units (10-33 esu) of order of magnitude, showing good third-order NLO properties. Last, the third-order NLO properties of the azobenzene ring can be improved by introducing strong electron donor groups (e.g. -N(CH3)2 or -NHCH3) in the azobenzene ring, so that the third-order NLO materials with good performance can be obtained.

Synthesis of Poly[(3-octanoylpyrrole-2,5-diyl)-p-nitrobenzylidene] and Its Third-order Nonlinear Optical Property

A novel soluble π-conjugated polymer, poly[(3-octanoylpyrrole-2,5-diyl)-p-nitrobenzylidene] (POPNBE), was synthesized by a five-step reaction. As a result of introducing a long acyl to pyrrole at 3-position, the polymer can be easily solved in polar solvents and be processed into films. The chemical structures of the intermediates and POPNBE were characterized by FTIR, 1H NMR, and UV-Vis-NIR spectrometries. The absorption band in the range of 400-600 nm in the UV-Vis-NIR spectrum is due to the π-π * band gap transition. The optical band gap of POPNBE obtained from the optical absorption is about 1.72 eV. The resonant third-order nonlinear optical property of POPNBE at 532 nm was studied by using the degenerate four-wave mixing(DFWM) technique. The resonant third-order nonlinear optical susceptibility of POPNBE is about 3.42×10 -8 esu.

Crystal Structure and Third-order Nonlinear Optical Property Studies on Copper(Ⅱ) Complex Containing NS Donor Ligand

A novel Schiff base ligand (HL) derived from S-methyldithiocabazate and pmethoxylbenzaldehyde was prepared and characterized. The Schiff base ligand acts as a single negatively charged bidentate ligand fondng D-M-D type comPlex (D=donor, M=metal). Single crystal X-ray diffraction analysis of the copper(Ⅱ) complex established that the geometry around Cu (Ⅱ) is square-planar with two equivalent M-N and M-S bonds. The two phenyl rings and the coordinated plane are almost in one plane fotheng an electronic delocalization system. Their thirdorder response was also studied.

Third-Order Nonlinear Optical Response near the Plasmon Resonance Band of Cu2-xSe Nanocrystals

The third-order nonlinear optical properties of water-soluble Cu Se nanocrystals are studied in the near infrared range of 700-980 nm using a femtosecond pulsed laser by the Z-scan technique. It is observed that the nonlinear optical response of Cu Se nanocrystals is sensitively dependent on the excitation wavelength and exhibits the enhanced nonlinearity compared with other selenides such as ZnSe and CdSe. The W-shaped Z-scan trace, a mixture of the reversed saturated absorption and saturated absorption, is observed near the plasmon resonance band of Cu Se nanocrystals, which is attributed to the state-filling of free carriers generated by copper vacancies (self-doping effect) of Cu Se nanocrystals as well as the hot carrier thermal effect upon intense femtosecond laser excitation. The large nonlinear optical response and tunable plasmonic band make Cu Se nanocrystals promising materials for applications in ultra-fast all-optical switching devices as well as nonlinear nanosensors.

Big Third-order Nonlinear Optical Properties of Two Poly[(3-alkylthiophene-2, 5-diyl)-(benzylidenequinomethane-2, 5-diyl)] Derivatives

Two novel poly[(3-alkylthiophene-2,5-diyl)-(benzylidenequinomethane-2,5-diyl)s] derivatives, poly[ (3-butylthiophene-2,5-diyl)-( p-N,N-dimethylamino) benzylidenequinomethane-2,5-diyl) ] ( PBTDMABQ) and poly [( 3-octylthiophene2,5-diyl) -(p-N, N-dimethylamino ) benzylidenequinomethane-2, 5-diyl)] (POTDMABQ), were synthesized.The band gaps of the two polymers are calculated as 1. 75 eV for PBTDMABQ and 1. 69 eV for POTDMABQ,respectively. The homogenous films of the two polymers were prepared and their third-ordernonlinear optical properties were studied by the backward degenerate four-wave mixing at 532 nm. Byusing the relative measurement technique, the third-order nonlinear optical susceptibilities ofPBTDMABQ and POTDMABQ are calculated as 5. 62 X 10^(-9) and 1. 22 X 10^(-8) ESU, respectively. It isfound that substituted alky groups have strong effects on the band gap and nonlinear opticalproperties of the two polymers. The relatively big third-order nonlinear optical susceptibilitiesand small band gap of POTDMABQ resulted mainly from the longer alkyl with strong electron-donatingability can enhance the delocation degree of conjugated π electronics.

Oscillation Criteria for Third-order NonlinearNeutral Dynamic Equations on Time Scales

Many practical problems, such as those from electronic engineering, mechanicalengineering, ecological engineering, aerospace engineering and so on, need to bedescribed by dynamic equations on time scales, so it is important in theory andpractical significance to study these equations. In this paper, the oscillation andasymptotic behavior of third-order nonlinear neutral delay dynamic equations ontime scales are studied by using generalized Riccati transformation technique, integralaveraging methods and comparison theorems. The main purpose of this paperis to establish some new oscillation criteria for such dynamic equations. The newKamenev criteria and Philos criteria are given, and an example is considered toillustrate our main results.

Third-order nonlinear differential operators preserving invariant subspaces of maximal dimension

In this paper, third-order nonlinear differential operators are studied. It is shown that they are quadratic forms when they preserve invariant subspaces of maximal dimension. A complete description of third-order quadratic operators with constant coefficients is obtained. One example is given to derive special solutions for evolution equations with third-order quadratic operators.

Crystal Structure and the Third-order Nonlinear Optical Property of a Cadmium(Ⅱ) Complex Based on a D-π-A Structural Organic Ligand

A triphenylamine-containing Schiff base ligand(L), namely, N-(4-(1 H-imidazol-1-yl)benzylidene)-N,N-diphenylbenzene-1,4-diamine, was synthesized and characterized. By coordination of the ligand with CdI2, a complex CdI2 L2 was obtained. The structure of the complex was analyzed through single-crystal X-ray diffraction. It crystallizes in monoclinic, space group P21/c with a = 25.277(5), b = 11.176(5), c = 17.912(5) ?, β = 106.056(5)°, V = 4863(3) ?3, Z = 4, Dc =1.633 Mg/m3, F(000) = 2360, Μr = 1195.19, μ = 1.766 mm-1, the final R = 0.0323 and wR = 0.0758 for 33414 observed reflections with I > 2?(I). The linear absorption spectra of the complex were experimentally and theoretically studied. And the third-order nonlinear optical(NLO) property of the complex was also measured through Z-scan technique.

Third-order nonlinear optical properties of graphene composites:A review

Graphene has excellent thirdorder nonlinear optical(NLO)properties due to its unique electronic band structure and wideband gap tunability.This paper focuses on the research progress of graphene and its composite materials in nonlinear optics in recent years.In this review,recent results on graphene(or graphene oxide)-metal nanoparticles(G-MNPs),graphene-metal-oxide nanoparticles(G-MONPs),graphene-metal sulfide nanoparticles(G-MSNPs),and graphene-organic molecular composites(G-OM)have been discussed.In addition,the enhancement mechanism of nonlinear absorption(NLA)and optical limiting(OL)have also been covered.

Investigation of third-order nonlinear optical properties of two azo-nickel chelate compounds

This paper investigates the third-order nonlinear optical properties of two azo-nickel chelate compounds by the optical Kerr gate method at 830 nm wavelength with pulse duration of 120 fs. Both of the two compounds exhibited large third-order optical nonlinearity. The second-order hyperpolarizability,γ, of Compound 1 is of 1.0 × 10^-31 esu. Due to the charge transfer, the γ of Compound 2 with electron donor and acceptor group is 4.9 × 10^-31 esu, which is a four-time enhancement in comparison with Compound i. The absorption spectra show that the electron push-pull effect, which induces intramolecular charge transfer, leads to the increased optical nonlinearity.

Crystal Structure,Electrochemical Properties and Third-order Nonlinear Optical Response of 4-Methyloxy benzaldehyde Thiosemicarbazone and Its Copper(Ⅱ)Complex

The crystal structUre of the title ligand and its copper (Ⅱ) complex was determined bysingle crystal X-ray diffraction. The electrochendcal properties and the third-order nonlinearoptical response of the ligand and its copper (Ⅱ) complex were also studied.

Large Third-Order Nonlinear Optical Susceptibility of Rh-Doped BaTiO_(3) Thin Films Prepared by Pulsed Laser Deposition

Rh-doped barium titanate (BaTiO3) epitaxial thin films have been fabricated on SrTiO_(3) (100) substrates by pulsed laser deposition. The nonlinear optical properties of the films were determined using the z-scan method at a wavelength of 532nm with a laser duration of 10hs. The real and imaginary parts of the third-order nonlinear susceptibility χ^((3)) were 5.71 × 10^(-7) esu and 9.59×10^(-8) esu, respectively. The real part value of x(3) of the Rh:BaTiO3 films is about one order larger than that of Ce-doped BaTiO3 thin films. The results show that Rh:BaTiO3 thin films have great potential applications for nonlinear optical devices.

Study on the Third-order Nonlinear Optical Properties of bis(tetraethylammonium) bis(1,3-dithiole-2-thione-4,5-dithiolato)cadium

A dmit2- salt: bis(tetraethylammonium)bis(1,3-dithiole-2-thione-4,5-dithiolato)cadium (CADMIT) was synthesized. The Optical Kerr Effect (OKE) signal of its acetonitrile solution was measured by femtosecond optical Kerr gate technique. Using CS2 OKE signal as reference signal measured under identical conditions, the third-order optical nonlinear susceptibility, c(3), of the sample solution was obtained to be about 2.98 x 10-14 esu at the concentration of 1.57 x 10-3 M. The second-order hyperpolarizability of its molecular was estimated to be as large as 1.23 x 10-32 esu. Its response time was about 195 fs, which is believed to be the contribution from the delocalized electrons.

Bi-Hamiltonian Structure of a Third-Order Nonlinear Evolution Equation on Plane Curve Motions

In the present paper, we identify the integrability of the third-order nonlinear evolution equation ut = （1/2）（（uxz ＋ u）^-2）z in a Hamiltonian viewpoint. We prove that the recursion operator obtained by S.Yu. Sakovich is hereditary, and then deduce a bi-Hamiltonian structure of the equation by using some decomposition of the hereditary operator. A hierarchy associated to the equation is also shown.

Result on the Convergence Behavior of Solutions of Certain System of Third-Order Nonlinear Differential Equations

Convergence behaviors of solutions arising from certain system of third-order nonlinear differential equations are studied. Such convergence of solutions corresponding to extreme stability of solutions when relates a pair of solutions of the system considered. Using suitable Lyapunov functionals, we prove that the solutions of the nonlinear differential equation are convergent. Result obtained generalizes and improves some known results in the literature. Example is included to illustrate the result.