EXISTENCE AND UNIQUENESS FOR THIRD ORDER NONLINEAR BOUNDARY VALUE PROBLEMS

In this paper, the existence and uniqueness of solutions for boundary valueproblem x′′′=f(t, x, x′, x″), x(0)=A, x′(0)=B, g(x′(1), x″(1))=0 are studied byusing Volterra type operator and upper and lower solutions. Our results improve someknown works.

THIRD ORDER NONLINEAR OPTICAL PROPERTIES OF PERNIGRANILINE SOLUTION IN CONCENTRATED SULFURIC ACID

The fully oxidized state of polyaniline (PANI), pernigraniline (PN), was synthesized by oxidation of the emeraldine base form of PAM dissolved in NMP solvent with m-chloroperoxybenzoic acid. The resulted PN was characterized by FTIR and UV-Visible spectra. The third-order nonlinear optical properties of PN in concentrated sulfuric acid (PN/H2SO4) were measured by a degenerated four wave mixing (DFWM) method. The results obtained suggest that the soliton-pair intermediate state associated with the degenerated state does not play a major role in the third order nonlinear optical properties of PANI.

Growth, Crystal Structure and Characterization of a New Single Crystal: Lithium Sodium Acid Phthalate (LiNaP), for Third Order Nonlinear Optical Applications

A new alkali metallo-organic single crystal of Lithium Sodium Acid Phthlate (LiNaP) complex has been synthesized from aqueous solution in the equimolar ratio 3:1:2. Transparent and bulk single crystals of dimension 9 × 4 ×2 mm3 have been grown from the conventional slow-cooling technique. The crystal structure of the compound has been solved from single crystal X-ray diffraction. The compound 2[C8H4O3]4-Li3+Na+ crystallizes in triclinic system with a space group of Pī having cell dimensions a = 7.5451(2) ? b = 9.8422(3) ? c = 25.2209(7) ? α = 80.299(2);β = 89.204(2);γ = 82.7770(10). FTIR measurement was carried out fo? LiNaP to study the vibrational structure of the compound. The various functional groups present in the molecule and the role of H-bonds in stabilizing the crystal structure of the compound have been explained. Optical absorption properties were studied for the grown crystal using UV-Vis-NIR spectrum. Thermal measurements were carried out for LiNaP to determine the thermal strength as well as to ascertain the hydrated nature of the crystal. Third order nonliner optical studies have also studied by Z-scan techniques. Nonlinear absorption and nonlinear refractive index were found out and the third order bulk susceptibility of compound was also estimated. The results of all studies have been discussed in detail.

Stability and Boundedness of Solutions of Certain Non-Autonomous Third Order Nonlinear Differential Equations

In this paper, by defining an appropriate Lyapunov functional, we obtain sufficient conditions for which all solutions of certain real non-autonomous third order nonlinear differential equations are asymptotically stable and bounded. The results obtained improve and extend some known results in the literature.

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. .

Hybrid silicon slotted photonic crystal waveguides:how does third order nonlinear performance scale with slow light?

We investigate in this paper the influence of slow light on the balance between the Kerr and two-photon absorption(TPA) processes in silicon slotted hybrid nonlinear waveguides. Three typical silicon photonic waveguide geometries are studied to estimate the influence of the light slow-down factor on the mode field overlap with the silicon region, as well as on the complex effective nonlinear susceptibility. It is found that slotted photonic crystal modes tend to focalize in their hollow core with increasing group index(n_G) values. Considering a hybrid integration of nonlinear polymers in such slotted waveguides, a relative decrease of the TPA process by more factor of 2 is predicted from n_G=10 to n_G=50. As a whole, this work shows that the relative influence of TPA decreases for slotted waveguides operating in the slow light regime, making them a suitable platform for third-order nonlinear optics.

Crystal Structure and Third order Nonlinear Optical Properties of cis-Configuration Biferrocene Trinuclear Complexes

The syntheses and characterization of novel biferrocene trinuclear complexes of Schiff base ligand, S methyl N (ferrocenyl 1 methyl methylidene)dithiocarbazate [hereafter abbreviated as HL (1)], are described. The X ray diffraction studies have established the structures of PdL 2 (3) and CuL 2 (4). The geometry of Pd(Ⅱ) in (3) is close to a square plane and in the novel cis configuration the two ferrocene moieties are in cis configuration, while that of Cu(Ⅱ) in (4) is close to a tetrahedral configuration. The electrochemical measurements suggested that the square planar configuration of Ni(Ⅱ) and Pd(Ⅱ) moieties can effectively transmit the redox effects of the ferrocene moieties, while the tetrahedral configuration of the Cu(Ⅱ) complex can not transmit the redox effects. The degenerate four wave mixing (DFWM) measurements showed that the complexes exhibit obviously nonlinear optical (NLO) response. The magnitude of their third order NLO susceptibility can be explained by their structure and electrochemical properties.

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.

Numerical Treatment of Nonlinear Third Order Boundary Value Problem

In this paper, the boundary value problems for nonlinear third order differential equations are treated. A generic approach based on nonpolynomial quintic spline is developed to solve such boundary value problem. We show that the approximate solutions of such problems obtained by the numerical algorithm developed using nonpolynomial quintic spline functions are better than those produced by other numerical methods. The algorithm is tested on a problem to demonstrate the practical usefulness of the approach.

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.

Ultrafast third-order optical nonlinearity of several sandwich-type phthalocyaninato and porphyrinato europium complexes

The third-order optical nonlinearity of two sandwich-type phthalocyaninato and porphyrinato europium com- plexes, including double- and triple-deckers （Eu[Pc（OC5H11）8]2, Eu2（Pc）（TPP）2, Pc=phthalocyanine, TPP=5, 10, 15, 20- tetraphenylporphyrinate）, was investigated by using the femtosecond time-resolved optical Kerr gate method at 830 nm wavelength. Their second-order hyperpolarizability is estimated to be 0.74× 10^-30esu and 3.0× 10^-3esu respectively. This exhibits an evident enhancement in comparison with 0.47×10^-30esu for one-decker Eu（Pc）（acac） （acac=acetylacetonate）, which is also measured under the same conditions. The enhancement is attributed to the introduction of lanthanide metal to the large π-conjugated system, intermacrocycle interaction and two-photon resonance etc.

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.

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-order nonlinear optical properties were studied by the backward degenerate four-wave mixing at 532nm. By using the relative measurement technique, the third-order nonlinear optical susceptibilities of PBTDMABQ and POTDMABQ are calculated as 5.62×10-9 and 1.22×10-8ESU, respectively. It is found that substituted alky groups have strong effects on the band gap and nonlinear optical properties of the two polymers. The relatively big third-order nonlinear optical susceptibilities and small band gap of POTDMABQ resulted mainly from the longer alkyl with strong electron-donating ability can enhance the delocation degree of conjugated π electronics.

Investigation on Third-Order Optical Nonlinearities of Two Organometallic Dmit2- Complexes Using Z-Scan Technique

The third-order nonlinear optical properties of two dmit organometallic complexes, [(CH3)4N] [Au(C3S5)2] (MeAu) and [(CH3)4N][Ni(C3S5)2] (Me Ni) in acetone solutions, were characterized us- ing a short pulse Z-scan technique at 1064 nm wavelength. Self-defocusing effects were found in both samples and stronger saturable absorp-tion was observed in MeNi solution comparing with that of MeAu. The origins were analyzed for the differences between the results. Two figures of merit W and T were also calculated to evalu-ate the suitability of two materials for all-optical integrated devices. The results of W=22.84 and T≈0 of MeAu make it an excellent candidate for the all-optical applications.

Ultrashort pulse breaking in optical fiber with third-order dispersion and quintic nonlinearity

The optical wave breaking （OWB） characteristics in terms of the pulse shape, spectrum, and frequency chirp, in the normal dispersion regime of an optical fiber with both the third-order dispersion （TOD） and quintic nonlinearity （QN） are numerically calculated. The results show that the TOD causes the asymmetry of the temporal- and spectral-domain, and the chirp characteristics. The OWB generally appears near the pulse center and at the trailing edge of the pulse, instead of at the two edges of the pulse symmetrically in the case of no TOD. With the increase of distance, the relation of OWB to the TOD near the pulse center increases quickly, leading to the generation of ultra-short pulse trains, while the OWB resulting from the case of no TOD at the trailing edge of the pulse disappears gradually. In addition, the positive （negative） QN enhances （weakens） the chirp amount and the fine structures, thereby inducing the OWB phenomena to appear earlier （later）. Thus, the TOD and the positive （negative） QN are beneficial （detrimental） to the OWB and the generation of ultra-short pulse trains.

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.

A New Newton-Type Method with Third-Order for Solving Systems of Nonlinear Equations

In this paper, a new two-step Newton-type method with third-order convergence for solving systems of nonlinear equations is proposed. We construct the new method based on the integral interpolation of Newton’s method. Its cubic convergence and error equation are proved theoretically, and demonstrated numerically. Its application to systems of nonlinear equations and boundary-value problems of nonlinear ODEs are shown as well in the numerical examples.

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.

Third-order elastic, piezoelectric, and dielectric constants

The definitions of the third-order elastic,piezoelectric,and dielectric constants and the properties of the associated tensors are discussed.Based on the energy conservation and coordinate transformation,the relations among the third-order constants are obtained.Furthermore,the relations among the third-order elastic,piezoelectric,and dielectric constants of the seven crystal systems and isotropic materials are listed in detail.These third-order constants relations play an important role in solving nonlinear problems of elastic and piezoelectric materials.It is further found that all third-order piezoelectric constants are 0 for 15 kinds of point groups,while all third-order dielectric constants are 0 for 16 kinds of point groups as well as isotropic material.The reason is that some of the point groups are centrally symmetric,and the other point groups are high symmetry.These results provide the foundation to measure these constants,to choose material,and to research nonlinear problems.Moreover,these results are helpful not only for the study of nonlinear elastic and piezoelectric problems,but also for the research on flexoelectric effects and size effects.