Riemann–Hilbert problem for the defocusing Lakshmanan–Porsezian–Daniel equation with fully asymmetric nonzero boundary conditions

The Riemann–Hilbert approach is demonstrated to investigate the defocusing Lakshmanan–Porsezian–Daniel equation under fully asymmetric nonzero boundary conditions.In contrast to the symmetry case,this paper focuses on the branch points related to the scattering problem rather than using the Riemann surfaces.For the direct problem,we analyze the Jost solution of lax pairs and some properties of scattering matrix,including two kinds of symmetries.The inverse problem at branch points can be presented,corresponding to the associated Riemann–Hilbert.Moreover,we investigate the time evolution problem and estimate the value of solving the solutions by Jost function.For the inverse problem,we construct it as a Riemann–Hilbert problem and formulate the reconstruction formula for the defocusing Lakshmanan–Porsezian–Daniel equation.The solutions of the Riemann–Hilbert problem can be constructed by estimating the solutions.Finally,we work out the solutions under fully asymmetric nonzero boundary conditions precisely via utilizing the Sokhotski–Plemelj formula and the square of the negative column transformation with the assistance of Riemann surfaces.These results are valuable for understanding physical phenomena and developing further applications of optical problems.

Effect of Thermal Defocusing on Backward Stimulated Raman Scattering in CH4

The conversion efficiency of stimulated Raman scattering （SRS） in CH4 is studied by using a single longitudinal mode second-harmonic Nd：YAG laser （532 nm, linewidth 0.003 cm^-1, pulse-width （FWHM） 6.5 ns）. Due to the heat release from vibrationally excited particles, SRS processes often suffer from the thermal defocusing effect （TDE）. In view of 6.5 ns laser pulse width is much shorter than the vibrational relaxation time of CH4 molecules, TDE can only affect the SRS processes afterwards. In the cases of low laser repetition, TDE will be not serious, because it will be removed by the thermal diffusion in Raman medium before the next pulse arrives. At the laser repetition rate 2 Hz, CH4 pressure 1.1 MPa and pump laser energy 95 mJ, the quantum conversion efficiency of backward first-Stokes （BS1） has attained 73%. This represents the highest first-stokes conversion efficiency in CH4. Furthermore, due to the relaxation oscillation, the BS1 pulses are narrowed to about 1.2 ns. As a result, the BS1 peak power turns out to be 2.7 times that of the pump. Its beam quality is also much better and is only slightly affected by TDE. This reason is that BS1 represents a wave-front-reversed replica of the pump beam, which can compensate the thermal distortions in Raman amplify process. Under the same conditions, but pump laser repetition rate as 10 Hz, the conversion efficiency of BS1 goes down to 36% due to TDE. From this study, we expect that a well-behaved 630 nm Raman laser may be designed by using a closed CH4/He circulating-cooling system, which may have some important applications.

Defocusing role in femtosecond filamentation: Higher-order Kerr effect or plasma effect?

The femtosecond filamentation in the classical and high-order Kerr （HOK） models is numerically investigated by adopting multi-photon ionization （MPI） cross section with different values. It is found that in the case that the MPI cross section is relatively small, there exists a big difference between the electron density as well as clamped intensity calculated in the classical model and those calculated in the HOK one, while in the case that the MPI cross section is relatively large, the electron density and clamped intensity calculated in the two models are nearly in agreement with each other, and under this circumstance, even if the higher-order nonlinear terms do exist, the free-charge generation and the associated defocusing in a filament are enough to mask their effects. The different behaviors of the maximum intensity and on-axis electron density at the collapse position with the pulse duration provides an approach to determine which effect plays the dominant defocusing role. These results demonstrate that it is ionization that results in the difference between the two models.

ALMOST CONSERVATION LAWS AND GLOBAL ROUGH SOLUTIONS OF THE DEFOCUSING NONLINEAR WAVE EQUATION ON R^2

In this article, we investigate the initial value problem（IVP） associated with the defocusing nonlinear wave equation on R^2 as follows：{δttu-△u=-u^3 u（0,x）=u0（x）,δtu＊（0,x）=u1（x,）where the initial data （uo,ul）∈H^s-1（R^2）It is shown that the IVP is global well-posedness in H^s（R^2）×H^s-1×H^s-1（R^2）for any 1 〉 s 〉2/5.The proof relies upon the almost conserved quantity in using multilinear correction term. The main difficulty is to control the growth of the variation of the almost conserved quantity. Finally, we utilize linear-nonlinear decomposition benefited from the ideas of Roy [1].

An Integrable Decomposition of Defocusing Nonlinear Schr(o|¨)dinger Equation

The method of nonlinearization of spectral problems is developed to the defocusing nonlinear Schr(o|¨)dingerequation.As an application,an integrable decomposition of the defocusing nonlinear Schr(o|¨)dinger equation is presented.

Localized Optical Waves in Defocusing Regime of Negative-Index Materials

We study optical localized waves on a plane-wave background in negative-index materials governed by the defocusing nonlinear Schr6dinger equation with self-steepening effect. Important characteristics of localized waves, such as the excitations, transitions, propagation stability, and mechanism, are revealed in detail. An intrigu- ing sequential transition that involves the rogue wave, antidark-dark soliton pair, antidark soliton and antidark soliton pair can be triggered as the self-steepening effect attenuates. The corresponding phase diagram is estab- lished in the defocusing regime of negative-index materials. The propagation stability of the localized waves is confirmed numerically. In particular, our results illuminate the transition mechanism by establishing the exact correspondence between the transition and the modulation instability analysis.

Soliton resolution and asymptotic stability of N-solutions for the defocusing Kundu–Eckhaus equation with nonzero boundary conditions

In this paper,we address interesting soliton resolution,asymptotic stability of N-soliton solutions and the Painleve asymptotics for the Kundu-Eckhaus(KE)equation with nonzero boundary conditions iq_(t)+q_(xx)-2(l|q|^(2)-1)q+4β^(2)(lql^(4)-1)q+4iβ(lql^(2))_(x)q=0,q(x,0)=q_(0)(x)-±1,x→±∞.The key to proving these results is to establish the formulation of a Riemann-Hilbert(RH)problem associated with the above Cauchy problem and find its connection with the RH problem of the defocusing NLS equation.With the■-steepest descent method and the results of the defocusing NLS equation,we find complete leading order approximation formulas for the defocusing KE equation on the whole(x,t)half-plane including soliton resolution and asymptotic stability of N-soliton solutions in a solitonic region,Zakharov-Shabat asymptotics in a solitonless region and the Painlevéasymptotics in two transition regions.

Depressing defocusing effect in microscopy by bi-orthogonal wavelet transform

For different dimensions, heights and irregular shapes, small and large particles cannot be viewed simultaneously as sharp images under the microscope due to optical defocusing effects. Bi-orthogonal wavelet transform is used in this paper to depress such defocusing effect by merging the wavelet coefficients of the image series. The algorithm is presented in detail, using SWE（13,7） and CDF（2,2）, and a few particle images are provided to validate the proposed method.

Impact of multifocal gas-permeable lens designs on short-term choroidal response, axial length, and retinal defocus profile

AIM:To investigate the impact of multifocal gas permeable contact lens(MFGPCL)in various add power and distance/near area allocation on short-term changes of choroidal thickness(ChT),axial length(AL),and retinal defocus profile in young adults.METHODS:Seventeen young adults(2 males and 15 females;age 23.17±4.48y)were randomly assigned to wear two designs binocularly with a one-week washout period in between.Total of four MFGPCL designs were assessed.All designs were distance-center that varied in two add power(+1.50 and 3.00 D)and/or two distance zone(DZ)diameters(1.50 and 3.00 mm;design A:DZ 1.5/add 3.0,B:DZ 1.5/add 1.5,C:DZ 3.0/add 3.0,D:DZ 3.0/add 1.5).ChT,AL,and peripheral refraction data were collected on each subject at baseline,on days 1 and 7 of MFGPCL daily wear.ChT was assessed in four quadrants using a spectraldomain optical coherence tomography.RESULTS:AL was shortened by-26±44μm with lens C,-18±27μm with lens D,-13±29μm with lens A,and-8±30μm with lens B(all P<0.05).A significant overall increase in ChT was observed with all 4 designs(lens A:+6±6μm,B:+3±7μm,C:+8±7μm,and D:+8±7μm).Temporal and superior choroid exhibited more choroidal thickening associated with MFGPCL.All designs induced significant relative peripheral myopia(RPM)beyond the central 20o across the horizontal meridian in both nasal and temporal fields(P<0.05).CONCLUSION:MFGPCLs show a significant influence on ChT and AL,which are associated with significant increase in RPM after short-term wear.The reliability and feasibility of quantifying short-term changes in ChT support its use as a promising marker for the long-term efficacy of myopia-controlling treatments.

Dark Solitons for the Defocusing Cubic Nonlinear Schr?dinger Equation with the Spatially Periodic Potential and Nonlinearity

We study the existence of dark solitons of the defocusing cubic nonlinear Schr¨odinger(NLS) eqaution with the spatially-periodic potential and nonlinearity. Firstly, we propose six families of upper and lower solutions of the dynamical systems arising from the stationary defocusing NLS equation. Secondly, by regarding a dark soliton as a heteroclinic orbit of the Poincar′e map, we present some constraint conditions for the periodic potential and nonlinearity to show the existence of stationary dark solitons of the defocusing NLS equation for six different cases in terms of the theory of strict lower and upper solutions and the dynamics of planar homeomorphisms. Finally, we give the explicit dark solitons of the defocusing NLS equation with the chosen periodic potential and nonlinearity.

Self-Focusing/Defocusing of Chirped Gaussian Laser Beam in Collisional Plasma with Linear Absorption

This paper presents an investigation on the self-focusing/defocusing of chirped Gaussian laser beam in collisional plasma with linear absorption. We have derived the differential equation for the beam width parameter by using WKB and paraxial approximations and solved it numerically. The effect of chirp and other laser plasma parameters is seen on the behavior of beam width parameter with dimensionless distance of propagation. The results are discussed and presented graphically. Our simulation results show that the amplitude of oscillations decreases with the distance of propagation. Due to collisional frequency, the laser beam shows fast divergence which can be minimized by the introduction of chirp parameter. The chirp decreases the effect of defocusing and increases the ability of self-focusing of laser beam in collisional plasma.

Image defocus deblurring method based on gradient difference of boundary neighborhood

Background For static scenes with multiple depth layers,existing defocused image deblurring methods have the problems of edge-ringing artifacts or insufficient deblurring owing to inaccurate estimation of the blur amount,and prior knowledge in nonblind deconvolution is not strong,which leads to image detail recovery challenges.Methods To this end,this study proposes a blur map estimation method for defocused images based on the gradient difference of the boundary neighborhood,which uses the gradient difference of the boundary neighborhood to accurately obtain the amount of blurring,thereby preventing boundary ringing artifacts.The obtained blur map is then used for blur detection to determine whether the image needs to be deblurred,thereby improving the efficiency of deblurring without manual intervention and judgment.Finally,a nonblind deconvolution algorithm was designed to achieve image deblurring based on the blur amount selection strategy and sparse prior.Results Experimental results showed that our method improves PSNR(Peak Signal-to-Noise Ratio)and SSIM(Structural Similarity Index)by an average of 4.6%and 7.3%,respectively,compared to existing methods.Conclusions Experimental results showed that the proposed method outperforms existing methods.Compared to existing methods,our method can better solve the problems of boundary ringing artifacts and detail information preservation in defocused image deblurring.

A spawning particle filter for defocused moving target detection in GNSS-based passive radar

Global Navigation Satellite System(GNSS)-based passive radar(GBPR)has been widely used in remote sensing applications.However,for moving target detection(MTD),the quadratic phase error(QPE)introduced by the non-cooperative target motion is usually difficult to be compensated,as the low power level of the GBPR echo signal renders the estimation of the Doppler rate less effective.Consequently,the moving target in GBPR image is usually defocused,which aggravates the difficulty of target detection even further.In this paper,a spawning particle filter(SPF)is proposed for defocused MTD.Firstly,the measurement model and the likelihood ratio function(LRF)of the defocused point-like target image are deduced.Then,a spawning particle set is generated for subsequent target detection,with reference to traditional particles in particle filter(PF)as their parent.After that,based on the PF estimator,the SPF algorithm and its sequential Monte Carlo(SMC)implementation are proposed with a novel amplitude estimation method to decrease the target state dimension.Finally,the effectiveness of the proposed SPF is demonstrated by numerical simulations and pre-liminary experimental results,showing that the target range and Doppler can be estimated accurately.

Analysis of inverse synthetic aperture radar imaging in the presence of time-varying plasma sheath

The plasma sheath can induce radar signal modulation,causing not only ineffective target detection,but also defocusing in inverse synthetic aperture radar(ISAR)imaging.In this paper,through establishing radar echo models of the reentry object enveloped with time-varying plasma sheath,we simulated the defocusing of ISAR images in typical environment.Simulation results suggested that the ISAR defocusing is caused by false scatterings,upon which the false scatterings’formation mechanism and distribution property are analyzed and studied.The range of false scattering correlates with the electron density fluctuation frequency.The combined value of the electron density fluctuation and the pulse repetition frequency jointly determines the Doppler of false scattering.Two measurement metrics including peak signal-to-noise ratio and structural similarity are used to evaluate the influence of ISAR imaging.

Effect of Laser Beam Welding Parameters on Microstructure and Properties of Duplex Stainless Steel

The present study is concerned with laser beam welding and its effect on size and microstructure of fusion zone then, on mechanical and corrosion properties of duplex stainless steel welded joints. In this regard, influence of different laser welding parameters was clarified. Both bead-on-plate and autogenously butt welded joints were made using carbon dioxide laser with a maximum output of 9 kW in the continuous wave mode. Welded joints were subjected to visual, dye penetrant and radiography tests before sectioning it for different destructive tests. Accelerated corrosion test was carried out based on tafel plot technique. The results achieved in this investigation disclosed that welding parameters play an important role in obtaining satisfactory properties of welded joint. High laser power and/or high welding speed together with adjusting laser focused spot at specimen surface have produced welded joints with a remarkable decrease in fusion zone size and an acceptable weld profile with higher weld depth/width ratio. Besides, acceptable mechanical and corrosion properties were obtained. Using nitrogen as a shielding gas has resulted in improving mechanical and corrosion properties of welded joints in comparison with argon shielding. This is related to maintaining proper ferrite/austenite balance in both weld metal and HAZ in case of nitrogen shielding. As a conclusion, laser power, welding speed, defocusing distance and type of shielding gas combination have to be optimized for obtaining welded joints with acceptable profile as well as mechanical and corrosion properties.

An Analytical Solution for Nearshore Circulation Driven by Focused/Defocused Waves

An offshore shoal or bar refracts ocean surface waves and causes wave focusing/defocusing on the adjacent beach. Wave focal patterns characterized by alongshore variations in wave height, wave angle, and breaking location induce alongshore non-uniformities of wave setup and nearshore circulation, e.g., rip currents and alongshore currents, in the surfzone. A simplified analytic model for nearshore circulation generated by focused/defocused waves on a planar beach is developed and theoretical solutions are obtained using transport stream function and perturbations in alongshore distributions of wave height and wave angle at the breaker line. The analytic model suggests that alongshore currents are strongly affected by a pair of counter-rotating vortices generated shoreward of the wave focal zone. The vortices are persistent, and their strengths depend on the amplitudes of alongshore variations in wave height and wave angle. The alongshore gradient in wave height tends to intensify the vortices while the convergence of wave angle tends to weaken the vortices. Divergent flows associated with the vortices in the surfzone are intense, strengthening alongshore currents in the downstream of the wave focal zone and weakening alongshore currents or causing flows reversal in the upstream. Alongshore currents are modulated by rip currents associated with the wave focusing/defocusing patterns.

Propagation of Intense Femtosecond Laser Pulses in Deuterated Methane and Deuterium Cluster Jets

Propagation of intense femtosecond laser pulses （60 fs, 800 nm, 120 m J, 6 × 10^17 W/cm^2 in vacuum） in supersonic （CD4）N and （D2）N cluster jets at different backing pressures was studied. Pump-probe interferometry is employed to investigate the propagation of laser beams in dense cluster jets by examining the electron density distribution of plasma chan- nels. It was found that propagation effects, including ionization-induced defocusing and laser attenuation of incident pulses, are very different in the （CD4）N and （D2）N cluster jets. Different ionization states of CD4 and D2 molecules were observed by analyzing the transverse electron density profiles of the plasma channels and should be considered as a major reason for the differences in the propagation effects. Numerical simulations of the nonlinear propagation of femtosecond laser pulses in （CD4）N and （D2）N cluster jets were performed, and the results indicated a good reproduction of the experimental data.

ON THE MODIFIED NONLINEAR SCHRDINGER EQUATION IN THE SEMICLASSICAL LIMIT:SUPERSONIC,SUBSONIC,AND TRANSSONIC BEHAVIOR

The purpose of this paper is to present a comparison between the modified nonlinear SchrSdinger （MNLS） equation and the focusing and defocusing variants of the （unmodified） nonlinear SchrSdinger （NLS） equation in the semiclassical limit. We describe aspects of the limiting dynamics and discuss how the nature of the dynamics is evident theoretically through inverse-scattering and noncommutative steepest descent methods. The main message is that, depending on initial data, the MNLS equation can behave either like the defocusing NLS equation, like the focusing NLS equation （in both cases the analogy is asymptotically accurate in the semiclassical limit when the NLS equation is posed with appropriately modified initial data）, or like an interesting mixture of the two. In the latter case, we identify a feature of the dynamics analogous to a sonic line in gas dynamics, a free boundary separating subsonic flow from supersonic flow.

Spectral attenuation of a 400-nm laser pulse propagating through a plasma filament induced by an intense femtosecond laser pulse

The spectral attenuation of a 400-nm probe laser propagating through a femtosecond plasma in air is studied.Defocusing effect of the low-density plasma is an obvious effect by examining the far-field patterns of the 400-nm pulse.Besides,the energy of 400-nm pulse drops after interaction with the plasma,which is found to be another effect leading to the attenuation.To reveal the physical origin behind the energy loss,we measure fluorescence emissions of the interaction area.The fluorescence is hardly detected with the weak 400-nm laser pulse,and the line spectra from the plasma filament induced by the 800-nm pump pulse are clearly shown.However,when the 400-nm pulse propagates through the plasma filament,the fluorescence at 391 nm from the first negative band system of N2+is enhanced,while that from the second positive band of neutral N2 at 337 nm remains constant.Efficient near-resonant absorption of the 400-nm pulse by the first negative band system occurs inside the plasma,which results in the enhanced fluorescence.Furthermore,the spectral attenuation of the 400-nm probe laser is measured as a function of the pump–probe time delay as well as the pump-pulse energy.

Scanning Measurement Method of Dispersion Mini-spot by Quadrant Photodetector

Quadrant photodetector is a new type position detector, which has already been applied to many fields such as measurement, target tracking, control, laser collimation, guidance, etc. System performance is related to laser spot area received by the quadrant photodetector. The optimum linear output signal of system is gotten only when the size of laser dispersion spot is well-proportioned to the detector photosensitive area. A method to measure the mini-spot received by detector and to adjust it in real time is presented, which is based on the quadrant geometrical and optical structure, making use of tilting mirror scanning, the sequence output signal of detector is gotten, then measurement model is built with rate of signal change as characteristic variable, the radius of the laser dispersion spot and adjustable value of dispersion are calculated. The experiment result shows that the maximum error is 0.58μm in measurement range of 0.5mm±0.1mm.